(sqrt (+ (- 1/2 (* 1/2 (cos (* 2 (/ (- z0 z1) 2))))) (* (* (* (cos z0) (cos z1)) (sin (/ (- z2 z3) 2))) (sin (/ (- z2 z3) 2)))))

Percentage Accurate: 60.5% → 98.8%

Time: 20.8s

Alternatives: 20

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} t_0 := \sin \left(\frac{z2 - z3}{2}\right)\\ \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot t\_0\right) \cdot t\_0} \end{array} \]

FPCore

(FPCore (z0 z1 z2 z3)
  :precision binary64
  (let* ((t_0 (sin (/ (- z2 z3) 2))))
  (sqrt
   (+
    (- 1/2 (* 1/2 (cos (* 2 (/ (- z0 z1) 2)))))
    (* (* (* (cos z0) (cos z1)) t_0) t_0)))))

C

double code(double z0, double z1, double z2, double z3) {
	double t_0 = sin(((z2 - z3) / 2.0));
	return sqrt(((0.5 - (0.5 * cos((2.0 * ((z0 - z1) / 2.0))))) + (((cos(z0) * cos(z1)) * t_0) * t_0)));
}

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(z0, z1, z2, z3)
use fmin_fmax_functions
    real(8), intent (in) :: z0
    real(8), intent (in) :: z1
    real(8), intent (in) :: z2
    real(8), intent (in) :: z3
    real(8) :: t_0
    t_0 = sin(((z2 - z3) / 2.0d0))
    code = sqrt(((0.5d0 - (0.5d0 * cos((2.0d0 * ((z0 - z1) / 2.0d0))))) + (((cos(z0) * cos(z1)) * t_0) * t_0)))
end function

Java

public static double code(double z0, double z1, double z2, double z3) {
	double t_0 = Math.sin(((z2 - z3) / 2.0));
	return Math.sqrt(((0.5 - (0.5 * Math.cos((2.0 * ((z0 - z1) / 2.0))))) + (((Math.cos(z0) * Math.cos(z1)) * t_0) * t_0)));
}

Python

def code(z0, z1, z2, z3):
	t_0 = math.sin(((z2 - z3) / 2.0))
	return math.sqrt(((0.5 - (0.5 * math.cos((2.0 * ((z0 - z1) / 2.0))))) + (((math.cos(z0) * math.cos(z1)) * t_0) * t_0)))

Julia

function code(z0, z1, z2, z3)
	t_0 = sin(Float64(Float64(z2 - z3) / 2.0))
	return sqrt(Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(Float64(z0 - z1) / 2.0))))) + Float64(Float64(Float64(cos(z0) * cos(z1)) * t_0) * t_0)))
end

MATLAB

function tmp = code(z0, z1, z2, z3)
	t_0 = sin(((z2 - z3) / 2.0));
	tmp = sqrt(((0.5 - (0.5 * cos((2.0 * ((z0 - z1) / 2.0))))) + (((cos(z0) * cos(z1)) * t_0) * t_0)));
end

Wolfram

code[z0_, z1_, z2_, z3_] := Block[{t$95$0 = N[Sin[N[(N[(z2 - z3), $MachinePrecision] / 2), $MachinePrecision]], $MachinePrecision]}, N[Sqrt[N[(N[(1/2 - N[(1/2 * N[Cos[N[(2 * N[(N[(z0 - z1), $MachinePrecision] / 2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[Cos[z0], $MachinePrecision] * N[Cos[z1], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Initial Program: 60.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(\frac{z2 - z3}{2}\right)\\ \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot t\_0\right) \cdot t\_0} \end{array} \]

FPCore

(FPCore (z0 z1 z2 z3)
  :precision binary64
  (let* ((t_0 (sin (/ (- z2 z3) 2))))
  (sqrt
   (+
    (- 1/2 (* 1/2 (cos (* 2 (/ (- z0 z1) 2)))))
    (* (* (* (cos z0) (cos z1)) t_0) t_0)))))

C

double code(double z0, double z1, double z2, double z3) {
	double t_0 = sin(((z2 - z3) / 2.0));
	return sqrt(((0.5 - (0.5 * cos((2.0 * ((z0 - z1) / 2.0))))) + (((cos(z0) * cos(z1)) * t_0) * t_0)));
}

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(z0, z1, z2, z3)
use fmin_fmax_functions
    real(8), intent (in) :: z0
    real(8), intent (in) :: z1
    real(8), intent (in) :: z2
    real(8), intent (in) :: z3
    real(8) :: t_0
    t_0 = sin(((z2 - z3) / 2.0d0))
    code = sqrt(((0.5d0 - (0.5d0 * cos((2.0d0 * ((z0 - z1) / 2.0d0))))) + (((cos(z0) * cos(z1)) * t_0) * t_0)))
end function

Java

public static double code(double z0, double z1, double z2, double z3) {
	double t_0 = Math.sin(((z2 - z3) / 2.0));
	return Math.sqrt(((0.5 - (0.5 * Math.cos((2.0 * ((z0 - z1) / 2.0))))) + (((Math.cos(z0) * Math.cos(z1)) * t_0) * t_0)));
}

Python

def code(z0, z1, z2, z3):
	t_0 = math.sin(((z2 - z3) / 2.0))
	return math.sqrt(((0.5 - (0.5 * math.cos((2.0 * ((z0 - z1) / 2.0))))) + (((math.cos(z0) * math.cos(z1)) * t_0) * t_0)))

Julia

function code(z0, z1, z2, z3)
	t_0 = sin(Float64(Float64(z2 - z3) / 2.0))
	return sqrt(Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(Float64(z0 - z1) / 2.0))))) + Float64(Float64(Float64(cos(z0) * cos(z1)) * t_0) * t_0)))
end

MATLAB

function tmp = code(z0, z1, z2, z3)
	t_0 = sin(((z2 - z3) / 2.0));
	tmp = sqrt(((0.5 - (0.5 * cos((2.0 * ((z0 - z1) / 2.0))))) + (((cos(z0) * cos(z1)) * t_0) * t_0)));
end

Wolfram

code[z0_, z1_, z2_, z3_] := Block[{t$95$0 = N[Sin[N[(N[(z2 - z3), $MachinePrecision] / 2), $MachinePrecision]], $MachinePrecision]}, N[Sqrt[N[(N[(1/2 - N[(1/2 * N[Cos[N[(2 * N[(N[(z0 - z1), $MachinePrecision] / 2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[Cos[z0], $MachinePrecision] * N[Cos[z1], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Alternative 1: 98.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(\frac{z2 - z3}{2}\right)\\ t_1 := \cos z0 \cdot \cos z1\\ t_2 := \sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\\ t_3 := \left(t\_1 \cdot t\_2\right) \cdot t\_2\\ \mathbf{if}\;\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(t\_1 \cdot t\_0\right) \cdot t\_0} \leq \frac{3602879701896397}{72057594037927936}:\\ \;\;\;\;\sqrt{{\sin \left(\left(z0 - z1\right) \cdot \frac{1}{2}\right)}^{2} + t\_3}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\frac{1}{2} - \mathsf{304\_z0z1z2z3z4}\left(\frac{1}{2}, 1, \left(\cos z1 \cdot \cos z0\right), \sin z1, \sin \left(-z0\right)\right)\right) + t\_3}\\ \end{array} \]

FPCore

(FPCore (z0 z1 z2 z3)
  :precision binary64
  (let* ((t_0 (sin (/ (- z2 z3) 2)))
       (t_1 (* (cos z0) (cos z1)))
       (t_2
        (-
         (* (sin (* z2 1/2)) (cos (* z3 1/2)))
         (* (cos (* z2 1/2)) (sin (* z3 1/2)))))
       (t_3 (* (* t_1 t_2) t_2)))
  (if (<=
       (sqrt
        (+
         (- 1/2 (* 1/2 (cos (* 2 (/ (- z0 z1) 2)))))
         (* (* t_1 t_0) t_0)))
       3602879701896397/72057594037927936)
    (sqrt (+ (pow (sin (* (- z0 z1) 1/2)) 2) t_3))
    (sqrt
     (+
      (-
       1/2
       (304-z0z1z2z3z4
        1/2
        1
        (* (cos z1) (cos z0))
        (sin z1)
        (sin (- z0))))
      t_3)))))

Derivation

1. Split input into 2 regimes

2. if (sqrt.f64 (+.f64 (-.f64 #s(literal 1/2 binary64) (*.f64 #s(literal 1/2 binary64) (cos.f64 (*.f64 #s(literal 2 binary64) (/.f64 (-.f64 z0 z1) #s(literal 2 binary64)))))) (*.f64 (*.f64 (*.f64 (cos.f64 z0) (cos.f64 z1)) (sin.f64 (/.f64 (-.f64 z2 z3) #s(literal 2 binary64)))) (sin.f64 (/.f64 (-.f64 z2 z3) #s(literal 2 binary64)))))) < 0.050000000000000003

   1. Initial program 60.5%

      \[\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \color{blue}{\sin \left(\frac{z2 - z3}{2}\right)}\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \color{blue}{\left(\frac{z2 - z3}{2}\right)}\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      3. lift--.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \left(\frac{\color{blue}{z2 - z3}}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      4. div-sub N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \color{blue}{\left(\frac{z2}{2} - \frac{z3}{2}\right)}\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      5. sin-diff N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \color{blue}{\left(\sin \left(\frac{z2}{2}\right) \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)}\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      6. lower--.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \color{blue}{\left(\sin \left(\frac{z2}{2}\right) \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)}\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\color{blue}{\sin \left(\frac{z2}{2}\right) \cdot \cos \left(\frac{z3}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      8. lower-sin.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\color{blue}{\sin \left(\frac{z2}{2}\right)} \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      9. mult-flip N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      10. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      12. lower-cos.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{z3}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      13. mult-flip N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(z3 \cdot \frac{1}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      14. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      15. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(z3 \cdot \frac{1}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      16. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      17. lower-cos.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{z2}{2}\right)} \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      18. mult-flip N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      19. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      20. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      21. lower-sin.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{z3}{2}\right)}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      22. mult-flip N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(z3 \cdot \frac{1}{2}\right)}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      23. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      24. lower-*.f64 59.7%

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(z3 \cdot \frac{1}{2}\right)}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

   3. Applied rewrites 59.7%

      \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \color{blue}{\left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)}\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{z2 - z3}{2}\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{z2 - z3}{2}\right)}} \]

      3. lift--.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{z2 - z3}}{2}\right)} \]

      4. div-sub N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{z2}{2} - \frac{z3}{2}\right)}} \]

      5. sin-diff N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{z2}{2}\right) \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)}} \]

      6. lower--.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{z2}{2}\right) \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{z2}{2}\right) \cdot \cos \left(\frac{z3}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      8. lower-sin.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{z2}{2}\right)} \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      9. mult-flip N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      10. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      12. lower-cos.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{z3}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      13. mult-flip N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(z3 \cdot \frac{1}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      14. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      15. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(z3 \cdot \frac{1}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      16. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)}\right)} \]

      17. lower-cos.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{z2}{2}\right)} \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      18. mult-flip N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      19. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      20. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      21. lower-sin.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{z3}{2}\right)}\right)} \]

      22. mult-flip N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(z3 \cdot \frac{1}{2}\right)}\right)} \]

      23. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \color{blue}{\frac{1}{2}}\right)\right)} \]

      24. lower-*.f64 75.8%

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(z3 \cdot \frac{1}{2}\right)}\right)} \]

   5. Applied rewrites 75.8%

      \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)}} \]

   7. Applied rewrites 78.4%

      \[\leadsto \sqrt{\color{blue}{{\sin \left(\left(z0 - z1\right) \cdot \frac{1}{2}\right)}^{2}} + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)} \]

   if 0.050000000000000003 < (sqrt.f64 (+.f64 (-.f64 #s(literal 1/2 binary64) (*.f64 #s(literal 1/2 binary64) (cos.f64 (*.f64 #s(literal 2 binary64) (/.f64 (-.f64 z0 z1) #s(literal 2 binary64)))))) (*.f64 (*.f64 (*.f64 (cos.f64 z0) (cos.f64 z1)) (sin.f64 (/.f64 (-.f64 z2 z3) #s(literal 2 binary64)))) (sin.f64 (/.f64 (-.f64 z2 z3) #s(literal 2 binary64))))))

   1. Initial program 60.5%

      \[\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \color{blue}{\sin \left(\frac{z2 - z3}{2}\right)}\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \color{blue}{\left(\frac{z2 - z3}{2}\right)}\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      3. lift--.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \left(\frac{\color{blue}{z2 - z3}}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      4. div-sub N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \color{blue}{\left(\frac{z2}{2} - \frac{z3}{2}\right)}\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      5. sin-diff N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \color{blue}{\left(\sin \left(\frac{z2}{2}\right) \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)}\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      6. lower--.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \color{blue}{\left(\sin \left(\frac{z2}{2}\right) \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)}\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\color{blue}{\sin \left(\frac{z2}{2}\right) \cdot \cos \left(\frac{z3}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      8. lower-sin.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\color{blue}{\sin \left(\frac{z2}{2}\right)} \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      9. mult-flip N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      10. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      12. lower-cos.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{z3}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      13. mult-flip N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(z3 \cdot \frac{1}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      14. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      15. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(z3 \cdot \frac{1}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      16. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      17. lower-cos.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{z2}{2}\right)} \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      18. mult-flip N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      19. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      20. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      21. lower-sin.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{z3}{2}\right)}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      22. mult-flip N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(z3 \cdot \frac{1}{2}\right)}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      23. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      24. lower-*.f64 59.7%

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(z3 \cdot \frac{1}{2}\right)}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

   3. Applied rewrites 59.7%

      \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \color{blue}{\left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)}\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{z2 - z3}{2}\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{z2 - z3}{2}\right)}} \]

      3. lift--.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{z2 - z3}}{2}\right)} \]

      4. div-sub N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{z2}{2} - \frac{z3}{2}\right)}} \]

      5. sin-diff N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{z2}{2}\right) \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)}} \]

      6. lower--.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{z2}{2}\right) \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{z2}{2}\right) \cdot \cos \left(\frac{z3}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      8. lower-sin.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{z2}{2}\right)} \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      9. mult-flip N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      10. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      12. lower-cos.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{z3}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      13. mult-flip N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(z3 \cdot \frac{1}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      14. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      15. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(z3 \cdot \frac{1}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      16. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)}\right)} \]

      17. lower-cos.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{z2}{2}\right)} \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      18. mult-flip N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      19. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      20. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      21. lower-sin.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{z3}{2}\right)}\right)} \]

      22. mult-flip N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(z3 \cdot \frac{1}{2}\right)}\right)} \]

      23. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \color{blue}{\frac{1}{2}}\right)\right)} \]

      24. lower-*.f64 75.8%

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(z3 \cdot \frac{1}{2}\right)}\right)} \]

   5. Applied rewrites 75.8%

      \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)}} \]

   7. Applied rewrites 96.4%

      \[\leadsto \sqrt{\left(\frac{1}{2} - \color{blue}{\mathsf{304\_z0z1z2z3z4}\left(\frac{1}{2}, 1, \left(\cos z1 \cdot \cos z0\right), \sin z1, \sin \left(-z0\right)\right)}\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 98.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(\frac{z2 - z3}{2}\right)\\ t_1 := \cos z0 \cdot \cos z1\\ t_2 := \sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\\ t_3 := \left(t\_1 \cdot t\_2\right) \cdot t\_2\\ \mathbf{if}\;\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(t\_1 \cdot t\_0\right) \cdot t\_0} \leq \frac{3602879701896397}{72057594037927936}:\\ \;\;\;\;\sqrt{{\sin \left(\left(z0 - z1\right) \cdot \frac{1}{2}\right)}^{2} + t\_3}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\frac{1}{2} - \mathsf{304\_z0z1z2z3z4}\left(\frac{1}{2}, \cos z1, \cos z0, \sin z1, \sin \left(-z0\right)\right)\right) + t\_3}\\ \end{array} \]

FPCore

(FPCore (z0 z1 z2 z3)
  :precision binary64
  (let* ((t_0 (sin (/ (- z2 z3) 2)))
       (t_1 (* (cos z0) (cos z1)))
       (t_2
        (-
         (* (sin (* z2 1/2)) (cos (* z3 1/2)))
         (* (cos (* z2 1/2)) (sin (* z3 1/2)))))
       (t_3 (* (* t_1 t_2) t_2)))
  (if (<=
       (sqrt
        (+
         (- 1/2 (* 1/2 (cos (* 2 (/ (- z0 z1) 2)))))
         (* (* t_1 t_0) t_0)))
       3602879701896397/72057594037927936)
    (sqrt (+ (pow (sin (* (- z0 z1) 1/2)) 2) t_3))
    (sqrt
     (+
      (-
       1/2
       (304-z0z1z2z3z4 1/2 (cos z1) (cos z0) (sin z1) (sin (- z0))))
      t_3)))))

Derivation

1. Split input into 2 regimes

2. if (sqrt.f64 (+.f64 (-.f64 #s(literal 1/2 binary64) (*.f64 #s(literal 1/2 binary64) (cos.f64 (*.f64 #s(literal 2 binary64) (/.f64 (-.f64 z0 z1) #s(literal 2 binary64)))))) (*.f64 (*.f64 (*.f64 (cos.f64 z0) (cos.f64 z1)) (sin.f64 (/.f64 (-.f64 z2 z3) #s(literal 2 binary64)))) (sin.f64 (/.f64 (-.f64 z2 z3) #s(literal 2 binary64)))))) < 0.050000000000000003

   1. Initial program 60.5%

      \[\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \color{blue}{\sin \left(\frac{z2 - z3}{2}\right)}\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \color{blue}{\left(\frac{z2 - z3}{2}\right)}\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      3. lift--.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \left(\frac{\color{blue}{z2 - z3}}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      4. div-sub N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \color{blue}{\left(\frac{z2}{2} - \frac{z3}{2}\right)}\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      5. sin-diff N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \color{blue}{\left(\sin \left(\frac{z2}{2}\right) \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)}\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      6. lower--.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \color{blue}{\left(\sin \left(\frac{z2}{2}\right) \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)}\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\color{blue}{\sin \left(\frac{z2}{2}\right) \cdot \cos \left(\frac{z3}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      8. lower-sin.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\color{blue}{\sin \left(\frac{z2}{2}\right)} \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      9. mult-flip N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      10. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      12. lower-cos.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{z3}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      13. mult-flip N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(z3 \cdot \frac{1}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      14. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      15. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(z3 \cdot \frac{1}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      16. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      17. lower-cos.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{z2}{2}\right)} \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      18. mult-flip N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      19. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      20. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      21. lower-sin.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{z3}{2}\right)}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      22. mult-flip N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(z3 \cdot \frac{1}{2}\right)}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      23. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      24. lower-*.f64 59.7%

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(z3 \cdot \frac{1}{2}\right)}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

   3. Applied rewrites 59.7%

      \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \color{blue}{\left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)}\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{z2 - z3}{2}\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{z2 - z3}{2}\right)}} \]

      3. lift--.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{z2 - z3}}{2}\right)} \]

      4. div-sub N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{z2}{2} - \frac{z3}{2}\right)}} \]

      5. sin-diff N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{z2}{2}\right) \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)}} \]

      6. lower--.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{z2}{2}\right) \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{z2}{2}\right) \cdot \cos \left(\frac{z3}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      8. lower-sin.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{z2}{2}\right)} \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      9. mult-flip N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      10. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      12. lower-cos.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{z3}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      13. mult-flip N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(z3 \cdot \frac{1}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      14. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      15. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(z3 \cdot \frac{1}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      16. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)}\right)} \]

      17. lower-cos.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{z2}{2}\right)} \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      18. mult-flip N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      19. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      20. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      21. lower-sin.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{z3}{2}\right)}\right)} \]

      22. mult-flip N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(z3 \cdot \frac{1}{2}\right)}\right)} \]

      23. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \color{blue}{\frac{1}{2}}\right)\right)} \]

      24. lower-*.f64 75.8%

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(z3 \cdot \frac{1}{2}\right)}\right)} \]

   5. Applied rewrites 75.8%

      \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)}} \]

   7. Applied rewrites 78.4%

      \[\leadsto \sqrt{\color{blue}{{\sin \left(\left(z0 - z1\right) \cdot \frac{1}{2}\right)}^{2}} + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)} \]

   if 0.050000000000000003 < (sqrt.f64 (+.f64 (-.f64 #s(literal 1/2 binary64) (*.f64 #s(literal 1/2 binary64) (cos.f64 (*.f64 #s(literal 2 binary64) (/.f64 (-.f64 z0 z1) #s(literal 2 binary64)))))) (*.f64 (*.f64 (*.f64 (cos.f64 z0) (cos.f64 z1)) (sin.f64 (/.f64 (-.f64 z2 z3) #s(literal 2 binary64)))) (sin.f64 (/.f64 (-.f64 z2 z3) #s(literal 2 binary64))))))

   1. Initial program 60.5%

      \[\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \color{blue}{\sin \left(\frac{z2 - z3}{2}\right)}\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \color{blue}{\left(\frac{z2 - z3}{2}\right)}\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      3. lift--.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \left(\frac{\color{blue}{z2 - z3}}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      4. div-sub N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \color{blue}{\left(\frac{z2}{2} - \frac{z3}{2}\right)}\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      5. sin-diff N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \color{blue}{\left(\sin \left(\frac{z2}{2}\right) \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)}\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      6. lower--.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \color{blue}{\left(\sin \left(\frac{z2}{2}\right) \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)}\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\color{blue}{\sin \left(\frac{z2}{2}\right) \cdot \cos \left(\frac{z3}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      8. lower-sin.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\color{blue}{\sin \left(\frac{z2}{2}\right)} \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      9. mult-flip N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      10. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      12. lower-cos.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{z3}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      13. mult-flip N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(z3 \cdot \frac{1}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      14. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      15. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(z3 \cdot \frac{1}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      16. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      17. lower-cos.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{z2}{2}\right)} \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      18. mult-flip N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      19. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      20. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      21. lower-sin.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{z3}{2}\right)}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      22. mult-flip N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(z3 \cdot \frac{1}{2}\right)}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      23. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      24. lower-*.f64 59.7%

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(z3 \cdot \frac{1}{2}\right)}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

   3. Applied rewrites 59.7%

      \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \color{blue}{\left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)}\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{z2 - z3}{2}\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{z2 - z3}{2}\right)}} \]

      3. lift--.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{z2 - z3}}{2}\right)} \]

      4. div-sub N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{z2}{2} - \frac{z3}{2}\right)}} \]

      5. sin-diff N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{z2}{2}\right) \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)}} \]

      6. lower--.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{z2}{2}\right) \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{z2}{2}\right) \cdot \cos \left(\frac{z3}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      8. lower-sin.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{z2}{2}\right)} \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      9. mult-flip N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      10. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      12. lower-cos.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{z3}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      13. mult-flip N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(z3 \cdot \frac{1}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      14. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      15. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(z3 \cdot \frac{1}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      16. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)}\right)} \]

      17. lower-cos.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{z2}{2}\right)} \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      18. mult-flip N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      19. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      20. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      21. lower-sin.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{z3}{2}\right)}\right)} \]

      22. mult-flip N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(z3 \cdot \frac{1}{2}\right)}\right)} \]

      23. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \color{blue}{\frac{1}{2}}\right)\right)} \]

      24. lower-*.f64 75.8%

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(z3 \cdot \frac{1}{2}\right)}\right)} \]

   5. Applied rewrites 75.8%

      \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)}} \]

   7. Applied rewrites 96.4%

      \[\leadsto \sqrt{\left(\frac{1}{2} - \color{blue}{\mathsf{304\_z0z1z2z3z4}\left(\frac{1}{2}, \cos z1, \cos z0, \sin z1, \sin \left(-z0\right)\right)}\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 98.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(\frac{z2 - z3}{2}\right)\\ t_1 := \cos z0 \cdot \cos z1\\ t_2 := \sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\\ t_3 := \left(t\_1 \cdot t\_2\right) \cdot t\_2\\ \mathbf{if}\;\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(t\_1 \cdot t\_0\right) \cdot t\_0} \leq \frac{3602879701896397}{72057594037927936}:\\ \;\;\;\;\sqrt{{\sin \left(\left(z0 - z1\right) \cdot \frac{1}{2}\right)}^{2} + t\_3}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \left(\cos z1 \cdot \cos z0 + \sin z1 \cdot \sin z0\right)\right) + t\_3}\\ \end{array} \]

FPCore

(FPCore (z0 z1 z2 z3)
  :precision binary64
  (let* ((t_0 (sin (/ (- z2 z3) 2)))
       (t_1 (* (cos z0) (cos z1)))
       (t_2
        (-
         (* (sin (* z2 1/2)) (cos (* z3 1/2)))
         (* (cos (* z2 1/2)) (sin (* z3 1/2)))))
       (t_3 (* (* t_1 t_2) t_2)))
  (if (<=
       (sqrt
        (+
         (- 1/2 (* 1/2 (cos (* 2 (/ (- z0 z1) 2)))))
         (* (* t_1 t_0) t_0)))
       3602879701896397/72057594037927936)
    (sqrt (+ (pow (sin (* (- z0 z1) 1/2)) 2) t_3))
    (sqrt
     (+
      (- 1/2 (* 1/2 (+ (* (cos z1) (cos z0)) (* (sin z1) (sin z0)))))
      t_3)))))

C

double code(double z0, double z1, double z2, double z3) {
	double t_0 = sin(((z2 - z3) / 2.0));
	double t_1 = cos(z0) * cos(z1);
	double t_2 = (sin((z2 * 0.5)) * cos((z3 * 0.5))) - (cos((z2 * 0.5)) * sin((z3 * 0.5)));
	double t_3 = (t_1 * t_2) * t_2;
	double tmp;
	if (sqrt(((0.5 - (0.5 * cos((2.0 * ((z0 - z1) / 2.0))))) + ((t_1 * t_0) * t_0))) <= 0.05) {
		tmp = sqrt((pow(sin(((z0 - z1) * 0.5)), 2.0) + t_3));
	} else {
		tmp = sqrt(((0.5 - (0.5 * ((cos(z1) * cos(z0)) + (sin(z1) * sin(z0))))) + t_3));
	}
	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(z0, z1, z2, z3)
use fmin_fmax_functions
    real(8), intent (in) :: z0
    real(8), intent (in) :: z1
    real(8), intent (in) :: z2
    real(8), intent (in) :: z3
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = sin(((z2 - z3) / 2.0d0))
    t_1 = cos(z0) * cos(z1)
    t_2 = (sin((z2 * 0.5d0)) * cos((z3 * 0.5d0))) - (cos((z2 * 0.5d0)) * sin((z3 * 0.5d0)))
    t_3 = (t_1 * t_2) * t_2
    if (sqrt(((0.5d0 - (0.5d0 * cos((2.0d0 * ((z0 - z1) / 2.0d0))))) + ((t_1 * t_0) * t_0))) <= 0.05d0) then
        tmp = sqrt(((sin(((z0 - z1) * 0.5d0)) ** 2.0d0) + t_3))
    else
        tmp = sqrt(((0.5d0 - (0.5d0 * ((cos(z1) * cos(z0)) + (sin(z1) * sin(z0))))) + t_3))
    end if
    code = tmp
end function

Java

public static double code(double z0, double z1, double z2, double z3) {
	double t_0 = Math.sin(((z2 - z3) / 2.0));
	double t_1 = Math.cos(z0) * Math.cos(z1);
	double t_2 = (Math.sin((z2 * 0.5)) * Math.cos((z3 * 0.5))) - (Math.cos((z2 * 0.5)) * Math.sin((z3 * 0.5)));
	double t_3 = (t_1 * t_2) * t_2;
	double tmp;
	if (Math.sqrt(((0.5 - (0.5 * Math.cos((2.0 * ((z0 - z1) / 2.0))))) + ((t_1 * t_0) * t_0))) <= 0.05) {
		tmp = Math.sqrt((Math.pow(Math.sin(((z0 - z1) * 0.5)), 2.0) + t_3));
	} else {
		tmp = Math.sqrt(((0.5 - (0.5 * ((Math.cos(z1) * Math.cos(z0)) + (Math.sin(z1) * Math.sin(z0))))) + t_3));
	}
	return tmp;
}

Python

def code(z0, z1, z2, z3):
	t_0 = math.sin(((z2 - z3) / 2.0))
	t_1 = math.cos(z0) * math.cos(z1)
	t_2 = (math.sin((z2 * 0.5)) * math.cos((z3 * 0.5))) - (math.cos((z2 * 0.5)) * math.sin((z3 * 0.5)))
	t_3 = (t_1 * t_2) * t_2
	tmp = 0
	if math.sqrt(((0.5 - (0.5 * math.cos((2.0 * ((z0 - z1) / 2.0))))) + ((t_1 * t_0) * t_0))) <= 0.05:
		tmp = math.sqrt((math.pow(math.sin(((z0 - z1) * 0.5)), 2.0) + t_3))
	else:
		tmp = math.sqrt(((0.5 - (0.5 * ((math.cos(z1) * math.cos(z0)) + (math.sin(z1) * math.sin(z0))))) + t_3))
	return tmp

Julia

function code(z0, z1, z2, z3)
	t_0 = sin(Float64(Float64(z2 - z3) / 2.0))
	t_1 = Float64(cos(z0) * cos(z1))
	t_2 = Float64(Float64(sin(Float64(z2 * 0.5)) * cos(Float64(z3 * 0.5))) - Float64(cos(Float64(z2 * 0.5)) * sin(Float64(z3 * 0.5))))
	t_3 = Float64(Float64(t_1 * t_2) * t_2)
	tmp = 0.0
	if (sqrt(Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(Float64(z0 - z1) / 2.0))))) + Float64(Float64(t_1 * t_0) * t_0))) <= 0.05)
		tmp = sqrt(Float64((sin(Float64(Float64(z0 - z1) * 0.5)) ^ 2.0) + t_3));
	else
		tmp = sqrt(Float64(Float64(0.5 - Float64(0.5 * Float64(Float64(cos(z1) * cos(z0)) + Float64(sin(z1) * sin(z0))))) + t_3));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z0, z1, z2, z3)
	t_0 = sin(((z2 - z3) / 2.0));
	t_1 = cos(z0) * cos(z1);
	t_2 = (sin((z2 * 0.5)) * cos((z3 * 0.5))) - (cos((z2 * 0.5)) * sin((z3 * 0.5)));
	t_3 = (t_1 * t_2) * t_2;
	tmp = 0.0;
	if (sqrt(((0.5 - (0.5 * cos((2.0 * ((z0 - z1) / 2.0))))) + ((t_1 * t_0) * t_0))) <= 0.05)
		tmp = sqrt(((sin(((z0 - z1) * 0.5)) ^ 2.0) + t_3));
	else
		tmp = sqrt(((0.5 - (0.5 * ((cos(z1) * cos(z0)) + (sin(z1) * sin(z0))))) + t_3));
	end
	tmp_2 = tmp;
end

Wolfram

code[z0_, z1_, z2_, z3_] := Block[{t$95$0 = N[Sin[N[(N[(z2 - z3), $MachinePrecision] / 2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[z0], $MachinePrecision] * N[Cos[z1], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Sin[N[(z2 * 1/2), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(z3 * 1/2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[N[(z2 * 1/2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(z3 * 1/2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(t$95$1 * t$95$2), $MachinePrecision] * t$95$2), $MachinePrecision]}, If[LessEqual[N[Sqrt[N[(N[(1/2 - N[(1/2 * N[Cos[N[(2 * N[(N[(z0 - z1), $MachinePrecision] / 2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$1 * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 3602879701896397/72057594037927936], N[Sqrt[N[(N[Power[N[Sin[N[(N[(z0 - z1), $MachinePrecision] * 1/2), $MachinePrecision]], $MachinePrecision], 2], $MachinePrecision] + t$95$3), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(1/2 - N[(1/2 * N[(N[(N[Cos[z1], $MachinePrecision] * N[Cos[z0], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[z1], $MachinePrecision] * N[Sin[z0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if (sqrt.f64 (+.f64 (-.f64 #s(literal 1/2 binary64) (*.f64 #s(literal 1/2 binary64) (cos.f64 (*.f64 #s(literal 2 binary64) (/.f64 (-.f64 z0 z1) #s(literal 2 binary64)))))) (*.f64 (*.f64 (*.f64 (cos.f64 z0) (cos.f64 z1)) (sin.f64 (/.f64 (-.f64 z2 z3) #s(literal 2 binary64)))) (sin.f64 (/.f64 (-.f64 z2 z3) #s(literal 2 binary64)))))) < 0.050000000000000003

   1. Initial program 60.5%

      \[\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \color{blue}{\sin \left(\frac{z2 - z3}{2}\right)}\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \color{blue}{\left(\frac{z2 - z3}{2}\right)}\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      3. lift--.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \left(\frac{\color{blue}{z2 - z3}}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      4. div-sub N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \color{blue}{\left(\frac{z2}{2} - \frac{z3}{2}\right)}\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      5. sin-diff N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \color{blue}{\left(\sin \left(\frac{z2}{2}\right) \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)}\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      6. lower--.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \color{blue}{\left(\sin \left(\frac{z2}{2}\right) \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)}\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\color{blue}{\sin \left(\frac{z2}{2}\right) \cdot \cos \left(\frac{z3}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      8. lower-sin.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\color{blue}{\sin \left(\frac{z2}{2}\right)} \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      9. mult-flip N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      10. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      12. lower-cos.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{z3}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      13. mult-flip N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(z3 \cdot \frac{1}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      14. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      15. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(z3 \cdot \frac{1}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      16. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      17. lower-cos.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{z2}{2}\right)} \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      18. mult-flip N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      19. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      20. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      21. lower-sin.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{z3}{2}\right)}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      22. mult-flip N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(z3 \cdot \frac{1}{2}\right)}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      23. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      24. lower-*.f64 59.7%

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(z3 \cdot \frac{1}{2}\right)}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

   3. Applied rewrites 59.7%

      \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \color{blue}{\left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)}\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{z2 - z3}{2}\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{z2 - z3}{2}\right)}} \]

      3. lift--.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{z2 - z3}}{2}\right)} \]

      4. div-sub N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{z2}{2} - \frac{z3}{2}\right)}} \]

      5. sin-diff N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{z2}{2}\right) \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)}} \]

      6. lower--.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{z2}{2}\right) \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{z2}{2}\right) \cdot \cos \left(\frac{z3}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      8. lower-sin.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{z2}{2}\right)} \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      9. mult-flip N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      10. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      12. lower-cos.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{z3}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      13. mult-flip N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(z3 \cdot \frac{1}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      14. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      15. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(z3 \cdot \frac{1}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      16. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)}\right)} \]

      17. lower-cos.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{z2}{2}\right)} \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      18. mult-flip N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      19. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      20. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      21. lower-sin.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{z3}{2}\right)}\right)} \]

      22. mult-flip N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(z3 \cdot \frac{1}{2}\right)}\right)} \]

      23. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \color{blue}{\frac{1}{2}}\right)\right)} \]

      24. lower-*.f64 75.8%

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(z3 \cdot \frac{1}{2}\right)}\right)} \]

   5. Applied rewrites 75.8%

      \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)}} \]

   7. Applied rewrites 78.4%

      \[\leadsto \sqrt{\color{blue}{{\sin \left(\left(z0 - z1\right) \cdot \frac{1}{2}\right)}^{2}} + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)} \]

   if 0.050000000000000003 < (sqrt.f64 (+.f64 (-.f64 #s(literal 1/2 binary64) (*.f64 #s(literal 1/2 binary64) (cos.f64 (*.f64 #s(literal 2 binary64) (/.f64 (-.f64 z0 z1) #s(literal 2 binary64)))))) (*.f64 (*.f64 (*.f64 (cos.f64 z0) (cos.f64 z1)) (sin.f64 (/.f64 (-.f64 z2 z3) #s(literal 2 binary64)))) (sin.f64 (/.f64 (-.f64 z2 z3) #s(literal 2 binary64))))))

   1. Initial program 60.5%

      \[\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \color{blue}{\sin \left(\frac{z2 - z3}{2}\right)}\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \color{blue}{\left(\frac{z2 - z3}{2}\right)}\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      3. lift--.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \left(\frac{\color{blue}{z2 - z3}}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      4. div-sub N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \color{blue}{\left(\frac{z2}{2} - \frac{z3}{2}\right)}\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      5. sin-diff N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \color{blue}{\left(\sin \left(\frac{z2}{2}\right) \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)}\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      6. lower--.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \color{blue}{\left(\sin \left(\frac{z2}{2}\right) \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)}\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\color{blue}{\sin \left(\frac{z2}{2}\right) \cdot \cos \left(\frac{z3}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      8. lower-sin.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\color{blue}{\sin \left(\frac{z2}{2}\right)} \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      9. mult-flip N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      10. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      12. lower-cos.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{z3}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      13. mult-flip N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(z3 \cdot \frac{1}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      14. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      15. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(z3 \cdot \frac{1}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      16. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      17. lower-cos.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{z2}{2}\right)} \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      18. mult-flip N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      19. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      20. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      21. lower-sin.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{z3}{2}\right)}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      22. mult-flip N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(z3 \cdot \frac{1}{2}\right)}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      23. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      24. lower-*.f64 59.7%

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(z3 \cdot \frac{1}{2}\right)}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

   3. Applied rewrites 59.7%

      \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \color{blue}{\left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)}\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{z2 - z3}{2}\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{z2 - z3}{2}\right)}} \]

      3. lift--.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{z2 - z3}}{2}\right)} \]

      4. div-sub N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{z2}{2} - \frac{z3}{2}\right)}} \]

      5. sin-diff N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{z2}{2}\right) \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)}} \]

      6. lower--.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{z2}{2}\right) \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{z2}{2}\right) \cdot \cos \left(\frac{z3}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      8. lower-sin.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{z2}{2}\right)} \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      9. mult-flip N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      10. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      12. lower-cos.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{z3}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      13. mult-flip N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(z3 \cdot \frac{1}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      14. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      15. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(z3 \cdot \frac{1}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      16. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)}\right)} \]

      17. lower-cos.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{z2}{2}\right)} \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      18. mult-flip N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      19. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      20. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      21. lower-sin.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{z3}{2}\right)}\right)} \]

      22. mult-flip N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(z3 \cdot \frac{1}{2}\right)}\right)} \]

      23. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \color{blue}{\frac{1}{2}}\right)\right)} \]

      24. lower-*.f64 75.8%

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(z3 \cdot \frac{1}{2}\right)}\right)} \]

   5. Applied rewrites 75.8%

      \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)}} \]
   6. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot \frac{z0 - z1}{2}\right)}\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)} \]

      2. cos-neg-rev N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(\mathsf{neg}\left(2 \cdot \frac{z0 - z1}{2}\right)\right)}\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\color{blue}{2 \cdot \frac{z0 - z1}{2}}\right)\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)} \]

      4. count-2-rev N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\color{blue}{\left(\frac{z0 - z1}{2} + \frac{z0 - z1}{2}\right)}\right)\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)} \]

      5. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\left(\color{blue}{\frac{z0 - z1}{2}} + \frac{z0 - z1}{2}\right)\right)\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)} \]

      6. mult-flip N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\left(\color{blue}{\left(z0 - z1\right) \cdot \frac{1}{2}} + \frac{z0 - z1}{2}\right)\right)\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)} \]

      7. metadata-eval N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\left(\left(z0 - z1\right) \cdot \color{blue}{\frac{1}{2}} + \frac{z0 - z1}{2}\right)\right)\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)} \]

      8. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\left(\left(z0 - z1\right) \cdot \frac{1}{2} + \color{blue}{\frac{z0 - z1}{2}}\right)\right)\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)} \]

      9. mult-flip N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\left(\left(z0 - z1\right) \cdot \frac{1}{2} + \color{blue}{\left(z0 - z1\right) \cdot \frac{1}{2}}\right)\right)\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)} \]

      10. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\left(\left(z0 - z1\right) \cdot \frac{1}{2} + \left(z0 - z1\right) \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)} \]

      11. distribute-lft-out N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\color{blue}{\left(z0 - z1\right) \cdot \left(\frac{1}{2} + \frac{1}{2}\right)}\right)\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)} \]

      12. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\left(z0 - z1\right) \cdot \color{blue}{1}\right)\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)} \]

      13. *-rgt-identity N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\color{blue}{\left(z0 - z1\right)}\right)\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)} \]

      14. lift--.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\color{blue}{\left(z0 - z1\right)}\right)\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)} \]

      15. sub-negate-rev N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(z1 - z0\right)}\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)} \]

      16. cos-diff N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\left(\cos z1 \cdot \cos z0 + \sin z1 \cdot \sin z0\right)}\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)} \]

      17. lift-cos.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \left(\color{blue}{\cos z1} \cdot \cos z0 + \sin z1 \cdot \sin z0\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)} \]

      18. lift-cos.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \left(\cos z1 \cdot \color{blue}{\cos z0} + \sin z1 \cdot \sin z0\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)} \]

      19. *-commutative N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \left(\color{blue}{\cos z0 \cdot \cos z1} + \sin z1 \cdot \sin z0\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)} \]

      20. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \left(\color{blue}{\cos z0 \cdot \cos z1} + \sin z1 \cdot \sin z0\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)} \]

      21. lower-+.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\left(\cos z0 \cdot \cos z1 + \sin z1 \cdot \sin z0\right)}\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)} \]

   7. Applied rewrites 96.4%

      \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\left(\cos z1 \cdot \cos z0 + \sin z1 \cdot \sin z0\right)}\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 88.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \cos z1 \cdot \cos z0\\ t_1 := \sin \left(\left(z2 - z3\right) \cdot \frac{1}{2}\right)\\ t_2 := \sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\\ \mathbf{if}\;z0 \leq \frac{-944473296573929}{1180591620717411303424}:\\ \;\;\;\;\sqrt{\left(\frac{1}{2} - \mathsf{304\_z0z1z2z3z4}\left(\frac{1}{2}, \cos z1, \cos z0, \sin z1, \sin \left(-z0\right)\right)\right) + \left(\left(t\_1 \cdot \cos z0\right) \cdot \cos z1\right) \cdot t\_1}\\ \mathbf{elif}\;z0 \leq 280000000000000:\\ \;\;\;\;\sqrt{{\sin \left(\left(z0 - z1\right) \cdot \frac{1}{2}\right)}^{2} + \left(\left(\cos z0 \cdot \cos z1\right) \cdot t\_2\right) \cdot t\_2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{2} - \left(\mathsf{304\_z0z1z2z3z4}\left(\frac{1}{2}, t\_0, 1, \sin z1, \left(-\sin z0\right)\right) + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot t\_0\right)}\\ \end{array} \]

FPCore

(FPCore (z0 z1 z2 z3)
  :precision binary64
  (let* ((t_0 (* (cos z1) (cos z0)))
       (t_1 (sin (* (- z2 z3) 1/2)))
       (t_2
        (-
         (* (sin (* z2 1/2)) (cos (* z3 1/2)))
         (* (cos (* z2 1/2)) (sin (* z3 1/2))))))
  (if (<= z0 -944473296573929/1180591620717411303424)
    (sqrt
     (+
      (-
       1/2
       (304-z0z1z2z3z4 1/2 (cos z1) (cos z0) (sin z1) (sin (- z0))))
      (* (* (* t_1 (cos z0)) (cos z1)) t_1)))
    (if (<= z0 280000000000000)
      (sqrt
       (+
        (pow (sin (* (- z0 z1) 1/2)) 2)
        (* (* (* (cos z0) (cos z1)) t_2) t_2)))
      (sqrt
       (-
        1/2
        (+
         (304-z0z1z2z3z4 1/2 t_0 1 (sin z1) (- (sin z0)))
         (* (- (* (cos (* (- z2 z3) 1)) 1/2) 1/2) t_0))))))))

Derivation

1. Split input into 3 regimes

2. if z0 < -7.9999999999999996e-7

   1. Initial program 60.5%

      \[\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \color{blue}{\sin \left(\frac{z2 - z3}{2}\right)}\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \color{blue}{\left(\frac{z2 - z3}{2}\right)}\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      3. lift--.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \left(\frac{\color{blue}{z2 - z3}}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      4. div-sub N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \color{blue}{\left(\frac{z2}{2} - \frac{z3}{2}\right)}\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      5. sin-diff N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \color{blue}{\left(\sin \left(\frac{z2}{2}\right) \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)}\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      6. lower--.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \color{blue}{\left(\sin \left(\frac{z2}{2}\right) \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)}\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\color{blue}{\sin \left(\frac{z2}{2}\right) \cdot \cos \left(\frac{z3}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      8. lower-sin.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\color{blue}{\sin \left(\frac{z2}{2}\right)} \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      9. mult-flip N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      10. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      12. lower-cos.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{z3}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      13. mult-flip N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(z3 \cdot \frac{1}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      14. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      15. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(z3 \cdot \frac{1}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      16. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      17. lower-cos.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{z2}{2}\right)} \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      18. mult-flip N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      19. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      20. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      21. lower-sin.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{z3}{2}\right)}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      22. mult-flip N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(z3 \cdot \frac{1}{2}\right)}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      23. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      24. lower-*.f64 59.7%

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(z3 \cdot \frac{1}{2}\right)}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

   3. Applied rewrites 59.7%

      \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \color{blue}{\left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)}\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{z2 - z3}{2}\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{z2 - z3}{2}\right)}} \]

      3. lift--.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{z2 - z3}}{2}\right)} \]

      4. div-sub N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{z2}{2} - \frac{z3}{2}\right)}} \]

      5. sin-diff N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{z2}{2}\right) \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)}} \]

      6. lower--.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{z2}{2}\right) \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{z2}{2}\right) \cdot \cos \left(\frac{z3}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      8. lower-sin.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{z2}{2}\right)} \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      9. mult-flip N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      10. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      12. lower-cos.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{z3}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      13. mult-flip N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(z3 \cdot \frac{1}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      14. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      15. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(z3 \cdot \frac{1}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      16. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)}\right)} \]

      17. lower-cos.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{z2}{2}\right)} \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      18. mult-flip N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      19. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      20. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      21. lower-sin.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{z3}{2}\right)}\right)} \]

      22. mult-flip N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(z3 \cdot \frac{1}{2}\right)}\right)} \]

      23. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \color{blue}{\frac{1}{2}}\right)\right)} \]

      24. lower-*.f64 75.8%

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(z3 \cdot \frac{1}{2}\right)}\right)} \]

   5. Applied rewrites 75.8%

      \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)}} \]

   7. Applied rewrites 96.4%

      \[\leadsto \sqrt{\left(\frac{1}{2} - \color{blue}{\mathsf{304\_z0z1z2z3z4}\left(\frac{1}{2}, \cos z1, \cos z0, \sin z1, \sin \left(-z0\right)\right)}\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)} \]

   9. Applied rewrites 76.1%

      \[\leadsto \sqrt{\left(\frac{1}{2} - \mathsf{304\_z0z1z2z3z4}\left(\frac{1}{2}, \cos z1, \cos z0, \sin z1, \sin \left(-z0\right)\right)\right) + \color{blue}{\left(\left(\sin \left(\left(z2 - z3\right) \cdot \frac{1}{2}\right) \cdot \cos z0\right) \cdot \cos z1\right) \cdot \sin \left(\left(z2 - z3\right) \cdot \frac{1}{2}\right)}} \]

   if -7.9999999999999996e-7 < z0 < 2.8e14

   1. Initial program 60.5%

      \[\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \color{blue}{\sin \left(\frac{z2 - z3}{2}\right)}\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \color{blue}{\left(\frac{z2 - z3}{2}\right)}\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      3. lift--.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \left(\frac{\color{blue}{z2 - z3}}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      4. div-sub N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \color{blue}{\left(\frac{z2}{2} - \frac{z3}{2}\right)}\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      5. sin-diff N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \color{blue}{\left(\sin \left(\frac{z2}{2}\right) \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)}\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      6. lower--.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \color{blue}{\left(\sin \left(\frac{z2}{2}\right) \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)}\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\color{blue}{\sin \left(\frac{z2}{2}\right) \cdot \cos \left(\frac{z3}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      8. lower-sin.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\color{blue}{\sin \left(\frac{z2}{2}\right)} \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      9. mult-flip N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      10. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      12. lower-cos.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{z3}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      13. mult-flip N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(z3 \cdot \frac{1}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      14. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      15. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(z3 \cdot \frac{1}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      16. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      17. lower-cos.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{z2}{2}\right)} \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      18. mult-flip N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      19. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      20. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      21. lower-sin.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{z3}{2}\right)}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      22. mult-flip N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(z3 \cdot \frac{1}{2}\right)}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      23. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      24. lower-*.f64 59.7%

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(z3 \cdot \frac{1}{2}\right)}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

   3. Applied rewrites 59.7%

      \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \color{blue}{\left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)}\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{z2 - z3}{2}\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{z2 - z3}{2}\right)}} \]

      3. lift--.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{z2 - z3}}{2}\right)} \]

      4. div-sub N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{z2}{2} - \frac{z3}{2}\right)}} \]

      5. sin-diff N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{z2}{2}\right) \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)}} \]

      6. lower--.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{z2}{2}\right) \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{z2}{2}\right) \cdot \cos \left(\frac{z3}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      8. lower-sin.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{z2}{2}\right)} \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      9. mult-flip N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      10. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      12. lower-cos.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{z3}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      13. mult-flip N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(z3 \cdot \frac{1}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      14. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      15. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(z3 \cdot \frac{1}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      16. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)}\right)} \]

      17. lower-cos.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{z2}{2}\right)} \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      18. mult-flip N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      19. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      20. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      21. lower-sin.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{z3}{2}\right)}\right)} \]

      22. mult-flip N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(z3 \cdot \frac{1}{2}\right)}\right)} \]

      23. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \color{blue}{\frac{1}{2}}\right)\right)} \]

      24. lower-*.f64 75.8%

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(z3 \cdot \frac{1}{2}\right)}\right)} \]

   5. Applied rewrites 75.8%

      \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)}} \]

   7. Applied rewrites 78.4%

      \[\leadsto \sqrt{\color{blue}{{\sin \left(\left(z0 - z1\right) \cdot \frac{1}{2}\right)}^{2}} + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)} \]

   if 2.8e14 < z0

   1. Initial program 60.5%

      \[\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

   3. Applied rewrites 57.8%

      \[\leadsto \sqrt{\color{blue}{\frac{1}{2} - \left(\cos \left(\left(z0 - z1\right) \cdot 1\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(z2 - z3\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)}} \]

   5. Applied rewrites 57.8%

      \[\leadsto \sqrt{\frac{1}{2} - \color{blue}{\left(\cos \left(z1 - z0\right) \cdot \frac{1}{2} + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\color{blue}{\cos \left(z1 - z0\right) \cdot \frac{1}{2}} + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\color{blue}{\frac{1}{2} \cdot \cos \left(z1 - z0\right)} + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      3. lift-cos.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\frac{1}{2} \cdot \color{blue}{\cos \left(z1 - z0\right)} + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      4. lift--.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\frac{1}{2} \cdot \cos \color{blue}{\left(z1 - z0\right)} + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      5. cos-diff N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\frac{1}{2} \cdot \color{blue}{\left(\cos z1 \cdot \cos z0 + \sin z1 \cdot \sin z0\right)} + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      6. lift-cos.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\frac{1}{2} \cdot \left(\color{blue}{\cos z1} \cdot \cos z0 + \sin z1 \cdot \sin z0\right) + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      7. lift-cos.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\frac{1}{2} \cdot \left(\cos z1 \cdot \color{blue}{\cos z0} + \sin z1 \cdot \sin z0\right) + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\frac{1}{2} \cdot \left(\color{blue}{\cos z1 \cdot \cos z0} + \sin z1 \cdot \sin z0\right) + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      9. *-lft-identity N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\frac{1}{2} \cdot \left(\color{blue}{1 \cdot \left(\cos z1 \cdot \cos z0\right)} + \sin z1 \cdot \sin z0\right) + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      10. add-flip N/A

          \[\leadsto \sqrt{\frac{1}{2} - \left(\frac{1}{2} \cdot \color{blue}{\left(1 \cdot \left(\cos z1 \cdot \cos z0\right) - \left(\mathsf{neg}\left(\sin z1 \cdot \sin z0\right)\right)\right)} + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      11. *-lft-identity N/A

          \[\leadsto \sqrt{\frac{1}{2} - \left(\frac{1}{2} \cdot \left(1 \cdot \color{blue}{\left(1 \cdot \left(\cos z1 \cdot \cos z0\right)\right)} - \left(\mathsf{neg}\left(\sin z1 \cdot \sin z0\right)\right)\right) + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      12. *-commutative N/A

          \[\leadsto \sqrt{\frac{1}{2} - \left(\frac{1}{2} \cdot \left(\color{blue}{\left(1 \cdot \left(\cos z1 \cdot \cos z0\right)\right) \cdot 1} - \left(\mathsf{neg}\left(\sin z1 \cdot \sin z0\right)\right)\right) + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      13. lift-sin.f64 N/A

          \[\leadsto \sqrt{\frac{1}{2} - \left(\frac{1}{2} \cdot \left(\left(1 \cdot \left(\cos z1 \cdot \cos z0\right)\right) \cdot 1 - \left(\mathsf{neg}\left(\color{blue}{\sin z1} \cdot \sin z0\right)\right)\right) + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      14. distribute-rgt-neg-out N/A

          \[\leadsto \sqrt{\frac{1}{2} - \left(\frac{1}{2} \cdot \left(\left(1 \cdot \left(\cos z1 \cdot \cos z0\right)\right) \cdot 1 - \color{blue}{\sin z1 \cdot \left(\mathsf{neg}\left(\sin z0\right)\right)}\right) + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      15. sin-neg N/A

          \[\leadsto \sqrt{\frac{1}{2} - \left(\frac{1}{2} \cdot \left(\left(1 \cdot \left(\cos z1 \cdot \cos z0\right)\right) \cdot 1 - \sin z1 \cdot \color{blue}{\sin \left(\mathsf{neg}\left(z0\right)\right)}\right) + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      16. lift-neg.f64 N/A

          \[\leadsto \sqrt{\frac{1}{2} - \left(\frac{1}{2} \cdot \left(\left(1 \cdot \left(\cos z1 \cdot \cos z0\right)\right) \cdot 1 - \sin z1 \cdot \sin \color{blue}{\left(-z0\right)}\right) + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      17. lift-sin.f64 N/A

          \[\leadsto \sqrt{\frac{1}{2} - \left(\frac{1}{2} \cdot \left(\left(1 \cdot \left(\cos z1 \cdot \cos z0\right)\right) \cdot 1 - \sin z1 \cdot \color{blue}{\sin \left(-z0\right)}\right) + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      18. lower-304-z0z1z2z3z4 N/A

          \[\leadsto \sqrt{\frac{1}{2} - \left(\color{blue}{\mathsf{304\_z0z1z2z3z4}\left(\frac{1}{2}, \left(1 \cdot \left(\cos z1 \cdot \cos z0\right)\right), 1, \sin z1, \sin \left(-z0\right)\right)} + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

   7. Applied rewrites 73.4%

      \[\leadsto \sqrt{\frac{1}{2} - \left(\color{blue}{\mathsf{304\_z0z1z2z3z4}\left(\frac{1}{2}, \left(\cos z1 \cdot \cos z0\right), 1, \sin z1, \left(-\sin z0\right)\right)} + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 86.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{304\_z0z1z2z3z4}\left(1, \sin \left(z2 \cdot \frac{1}{2}\right), \cos \left(\frac{-1}{2} \cdot z3\right), \sin \left(z3 \cdot \frac{1}{2}\right), \cos \left(\frac{-1}{2} \cdot z2\right)\right)\\ t_1 := \sin \left(\left(z2 - z3\right) \cdot \frac{1}{2}\right)\\ t_2 := \cos z1 \cdot \cos z0\\ \mathbf{if}\;z0 \leq \frac{-944473296573929}{1180591620717411303424}:\\ \;\;\;\;\sqrt{\left(\frac{1}{2} - \mathsf{304\_z0z1z2z3z4}\left(\frac{1}{2}, \cos z1, \cos z0, \sin z1, \sin \left(-z0\right)\right)\right) + \left(\left(t\_1 \cdot \cos z0\right) \cdot \cos z1\right) \cdot t\_1}\\ \mathbf{elif}\;z0 \leq 8499999999999999588958208:\\ \;\;\;\;\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\frac{1}{\frac{2}{\cos \left(z1 - z0\right) + \cos \left(z1 + z0\right)}} \cdot t\_0\right) \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{2} - \left(\mathsf{304\_z0z1z2z3z4}\left(\frac{1}{2}, t\_2, 1, \sin z1, \left(-\sin z0\right)\right) + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot t\_2\right)}\\ \end{array} \]

FPCore

(FPCore (z0 z1 z2 z3)
  :precision binary64
  (let* ((t_0
        (304-z0z1z2z3z4
         1
         (sin (* z2 1/2))
         (cos (* -1/2 z3))
         (sin (* z3 1/2))
         (cos (* -1/2 z2))))
       (t_1 (sin (* (- z2 z3) 1/2)))
       (t_2 (* (cos z1) (cos z0))))
  (if (<= z0 -944473296573929/1180591620717411303424)
    (sqrt
     (+
      (-
       1/2
       (304-z0z1z2z3z4 1/2 (cos z1) (cos z0) (sin z1) (sin (- z0))))
      (* (* (* t_1 (cos z0)) (cos z1)) t_1)))
    (if (<= z0 8499999999999999588958208)
      (sqrt
       (+
        (- 1/2 (* 1/2 (cos (* 2 (/ (- z0 z1) 2)))))
        (*
         (* (/ 1 (/ 2 (+ (cos (- z1 z0)) (cos (+ z1 z0))))) t_0)
         t_0)))
      (sqrt
       (-
        1/2
        (+
         (304-z0z1z2z3z4 1/2 t_2 1 (sin z1) (- (sin z0)))
         (* (- (* (cos (* (- z2 z3) 1)) 1/2) 1/2) t_2))))))))

Derivation

1. Split input into 3 regimes

2. if z0 < -7.9999999999999996e-7

   1. Initial program 60.5%

      \[\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \color{blue}{\sin \left(\frac{z2 - z3}{2}\right)}\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \color{blue}{\left(\frac{z2 - z3}{2}\right)}\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      3. lift--.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \left(\frac{\color{blue}{z2 - z3}}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      4. div-sub N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \color{blue}{\left(\frac{z2}{2} - \frac{z3}{2}\right)}\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      5. sin-diff N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \color{blue}{\left(\sin \left(\frac{z2}{2}\right) \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)}\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      6. lower--.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \color{blue}{\left(\sin \left(\frac{z2}{2}\right) \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)}\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\color{blue}{\sin \left(\frac{z2}{2}\right) \cdot \cos \left(\frac{z3}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      8. lower-sin.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\color{blue}{\sin \left(\frac{z2}{2}\right)} \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      9. mult-flip N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      10. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      12. lower-cos.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{z3}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      13. mult-flip N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(z3 \cdot \frac{1}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      14. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      15. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(z3 \cdot \frac{1}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      16. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      17. lower-cos.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{z2}{2}\right)} \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      18. mult-flip N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      19. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      20. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      21. lower-sin.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{z3}{2}\right)}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      22. mult-flip N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(z3 \cdot \frac{1}{2}\right)}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      23. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      24. lower-*.f64 59.7%

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(z3 \cdot \frac{1}{2}\right)}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

   3. Applied rewrites 59.7%

      \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \color{blue}{\left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)}\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{z2 - z3}{2}\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{z2 - z3}{2}\right)}} \]

      3. lift--.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{z2 - z3}}{2}\right)} \]

      4. div-sub N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{z2}{2} - \frac{z3}{2}\right)}} \]

      5. sin-diff N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{z2}{2}\right) \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)}} \]

      6. lower--.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{z2}{2}\right) \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{z2}{2}\right) \cdot \cos \left(\frac{z3}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      8. lower-sin.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{z2}{2}\right)} \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      9. mult-flip N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      10. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      12. lower-cos.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{z3}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      13. mult-flip N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(z3 \cdot \frac{1}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      14. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      15. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(z3 \cdot \frac{1}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      16. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)}\right)} \]

      17. lower-cos.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{z2}{2}\right)} \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      18. mult-flip N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      19. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      20. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      21. lower-sin.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{z3}{2}\right)}\right)} \]

      22. mult-flip N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(z3 \cdot \frac{1}{2}\right)}\right)} \]

      23. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \color{blue}{\frac{1}{2}}\right)\right)} \]

      24. lower-*.f64 75.8%

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(z3 \cdot \frac{1}{2}\right)}\right)} \]

   5. Applied rewrites 75.8%

      \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)}} \]

   7. Applied rewrites 96.4%

      \[\leadsto \sqrt{\left(\frac{1}{2} - \color{blue}{\mathsf{304\_z0z1z2z3z4}\left(\frac{1}{2}, \cos z1, \cos z0, \sin z1, \sin \left(-z0\right)\right)}\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)} \]

   9. Applied rewrites 76.1%

      \[\leadsto \sqrt{\left(\frac{1}{2} - \mathsf{304\_z0z1z2z3z4}\left(\frac{1}{2}, \cos z1, \cos z0, \sin z1, \sin \left(-z0\right)\right)\right) + \color{blue}{\left(\left(\sin \left(\left(z2 - z3\right) \cdot \frac{1}{2}\right) \cdot \cos z0\right) \cdot \cos z1\right) \cdot \sin \left(\left(z2 - z3\right) \cdot \frac{1}{2}\right)}} \]

   if -7.9999999999999996e-7 < z0 < 8.4999999999999996e24

   1. Initial program 60.5%

      \[\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\color{blue}{\left(\cos z0 \cdot \cos z1\right)} \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      2. lift-cos.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\color{blue}{\cos z0} \cdot \cos z1\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      3. lift-cos.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \color{blue}{\cos z1}\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      4. cos-mult N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\color{blue}{\frac{\cos \left(z0 + z1\right) + \cos \left(z0 - z1\right)}{2}} \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      5. div-flip N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\color{blue}{\frac{1}{\frac{2}{\cos \left(z0 + z1\right) + \cos \left(z0 - z1\right)}}} \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      6. lower-unsound-/.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\color{blue}{\frac{1}{\frac{2}{\cos \left(z0 + z1\right) + \cos \left(z0 - z1\right)}}} \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      7. lower-unsound-/.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\frac{1}{\color{blue}{\frac{2}{\cos \left(z0 + z1\right) + \cos \left(z0 - z1\right)}}} \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      8. +-commutative N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\frac{1}{\frac{2}{\color{blue}{\cos \left(z0 - z1\right) + \cos \left(z0 + z1\right)}}} \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      9. lower-+.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\frac{1}{\frac{2}{\color{blue}{\cos \left(z0 - z1\right) + \cos \left(z0 + z1\right)}}} \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      10. lift--.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\frac{1}{\frac{2}{\cos \color{blue}{\left(z0 - z1\right)} + \cos \left(z0 + z1\right)}} \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      11. cos-neg-rev N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\frac{1}{\frac{2}{\color{blue}{\cos \left(\mathsf{neg}\left(\left(z0 - z1\right)\right)\right)} + \cos \left(z0 + z1\right)}} \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      12. lower-cos.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\frac{1}{\frac{2}{\color{blue}{\cos \left(\mathsf{neg}\left(\left(z0 - z1\right)\right)\right)} + \cos \left(z0 + z1\right)}} \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      13. lift--.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\frac{1}{\frac{2}{\cos \left(\mathsf{neg}\left(\color{blue}{\left(z0 - z1\right)}\right)\right) + \cos \left(z0 + z1\right)}} \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      14. sub-negate-rev N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\frac{1}{\frac{2}{\cos \color{blue}{\left(z1 - z0\right)} + \cos \left(z0 + z1\right)}} \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      15. lower--.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\frac{1}{\frac{2}{\cos \color{blue}{\left(z1 - z0\right)} + \cos \left(z0 + z1\right)}} \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      16. lower-cos.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\frac{1}{\frac{2}{\cos \left(z1 - z0\right) + \color{blue}{\cos \left(z0 + z1\right)}}} \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      17. +-commutative N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\frac{1}{\frac{2}{\cos \left(z1 - z0\right) + \cos \color{blue}{\left(z1 + z0\right)}}} \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      18. lower-+.f64 61.0%

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\frac{1}{\frac{2}{\cos \left(z1 - z0\right) + \cos \color{blue}{\left(z1 + z0\right)}}} \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

   3. Applied rewrites 61.0%

      \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\color{blue}{\frac{1}{\frac{2}{\cos \left(z1 - z0\right) + \cos \left(z1 + z0\right)}}} \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]
   4. Step-by-step derivation

      1. *-lft-identity N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\frac{1}{\frac{2}{\cos \left(z1 - z0\right) + \cos \left(z1 + z0\right)}} \cdot \color{blue}{\left(1 \cdot \sin \left(\frac{z2 - z3}{2}\right)\right)}\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      2. lift-sin.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\frac{1}{\frac{2}{\cos \left(z1 - z0\right) + \cos \left(z1 + z0\right)}} \cdot \left(1 \cdot \color{blue}{\sin \left(\frac{z2 - z3}{2}\right)}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\frac{1}{\frac{2}{\cos \left(z1 - z0\right) + \cos \left(z1 + z0\right)}} \cdot \left(1 \cdot \sin \color{blue}{\left(\frac{z2 - z3}{2}\right)}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      4. mult-flip N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\frac{1}{\frac{2}{\cos \left(z1 - z0\right) + \cos \left(z1 + z0\right)}} \cdot \left(1 \cdot \sin \color{blue}{\left(\left(z2 - z3\right) \cdot \frac{1}{2}\right)}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      5. metadata-eval N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\frac{1}{\frac{2}{\cos \left(z1 - z0\right) + \cos \left(z1 + z0\right)}} \cdot \left(1 \cdot \sin \left(\left(z2 - z3\right) \cdot \color{blue}{\frac{1}{2}}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      6. *-commutative N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\frac{1}{\frac{2}{\cos \left(z1 - z0\right) + \cos \left(z1 + z0\right)}} \cdot \left(1 \cdot \sin \color{blue}{\left(\frac{1}{2} \cdot \left(z2 - z3\right)\right)}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      7. lift--.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\frac{1}{\frac{2}{\cos \left(z1 - z0\right) + \cos \left(z1 + z0\right)}} \cdot \left(1 \cdot \sin \left(\frac{1}{2} \cdot \color{blue}{\left(z2 - z3\right)}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      8. distribute-rgt-out-- N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\frac{1}{\frac{2}{\cos \left(z1 - z0\right) + \cos \left(z1 + z0\right)}} \cdot \left(1 \cdot \sin \color{blue}{\left(z2 \cdot \frac{1}{2} - z3 \cdot \frac{1}{2}\right)}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\frac{1}{\frac{2}{\cos \left(z1 - z0\right) + \cos \left(z1 + z0\right)}} \cdot \left(1 \cdot \sin \left(\color{blue}{z2 \cdot \frac{1}{2}} - z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\frac{1}{\frac{2}{\cos \left(z1 - z0\right) + \cos \left(z1 + z0\right)}} \cdot \left(1 \cdot \sin \left(z2 \cdot \frac{1}{2} - \color{blue}{z3 \cdot \frac{1}{2}}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      11. sin-diff-rev N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\frac{1}{\frac{2}{\cos \left(z1 - z0\right) + \cos \left(z1 + z0\right)}} \cdot \left(1 \cdot \color{blue}{\left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      12. lift-sin.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\frac{1}{\frac{2}{\cos \left(z1 - z0\right) + \cos \left(z1 + z0\right)}} \cdot \left(1 \cdot \left(\color{blue}{\sin \left(z2 \cdot \frac{1}{2}\right)} \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      13. lift-cos.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\frac{1}{\frac{2}{\cos \left(z1 - z0\right) + \cos \left(z1 + z0\right)}} \cdot \left(1 \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(z3 \cdot \frac{1}{2}\right)} - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      14. lift-cos.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\frac{1}{\frac{2}{\cos \left(z1 - z0\right) + \cos \left(z1 + z0\right)}} \cdot \left(1 \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(z2 \cdot \frac{1}{2}\right)} \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      15. lift-sin.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\frac{1}{\frac{2}{\cos \left(z1 - z0\right) + \cos \left(z1 + z0\right)}} \cdot \left(1 \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(z3 \cdot \frac{1}{2}\right)}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      16. *-commutative N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\frac{1}{\frac{2}{\cos \left(z1 - z0\right) + \cos \left(z1 + z0\right)}} \cdot \left(1 \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \color{blue}{\sin \left(z3 \cdot \frac{1}{2}\right) \cdot \cos \left(z2 \cdot \frac{1}{2}\right)}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      17. lower-304-z0z1z2z3z4 60.0%

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\frac{1}{\frac{2}{\cos \left(z1 - z0\right) + \cos \left(z1 + z0\right)}} \cdot \color{blue}{\mathsf{304\_z0z1z2z3z4}\left(1, \sin \left(z2 \cdot \frac{1}{2}\right), \cos \left(z3 \cdot \frac{1}{2}\right), \sin \left(z3 \cdot \frac{1}{2}\right), \cos \left(z2 \cdot \frac{1}{2}\right)\right)}\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

   5. Applied rewrites 60.0%

      \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\frac{1}{\frac{2}{\cos \left(z1 - z0\right) + \cos \left(z1 + z0\right)}} \cdot \color{blue}{\mathsf{304\_z0z1z2z3z4}\left(1, \sin \left(z2 \cdot \frac{1}{2}\right), \cos \left(\frac{-1}{2} \cdot z3\right), \sin \left(z3 \cdot \frac{1}{2}\right), \cos \left(\frac{-1}{2} \cdot z2\right)\right)}\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]
   6. Step-by-step derivation

      1. *-lft-identity N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\frac{1}{\frac{2}{\cos \left(z1 - z0\right) + \cos \left(z1 + z0\right)}} \cdot \mathsf{304\_z0z1z2z3z4}\left(1, \sin \left(z2 \cdot \frac{1}{2}\right), \cos \left(\frac{-1}{2} \cdot z3\right), \sin \left(z3 \cdot \frac{1}{2}\right), \cos \left(\frac{-1}{2} \cdot z2\right)\right)\right) \cdot \color{blue}{\left(1 \cdot \sin \left(\frac{z2 - z3}{2}\right)\right)}} \]

      2. lift-sin.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\frac{1}{\frac{2}{\cos \left(z1 - z0\right) + \cos \left(z1 + z0\right)}} \cdot \mathsf{304\_z0z1z2z3z4}\left(1, \sin \left(z2 \cdot \frac{1}{2}\right), \cos \left(\frac{-1}{2} \cdot z3\right), \sin \left(z3 \cdot \frac{1}{2}\right), \cos \left(\frac{-1}{2} \cdot z2\right)\right)\right) \cdot \left(1 \cdot \color{blue}{\sin \left(\frac{z2 - z3}{2}\right)}\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\frac{1}{\frac{2}{\cos \left(z1 - z0\right) + \cos \left(z1 + z0\right)}} \cdot \mathsf{304\_z0z1z2z3z4}\left(1, \sin \left(z2 \cdot \frac{1}{2}\right), \cos \left(\frac{-1}{2} \cdot z3\right), \sin \left(z3 \cdot \frac{1}{2}\right), \cos \left(\frac{-1}{2} \cdot z2\right)\right)\right) \cdot \left(1 \cdot \sin \color{blue}{\left(\frac{z2 - z3}{2}\right)}\right)} \]

      4. mult-flip N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\frac{1}{\frac{2}{\cos \left(z1 - z0\right) + \cos \left(z1 + z0\right)}} \cdot \mathsf{304\_z0z1z2z3z4}\left(1, \sin \left(z2 \cdot \frac{1}{2}\right), \cos \left(\frac{-1}{2} \cdot z3\right), \sin \left(z3 \cdot \frac{1}{2}\right), \cos \left(\frac{-1}{2} \cdot z2\right)\right)\right) \cdot \left(1 \cdot \sin \color{blue}{\left(\left(z2 - z3\right) \cdot \frac{1}{2}\right)}\right)} \]

      5. metadata-eval N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\frac{1}{\frac{2}{\cos \left(z1 - z0\right) + \cos \left(z1 + z0\right)}} \cdot \mathsf{304\_z0z1z2z3z4}\left(1, \sin \left(z2 \cdot \frac{1}{2}\right), \cos \left(\frac{-1}{2} \cdot z3\right), \sin \left(z3 \cdot \frac{1}{2}\right), \cos \left(\frac{-1}{2} \cdot z2\right)\right)\right) \cdot \left(1 \cdot \sin \left(\left(z2 - z3\right) \cdot \color{blue}{\frac{1}{2}}\right)\right)} \]

      6. *-commutative N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\frac{1}{\frac{2}{\cos \left(z1 - z0\right) + \cos \left(z1 + z0\right)}} \cdot \mathsf{304\_z0z1z2z3z4}\left(1, \sin \left(z2 \cdot \frac{1}{2}\right), \cos \left(\frac{-1}{2} \cdot z3\right), \sin \left(z3 \cdot \frac{1}{2}\right), \cos \left(\frac{-1}{2} \cdot z2\right)\right)\right) \cdot \left(1 \cdot \sin \color{blue}{\left(\frac{1}{2} \cdot \left(z2 - z3\right)\right)}\right)} \]

      7. lift--.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\frac{1}{\frac{2}{\cos \left(z1 - z0\right) + \cos \left(z1 + z0\right)}} \cdot \mathsf{304\_z0z1z2z3z4}\left(1, \sin \left(z2 \cdot \frac{1}{2}\right), \cos \left(\frac{-1}{2} \cdot z3\right), \sin \left(z3 \cdot \frac{1}{2}\right), \cos \left(\frac{-1}{2} \cdot z2\right)\right)\right) \cdot \left(1 \cdot \sin \left(\frac{1}{2} \cdot \color{blue}{\left(z2 - z3\right)}\right)\right)} \]

      8. distribute-rgt-out-- N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\frac{1}{\frac{2}{\cos \left(z1 - z0\right) + \cos \left(z1 + z0\right)}} \cdot \mathsf{304\_z0z1z2z3z4}\left(1, \sin \left(z2 \cdot \frac{1}{2}\right), \cos \left(\frac{-1}{2} \cdot z3\right), \sin \left(z3 \cdot \frac{1}{2}\right), \cos \left(\frac{-1}{2} \cdot z2\right)\right)\right) \cdot \left(1 \cdot \sin \color{blue}{\left(z2 \cdot \frac{1}{2} - z3 \cdot \frac{1}{2}\right)}\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\frac{1}{\frac{2}{\cos \left(z1 - z0\right) + \cos \left(z1 + z0\right)}} \cdot \mathsf{304\_z0z1z2z3z4}\left(1, \sin \left(z2 \cdot \frac{1}{2}\right), \cos \left(\frac{-1}{2} \cdot z3\right), \sin \left(z3 \cdot \frac{1}{2}\right), \cos \left(\frac{-1}{2} \cdot z2\right)\right)\right) \cdot \left(1 \cdot \sin \left(\color{blue}{z2 \cdot \frac{1}{2}} - z3 \cdot \frac{1}{2}\right)\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\frac{1}{\frac{2}{\cos \left(z1 - z0\right) + \cos \left(z1 + z0\right)}} \cdot \mathsf{304\_z0z1z2z3z4}\left(1, \sin \left(z2 \cdot \frac{1}{2}\right), \cos \left(\frac{-1}{2} \cdot z3\right), \sin \left(z3 \cdot \frac{1}{2}\right), \cos \left(\frac{-1}{2} \cdot z2\right)\right)\right) \cdot \left(1 \cdot \sin \left(z2 \cdot \frac{1}{2} - \color{blue}{z3 \cdot \frac{1}{2}}\right)\right)} \]

      11. sin-diff-rev N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\frac{1}{\frac{2}{\cos \left(z1 - z0\right) + \cos \left(z1 + z0\right)}} \cdot \mathsf{304\_z0z1z2z3z4}\left(1, \sin \left(z2 \cdot \frac{1}{2}\right), \cos \left(\frac{-1}{2} \cdot z3\right), \sin \left(z3 \cdot \frac{1}{2}\right), \cos \left(\frac{-1}{2} \cdot z2\right)\right)\right) \cdot \left(1 \cdot \color{blue}{\left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)}\right)} \]

      12. lift-sin.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\frac{1}{\frac{2}{\cos \left(z1 - z0\right) + \cos \left(z1 + z0\right)}} \cdot \mathsf{304\_z0z1z2z3z4}\left(1, \sin \left(z2 \cdot \frac{1}{2}\right), \cos \left(\frac{-1}{2} \cdot z3\right), \sin \left(z3 \cdot \frac{1}{2}\right), \cos \left(\frac{-1}{2} \cdot z2\right)\right)\right) \cdot \left(1 \cdot \left(\color{blue}{\sin \left(z2 \cdot \frac{1}{2}\right)} \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right)} \]

      13. lift-cos.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\frac{1}{\frac{2}{\cos \left(z1 - z0\right) + \cos \left(z1 + z0\right)}} \cdot \mathsf{304\_z0z1z2z3z4}\left(1, \sin \left(z2 \cdot \frac{1}{2}\right), \cos \left(\frac{-1}{2} \cdot z3\right), \sin \left(z3 \cdot \frac{1}{2}\right), \cos \left(\frac{-1}{2} \cdot z2\right)\right)\right) \cdot \left(1 \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(z3 \cdot \frac{1}{2}\right)} - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right)} \]

      14. lift-cos.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\frac{1}{\frac{2}{\cos \left(z1 - z0\right) + \cos \left(z1 + z0\right)}} \cdot \mathsf{304\_z0z1z2z3z4}\left(1, \sin \left(z2 \cdot \frac{1}{2}\right), \cos \left(\frac{-1}{2} \cdot z3\right), \sin \left(z3 \cdot \frac{1}{2}\right), \cos \left(\frac{-1}{2} \cdot z2\right)\right)\right) \cdot \left(1 \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(z2 \cdot \frac{1}{2}\right)} \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right)} \]

      15. lift-sin.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\frac{1}{\frac{2}{\cos \left(z1 - z0\right) + \cos \left(z1 + z0\right)}} \cdot \mathsf{304\_z0z1z2z3z4}\left(1, \sin \left(z2 \cdot \frac{1}{2}\right), \cos \left(\frac{-1}{2} \cdot z3\right), \sin \left(z3 \cdot \frac{1}{2}\right), \cos \left(\frac{-1}{2} \cdot z2\right)\right)\right) \cdot \left(1 \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(z3 \cdot \frac{1}{2}\right)}\right)\right)} \]

      16. *-commutative N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\frac{1}{\frac{2}{\cos \left(z1 - z0\right) + \cos \left(z1 + z0\right)}} \cdot \mathsf{304\_z0z1z2z3z4}\left(1, \sin \left(z2 \cdot \frac{1}{2}\right), \cos \left(\frac{-1}{2} \cdot z3\right), \sin \left(z3 \cdot \frac{1}{2}\right), \cos \left(\frac{-1}{2} \cdot z2\right)\right)\right) \cdot \left(1 \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \color{blue}{\sin \left(z3 \cdot \frac{1}{2}\right) \cdot \cos \left(z2 \cdot \frac{1}{2}\right)}\right)\right)} \]

      17. lower-304-z0z1z2z3z4 76.3%

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\frac{1}{\frac{2}{\cos \left(z1 - z0\right) + \cos \left(z1 + z0\right)}} \cdot \mathsf{304\_z0z1z2z3z4}\left(1, \sin \left(z2 \cdot \frac{1}{2}\right), \cos \left(\frac{-1}{2} \cdot z3\right), \sin \left(z3 \cdot \frac{1}{2}\right), \cos \left(\frac{-1}{2} \cdot z2\right)\right)\right) \cdot \color{blue}{\mathsf{304\_z0z1z2z3z4}\left(1, \sin \left(z2 \cdot \frac{1}{2}\right), \cos \left(z3 \cdot \frac{1}{2}\right), \sin \left(z3 \cdot \frac{1}{2}\right), \cos \left(z2 \cdot \frac{1}{2}\right)\right)}} \]

   7. Applied rewrites 76.3%

      \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\frac{1}{\frac{2}{\cos \left(z1 - z0\right) + \cos \left(z1 + z0\right)}} \cdot \mathsf{304\_z0z1z2z3z4}\left(1, \sin \left(z2 \cdot \frac{1}{2}\right), \cos \left(\frac{-1}{2} \cdot z3\right), \sin \left(z3 \cdot \frac{1}{2}\right), \cos \left(\frac{-1}{2} \cdot z2\right)\right)\right) \cdot \color{blue}{\mathsf{304\_z0z1z2z3z4}\left(1, \sin \left(z2 \cdot \frac{1}{2}\right), \cos \left(\frac{-1}{2} \cdot z3\right), \sin \left(z3 \cdot \frac{1}{2}\right), \cos \left(\frac{-1}{2} \cdot z2\right)\right)}} \]

   if 8.4999999999999996e24 < z0

   1. Initial program 60.5%

      \[\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

   3. Applied rewrites 57.8%

      \[\leadsto \sqrt{\color{blue}{\frac{1}{2} - \left(\cos \left(\left(z0 - z1\right) \cdot 1\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(z2 - z3\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)}} \]

   5. Applied rewrites 57.8%

      \[\leadsto \sqrt{\frac{1}{2} - \color{blue}{\left(\cos \left(z1 - z0\right) \cdot \frac{1}{2} + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\color{blue}{\cos \left(z1 - z0\right) \cdot \frac{1}{2}} + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\color{blue}{\frac{1}{2} \cdot \cos \left(z1 - z0\right)} + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      3. lift-cos.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\frac{1}{2} \cdot \color{blue}{\cos \left(z1 - z0\right)} + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      4. lift--.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\frac{1}{2} \cdot \cos \color{blue}{\left(z1 - z0\right)} + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      5. cos-diff N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\frac{1}{2} \cdot \color{blue}{\left(\cos z1 \cdot \cos z0 + \sin z1 \cdot \sin z0\right)} + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      6. lift-cos.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\frac{1}{2} \cdot \left(\color{blue}{\cos z1} \cdot \cos z0 + \sin z1 \cdot \sin z0\right) + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      7. lift-cos.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\frac{1}{2} \cdot \left(\cos z1 \cdot \color{blue}{\cos z0} + \sin z1 \cdot \sin z0\right) + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\frac{1}{2} \cdot \left(\color{blue}{\cos z1 \cdot \cos z0} + \sin z1 \cdot \sin z0\right) + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      9. *-lft-identity N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\frac{1}{2} \cdot \left(\color{blue}{1 \cdot \left(\cos z1 \cdot \cos z0\right)} + \sin z1 \cdot \sin z0\right) + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      10. add-flip N/A

          \[\leadsto \sqrt{\frac{1}{2} - \left(\frac{1}{2} \cdot \color{blue}{\left(1 \cdot \left(\cos z1 \cdot \cos z0\right) - \left(\mathsf{neg}\left(\sin z1 \cdot \sin z0\right)\right)\right)} + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      11. *-lft-identity N/A

          \[\leadsto \sqrt{\frac{1}{2} - \left(\frac{1}{2} \cdot \left(1 \cdot \color{blue}{\left(1 \cdot \left(\cos z1 \cdot \cos z0\right)\right)} - \left(\mathsf{neg}\left(\sin z1 \cdot \sin z0\right)\right)\right) + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      12. *-commutative N/A

          \[\leadsto \sqrt{\frac{1}{2} - \left(\frac{1}{2} \cdot \left(\color{blue}{\left(1 \cdot \left(\cos z1 \cdot \cos z0\right)\right) \cdot 1} - \left(\mathsf{neg}\left(\sin z1 \cdot \sin z0\right)\right)\right) + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      13. lift-sin.f64 N/A

          \[\leadsto \sqrt{\frac{1}{2} - \left(\frac{1}{2} \cdot \left(\left(1 \cdot \left(\cos z1 \cdot \cos z0\right)\right) \cdot 1 - \left(\mathsf{neg}\left(\color{blue}{\sin z1} \cdot \sin z0\right)\right)\right) + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      14. distribute-rgt-neg-out N/A

          \[\leadsto \sqrt{\frac{1}{2} - \left(\frac{1}{2} \cdot \left(\left(1 \cdot \left(\cos z1 \cdot \cos z0\right)\right) \cdot 1 - \color{blue}{\sin z1 \cdot \left(\mathsf{neg}\left(\sin z0\right)\right)}\right) + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      15. sin-neg N/A

          \[\leadsto \sqrt{\frac{1}{2} - \left(\frac{1}{2} \cdot \left(\left(1 \cdot \left(\cos z1 \cdot \cos z0\right)\right) \cdot 1 - \sin z1 \cdot \color{blue}{\sin \left(\mathsf{neg}\left(z0\right)\right)}\right) + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      16. lift-neg.f64 N/A

          \[\leadsto \sqrt{\frac{1}{2} - \left(\frac{1}{2} \cdot \left(\left(1 \cdot \left(\cos z1 \cdot \cos z0\right)\right) \cdot 1 - \sin z1 \cdot \sin \color{blue}{\left(-z0\right)}\right) + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      17. lift-sin.f64 N/A

          \[\leadsto \sqrt{\frac{1}{2} - \left(\frac{1}{2} \cdot \left(\left(1 \cdot \left(\cos z1 \cdot \cos z0\right)\right) \cdot 1 - \sin z1 \cdot \color{blue}{\sin \left(-z0\right)}\right) + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      18. lower-304-z0z1z2z3z4 N/A

          \[\leadsto \sqrt{\frac{1}{2} - \left(\color{blue}{\mathsf{304\_z0z1z2z3z4}\left(\frac{1}{2}, \left(1 \cdot \left(\cos z1 \cdot \cos z0\right)\right), 1, \sin z1, \sin \left(-z0\right)\right)} + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

   7. Applied rewrites 73.4%

      \[\leadsto \sqrt{\frac{1}{2} - \left(\color{blue}{\mathsf{304\_z0z1z2z3z4}\left(\frac{1}{2}, \left(\cos z1 \cdot \cos z0\right), 1, \sin z1, \left(-\sin z0\right)\right)} + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 85.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(\left(z2 - z3\right) \cdot \frac{1}{2}\right)\\ t_1 := \cos z1 \cdot \cos z0\\ t_2 := \sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\\ \mathbf{if}\;z0 \leq \frac{-944473296573929}{1180591620717411303424}:\\ \;\;\;\;\sqrt{\left(\frac{1}{2} - \mathsf{304\_z0z1z2z3z4}\left(\frac{1}{2}, \cos z1, \cos z0, \sin z1, \sin \left(-z0\right)\right)\right) + \left(\left(t\_0 \cdot \cos z0\right) \cdot \cos z1\right) \cdot t\_0}\\ \mathbf{elif}\;z0 \leq 8499999999999999588958208:\\ \;\;\;\;\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\left(\cos \left(z1 + z0\right) + \cos \left(z1 - z0\right)\right) \cdot \frac{1}{2}\right) \cdot t\_2\right) \cdot t\_2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{2} - \left(\mathsf{304\_z0z1z2z3z4}\left(\frac{1}{2}, t\_1, 1, \sin z1, \left(-\sin z0\right)\right) + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot t\_1\right)}\\ \end{array} \]

FPCore

(FPCore (z0 z1 z2 z3)
  :precision binary64
  (let* ((t_0 (sin (* (- z2 z3) 1/2)))
       (t_1 (* (cos z1) (cos z0)))
       (t_2
        (-
         (* (sin (* z2 1/2)) (cos (* z3 1/2)))
         (* (cos (* z2 1/2)) (sin (* z3 1/2))))))
  (if (<= z0 -944473296573929/1180591620717411303424)
    (sqrt
     (+
      (-
       1/2
       (304-z0z1z2z3z4 1/2 (cos z1) (cos z0) (sin z1) (sin (- z0))))
      (* (* (* t_0 (cos z0)) (cos z1)) t_0)))
    (if (<= z0 8499999999999999588958208)
      (sqrt
       (+
        (- 1/2 (* 1/2 (cos (* 2 (/ (- z0 z1) 2)))))
        (* (* (* (+ (cos (+ z1 z0)) (cos (- z1 z0))) 1/2) t_2) t_2)))
      (sqrt
       (-
        1/2
        (+
         (304-z0z1z2z3z4 1/2 t_1 1 (sin z1) (- (sin z0)))
         (* (- (* (cos (* (- z2 z3) 1)) 1/2) 1/2) t_1))))))))

Derivation

1. Split input into 3 regimes

2. if z0 < -7.9999999999999996e-7

   1. Initial program 60.5%

      \[\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \color{blue}{\sin \left(\frac{z2 - z3}{2}\right)}\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \color{blue}{\left(\frac{z2 - z3}{2}\right)}\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      3. lift--.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \left(\frac{\color{blue}{z2 - z3}}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      4. div-sub N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \color{blue}{\left(\frac{z2}{2} - \frac{z3}{2}\right)}\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      5. sin-diff N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \color{blue}{\left(\sin \left(\frac{z2}{2}\right) \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)}\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      6. lower--.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \color{blue}{\left(\sin \left(\frac{z2}{2}\right) \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)}\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\color{blue}{\sin \left(\frac{z2}{2}\right) \cdot \cos \left(\frac{z3}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      8. lower-sin.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\color{blue}{\sin \left(\frac{z2}{2}\right)} \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      9. mult-flip N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      10. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      12. lower-cos.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{z3}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      13. mult-flip N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(z3 \cdot \frac{1}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      14. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      15. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(z3 \cdot \frac{1}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      16. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      17. lower-cos.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{z2}{2}\right)} \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      18. mult-flip N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      19. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      20. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      21. lower-sin.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{z3}{2}\right)}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      22. mult-flip N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(z3 \cdot \frac{1}{2}\right)}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      23. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      24. lower-*.f64 59.7%

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(z3 \cdot \frac{1}{2}\right)}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

   3. Applied rewrites 59.7%

      \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \color{blue}{\left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)}\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{z2 - z3}{2}\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{z2 - z3}{2}\right)}} \]

      3. lift--.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{z2 - z3}}{2}\right)} \]

      4. div-sub N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{z2}{2} - \frac{z3}{2}\right)}} \]

      5. sin-diff N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{z2}{2}\right) \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)}} \]

      6. lower--.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{z2}{2}\right) \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{z2}{2}\right) \cdot \cos \left(\frac{z3}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      8. lower-sin.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{z2}{2}\right)} \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      9. mult-flip N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      10. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      12. lower-cos.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{z3}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      13. mult-flip N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(z3 \cdot \frac{1}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      14. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      15. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(z3 \cdot \frac{1}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      16. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)}\right)} \]

      17. lower-cos.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{z2}{2}\right)} \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      18. mult-flip N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      19. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      20. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      21. lower-sin.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{z3}{2}\right)}\right)} \]

      22. mult-flip N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(z3 \cdot \frac{1}{2}\right)}\right)} \]

      23. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \color{blue}{\frac{1}{2}}\right)\right)} \]

      24. lower-*.f64 75.8%

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(z3 \cdot \frac{1}{2}\right)}\right)} \]

   5. Applied rewrites 75.8%

      \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)}} \]

   7. Applied rewrites 96.4%

      \[\leadsto \sqrt{\left(\frac{1}{2} - \color{blue}{\mathsf{304\_z0z1z2z3z4}\left(\frac{1}{2}, \cos z1, \cos z0, \sin z1, \sin \left(-z0\right)\right)}\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)} \]

   9. Applied rewrites 76.1%

      \[\leadsto \sqrt{\left(\frac{1}{2} - \mathsf{304\_z0z1z2z3z4}\left(\frac{1}{2}, \cos z1, \cos z0, \sin z1, \sin \left(-z0\right)\right)\right) + \color{blue}{\left(\left(\sin \left(\left(z2 - z3\right) \cdot \frac{1}{2}\right) \cdot \cos z0\right) \cdot \cos z1\right) \cdot \sin \left(\left(z2 - z3\right) \cdot \frac{1}{2}\right)}} \]

   if -7.9999999999999996e-7 < z0 < 8.4999999999999996e24

   1. Initial program 60.5%

      \[\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \color{blue}{\sin \left(\frac{z2 - z3}{2}\right)}\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \color{blue}{\left(\frac{z2 - z3}{2}\right)}\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      3. lift--.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \left(\frac{\color{blue}{z2 - z3}}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      4. div-sub N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \color{blue}{\left(\frac{z2}{2} - \frac{z3}{2}\right)}\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      5. sin-diff N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \color{blue}{\left(\sin \left(\frac{z2}{2}\right) \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)}\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      6. lower--.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \color{blue}{\left(\sin \left(\frac{z2}{2}\right) \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)}\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\color{blue}{\sin \left(\frac{z2}{2}\right) \cdot \cos \left(\frac{z3}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      8. lower-sin.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\color{blue}{\sin \left(\frac{z2}{2}\right)} \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      9. mult-flip N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      10. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      12. lower-cos.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{z3}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      13. mult-flip N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(z3 \cdot \frac{1}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      14. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      15. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(z3 \cdot \frac{1}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      16. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      17. lower-cos.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{z2}{2}\right)} \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      18. mult-flip N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      19. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      20. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      21. lower-sin.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{z3}{2}\right)}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      22. mult-flip N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(z3 \cdot \frac{1}{2}\right)}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      23. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      24. lower-*.f64 59.7%

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(z3 \cdot \frac{1}{2}\right)}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

   3. Applied rewrites 59.7%

      \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \color{blue}{\left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)}\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{z2 - z3}{2}\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{z2 - z3}{2}\right)}} \]

      3. lift--.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{z2 - z3}}{2}\right)} \]

      4. div-sub N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{z2}{2} - \frac{z3}{2}\right)}} \]

      5. sin-diff N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{z2}{2}\right) \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)}} \]

      6. lower--.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{z2}{2}\right) \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{z2}{2}\right) \cdot \cos \left(\frac{z3}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      8. lower-sin.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{z2}{2}\right)} \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      9. mult-flip N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      10. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      12. lower-cos.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{z3}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      13. mult-flip N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(z3 \cdot \frac{1}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      14. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      15. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(z3 \cdot \frac{1}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      16. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)}\right)} \]

      17. lower-cos.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{z2}{2}\right)} \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      18. mult-flip N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      19. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      20. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      21. lower-sin.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{z3}{2}\right)}\right)} \]

      22. mult-flip N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(z3 \cdot \frac{1}{2}\right)}\right)} \]

      23. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \color{blue}{\frac{1}{2}}\right)\right)} \]

      24. lower-*.f64 75.8%

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(z3 \cdot \frac{1}{2}\right)}\right)} \]

   5. Applied rewrites 75.8%

      \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\color{blue}{\left(\cos z0 \cdot \cos z1\right)} \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\color{blue}{\left(\cos z1 \cdot \cos z0\right)} \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)} \]

      3. lift-cos.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\color{blue}{\cos z1} \cdot \cos z0\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)} \]

      4. lift-cos.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z1 \cdot \color{blue}{\cos z0}\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)} \]

      5. cos-mult N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\color{blue}{\frac{\cos \left(z1 + z0\right) + \cos \left(z1 - z0\right)}{2}} \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)} \]

      6. lift-+.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\frac{\cos \color{blue}{\left(z1 + z0\right)} + \cos \left(z1 - z0\right)}{2} \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)} \]

      7. lift-cos.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\frac{\color{blue}{\cos \left(z1 + z0\right)} + \cos \left(z1 - z0\right)}{2} \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)} \]

      8. lift--.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\frac{\cos \left(z1 + z0\right) + \cos \color{blue}{\left(z1 - z0\right)}}{2} \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)} \]

      9. lift-cos.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\frac{\cos \left(z1 + z0\right) + \color{blue}{\cos \left(z1 - z0\right)}}{2} \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)} \]

      10. +-commutative N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\frac{\color{blue}{\cos \left(z1 - z0\right) + \cos \left(z1 + z0\right)}}{2} \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)} \]

      11. lift-+.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\frac{\color{blue}{\cos \left(z1 - z0\right) + \cos \left(z1 + z0\right)}}{2} \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)} \]

      12. mult-flip N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\color{blue}{\left(\left(\cos \left(z1 - z0\right) + \cos \left(z1 + z0\right)\right) \cdot \frac{1}{2}\right)} \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)} \]

      13. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\left(\cos \left(z1 - z0\right) + \cos \left(z1 + z0\right)\right) \cdot \color{blue}{\frac{1}{2}}\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)} \]

      14. lower-*.f64 76.3%

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\color{blue}{\left(\left(\cos \left(z1 - z0\right) + \cos \left(z1 + z0\right)\right) \cdot \frac{1}{2}\right)} \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)} \]

      15. lift-+.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\color{blue}{\left(\cos \left(z1 - z0\right) + \cos \left(z1 + z0\right)\right)} \cdot \frac{1}{2}\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)} \]

      16. +-commutative N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\color{blue}{\left(\cos \left(z1 + z0\right) + \cos \left(z1 - z0\right)\right)} \cdot \frac{1}{2}\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)} \]

      17. lower-+.f64 76.3%

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\color{blue}{\left(\cos \left(z1 + z0\right) + \cos \left(z1 - z0\right)\right)} \cdot \frac{1}{2}\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)} \]

   7. Applied rewrites 76.3%

      \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\color{blue}{\left(\left(\cos \left(z1 + z0\right) + \cos \left(z1 - z0\right)\right) \cdot \frac{1}{2}\right)} \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)} \]

   if 8.4999999999999996e24 < z0

   1. Initial program 60.5%

      \[\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

   3. Applied rewrites 57.8%

      \[\leadsto \sqrt{\color{blue}{\frac{1}{2} - \left(\cos \left(\left(z0 - z1\right) \cdot 1\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(z2 - z3\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)}} \]

   5. Applied rewrites 57.8%

      \[\leadsto \sqrt{\frac{1}{2} - \color{blue}{\left(\cos \left(z1 - z0\right) \cdot \frac{1}{2} + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\color{blue}{\cos \left(z1 - z0\right) \cdot \frac{1}{2}} + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\color{blue}{\frac{1}{2} \cdot \cos \left(z1 - z0\right)} + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      3. lift-cos.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\frac{1}{2} \cdot \color{blue}{\cos \left(z1 - z0\right)} + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      4. lift--.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\frac{1}{2} \cdot \cos \color{blue}{\left(z1 - z0\right)} + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      5. cos-diff N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\frac{1}{2} \cdot \color{blue}{\left(\cos z1 \cdot \cos z0 + \sin z1 \cdot \sin z0\right)} + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      6. lift-cos.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\frac{1}{2} \cdot \left(\color{blue}{\cos z1} \cdot \cos z0 + \sin z1 \cdot \sin z0\right) + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      7. lift-cos.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\frac{1}{2} \cdot \left(\cos z1 \cdot \color{blue}{\cos z0} + \sin z1 \cdot \sin z0\right) + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\frac{1}{2} \cdot \left(\color{blue}{\cos z1 \cdot \cos z0} + \sin z1 \cdot \sin z0\right) + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      9. *-lft-identity N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\frac{1}{2} \cdot \left(\color{blue}{1 \cdot \left(\cos z1 \cdot \cos z0\right)} + \sin z1 \cdot \sin z0\right) + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      10. add-flip N/A

          \[\leadsto \sqrt{\frac{1}{2} - \left(\frac{1}{2} \cdot \color{blue}{\left(1 \cdot \left(\cos z1 \cdot \cos z0\right) - \left(\mathsf{neg}\left(\sin z1 \cdot \sin z0\right)\right)\right)} + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      11. *-lft-identity N/A

          \[\leadsto \sqrt{\frac{1}{2} - \left(\frac{1}{2} \cdot \left(1 \cdot \color{blue}{\left(1 \cdot \left(\cos z1 \cdot \cos z0\right)\right)} - \left(\mathsf{neg}\left(\sin z1 \cdot \sin z0\right)\right)\right) + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      12. *-commutative N/A

          \[\leadsto \sqrt{\frac{1}{2} - \left(\frac{1}{2} \cdot \left(\color{blue}{\left(1 \cdot \left(\cos z1 \cdot \cos z0\right)\right) \cdot 1} - \left(\mathsf{neg}\left(\sin z1 \cdot \sin z0\right)\right)\right) + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      13. lift-sin.f64 N/A

          \[\leadsto \sqrt{\frac{1}{2} - \left(\frac{1}{2} \cdot \left(\left(1 \cdot \left(\cos z1 \cdot \cos z0\right)\right) \cdot 1 - \left(\mathsf{neg}\left(\color{blue}{\sin z1} \cdot \sin z0\right)\right)\right) + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      14. distribute-rgt-neg-out N/A

          \[\leadsto \sqrt{\frac{1}{2} - \left(\frac{1}{2} \cdot \left(\left(1 \cdot \left(\cos z1 \cdot \cos z0\right)\right) \cdot 1 - \color{blue}{\sin z1 \cdot \left(\mathsf{neg}\left(\sin z0\right)\right)}\right) + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      15. sin-neg N/A

          \[\leadsto \sqrt{\frac{1}{2} - \left(\frac{1}{2} \cdot \left(\left(1 \cdot \left(\cos z1 \cdot \cos z0\right)\right) \cdot 1 - \sin z1 \cdot \color{blue}{\sin \left(\mathsf{neg}\left(z0\right)\right)}\right) + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      16. lift-neg.f64 N/A

          \[\leadsto \sqrt{\frac{1}{2} - \left(\frac{1}{2} \cdot \left(\left(1 \cdot \left(\cos z1 \cdot \cos z0\right)\right) \cdot 1 - \sin z1 \cdot \sin \color{blue}{\left(-z0\right)}\right) + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      17. lift-sin.f64 N/A

          \[\leadsto \sqrt{\frac{1}{2} - \left(\frac{1}{2} \cdot \left(\left(1 \cdot \left(\cos z1 \cdot \cos z0\right)\right) \cdot 1 - \sin z1 \cdot \color{blue}{\sin \left(-z0\right)}\right) + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      18. lower-304-z0z1z2z3z4 N/A

          \[\leadsto \sqrt{\frac{1}{2} - \left(\color{blue}{\mathsf{304\_z0z1z2z3z4}\left(\frac{1}{2}, \left(1 \cdot \left(\cos z1 \cdot \cos z0\right)\right), 1, \sin z1, \sin \left(-z0\right)\right)} + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

   7. Applied rewrites 73.4%

      \[\leadsto \sqrt{\frac{1}{2} - \left(\color{blue}{\mathsf{304\_z0z1z2z3z4}\left(\frac{1}{2}, \left(\cos z1 \cdot \cos z0\right), 1, \sin z1, \left(-\sin z0\right)\right)} + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 85.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \cos z1 \cdot \cos z0\\ t_1 := \sin \left(\left(z2 - z3\right) \cdot \frac{1}{2}\right)\\ t_2 := \sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\\ \mathbf{if}\;z0 \leq \frac{-944473296573929}{1180591620717411303424}:\\ \;\;\;\;\sqrt{\left(\frac{1}{2} - \mathsf{304\_z0z1z2z3z4}\left(\frac{1}{2}, \cos z1, \cos z0, \sin z1, \sin \left(-z0\right)\right)\right) + \left(\left(t\_1 \cdot \cos z0\right) \cdot \cos z1\right) \cdot t\_1}\\ \mathbf{elif}\;z0 \leq 280000000000000:\\ \;\;\;\;\sqrt{\left(\frac{1}{2} - \cos \left(z1 - z0\right) \cdot \frac{1}{2}\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot t\_2\right) \cdot t\_2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{2} - \left(\mathsf{304\_z0z1z2z3z4}\left(\frac{1}{2}, t\_0, 1, \sin z1, \left(-\sin z0\right)\right) + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot t\_0\right)}\\ \end{array} \]

FPCore

(FPCore (z0 z1 z2 z3)
  :precision binary64
  (let* ((t_0 (* (cos z1) (cos z0)))
       (t_1 (sin (* (- z2 z3) 1/2)))
       (t_2
        (-
         (* (sin (* z2 1/2)) (cos (* z3 1/2)))
         (* (cos (* z2 1/2)) (sin (* z3 1/2))))))
  (if (<= z0 -944473296573929/1180591620717411303424)
    (sqrt
     (+
      (-
       1/2
       (304-z0z1z2z3z4 1/2 (cos z1) (cos z0) (sin z1) (sin (- z0))))
      (* (* (* t_1 (cos z0)) (cos z1)) t_1)))
    (if (<= z0 280000000000000)
      (sqrt
       (+
        (- 1/2 (* (cos (- z1 z0)) 1/2))
        (* (* (* (cos z0) (cos z1)) t_2) t_2)))
      (sqrt
       (-
        1/2
        (+
         (304-z0z1z2z3z4 1/2 t_0 1 (sin z1) (- (sin z0)))
         (* (- (* (cos (* (- z2 z3) 1)) 1/2) 1/2) t_0))))))))

Derivation

1. Split input into 3 regimes

2. if z0 < -7.9999999999999996e-7

   1. Initial program 60.5%

      \[\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \color{blue}{\sin \left(\frac{z2 - z3}{2}\right)}\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \color{blue}{\left(\frac{z2 - z3}{2}\right)}\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      3. lift--.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \left(\frac{\color{blue}{z2 - z3}}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      4. div-sub N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \color{blue}{\left(\frac{z2}{2} - \frac{z3}{2}\right)}\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      5. sin-diff N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \color{blue}{\left(\sin \left(\frac{z2}{2}\right) \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)}\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      6. lower--.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \color{blue}{\left(\sin \left(\frac{z2}{2}\right) \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)}\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\color{blue}{\sin \left(\frac{z2}{2}\right) \cdot \cos \left(\frac{z3}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      8. lower-sin.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\color{blue}{\sin \left(\frac{z2}{2}\right)} \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      9. mult-flip N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      10. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      12. lower-cos.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{z3}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      13. mult-flip N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(z3 \cdot \frac{1}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      14. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      15. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(z3 \cdot \frac{1}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      16. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      17. lower-cos.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{z2}{2}\right)} \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      18. mult-flip N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      19. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      20. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      21. lower-sin.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{z3}{2}\right)}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      22. mult-flip N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(z3 \cdot \frac{1}{2}\right)}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      23. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      24. lower-*.f64 59.7%

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(z3 \cdot \frac{1}{2}\right)}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

   3. Applied rewrites 59.7%

      \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \color{blue}{\left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)}\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{z2 - z3}{2}\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{z2 - z3}{2}\right)}} \]

      3. lift--.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{z2 - z3}}{2}\right)} \]

      4. div-sub N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{z2}{2} - \frac{z3}{2}\right)}} \]

      5. sin-diff N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{z2}{2}\right) \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)}} \]

      6. lower--.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{z2}{2}\right) \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{z2}{2}\right) \cdot \cos \left(\frac{z3}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      8. lower-sin.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{z2}{2}\right)} \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      9. mult-flip N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      10. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      12. lower-cos.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{z3}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      13. mult-flip N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(z3 \cdot \frac{1}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      14. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      15. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(z3 \cdot \frac{1}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      16. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)}\right)} \]

      17. lower-cos.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{z2}{2}\right)} \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      18. mult-flip N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      19. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      20. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      21. lower-sin.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{z3}{2}\right)}\right)} \]

      22. mult-flip N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(z3 \cdot \frac{1}{2}\right)}\right)} \]

      23. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \color{blue}{\frac{1}{2}}\right)\right)} \]

      24. lower-*.f64 75.8%

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(z3 \cdot \frac{1}{2}\right)}\right)} \]

   5. Applied rewrites 75.8%

      \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)}} \]

   7. Applied rewrites 96.4%

      \[\leadsto \sqrt{\left(\frac{1}{2} - \color{blue}{\mathsf{304\_z0z1z2z3z4}\left(\frac{1}{2}, \cos z1, \cos z0, \sin z1, \sin \left(-z0\right)\right)}\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)} \]

   9. Applied rewrites 76.1%

      \[\leadsto \sqrt{\left(\frac{1}{2} - \mathsf{304\_z0z1z2z3z4}\left(\frac{1}{2}, \cos z1, \cos z0, \sin z1, \sin \left(-z0\right)\right)\right) + \color{blue}{\left(\left(\sin \left(\left(z2 - z3\right) \cdot \frac{1}{2}\right) \cdot \cos z0\right) \cdot \cos z1\right) \cdot \sin \left(\left(z2 - z3\right) \cdot \frac{1}{2}\right)}} \]

   if -7.9999999999999996e-7 < z0 < 2.8e14

   1. Initial program 60.5%

      \[\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \color{blue}{\sin \left(\frac{z2 - z3}{2}\right)}\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \color{blue}{\left(\frac{z2 - z3}{2}\right)}\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      3. lift--.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \left(\frac{\color{blue}{z2 - z3}}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      4. div-sub N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \color{blue}{\left(\frac{z2}{2} - \frac{z3}{2}\right)}\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      5. sin-diff N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \color{blue}{\left(\sin \left(\frac{z2}{2}\right) \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)}\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      6. lower--.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \color{blue}{\left(\sin \left(\frac{z2}{2}\right) \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)}\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\color{blue}{\sin \left(\frac{z2}{2}\right) \cdot \cos \left(\frac{z3}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      8. lower-sin.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\color{blue}{\sin \left(\frac{z2}{2}\right)} \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      9. mult-flip N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      10. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      12. lower-cos.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{z3}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      13. mult-flip N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(z3 \cdot \frac{1}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      14. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      15. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(z3 \cdot \frac{1}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      16. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      17. lower-cos.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{z2}{2}\right)} \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      18. mult-flip N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      19. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      20. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      21. lower-sin.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{z3}{2}\right)}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      22. mult-flip N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(z3 \cdot \frac{1}{2}\right)}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      23. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      24. lower-*.f64 59.7%

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(z3 \cdot \frac{1}{2}\right)}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

   3. Applied rewrites 59.7%

      \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \color{blue}{\left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)}\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{z2 - z3}{2}\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{z2 - z3}{2}\right)}} \]

      3. lift--.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{z2 - z3}}{2}\right)} \]

      4. div-sub N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{z2}{2} - \frac{z3}{2}\right)}} \]

      5. sin-diff N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{z2}{2}\right) \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)}} \]

      6. lower--.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{z2}{2}\right) \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{z2}{2}\right) \cdot \cos \left(\frac{z3}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      8. lower-sin.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{z2}{2}\right)} \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      9. mult-flip N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      10. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      12. lower-cos.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{z3}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      13. mult-flip N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(z3 \cdot \frac{1}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      14. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      15. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(z3 \cdot \frac{1}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      16. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)}\right)} \]

      17. lower-cos.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{z2}{2}\right)} \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      18. mult-flip N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      19. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      20. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      21. lower-sin.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{z3}{2}\right)}\right)} \]

      22. mult-flip N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(z3 \cdot \frac{1}{2}\right)}\right)} \]

      23. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \color{blue}{\frac{1}{2}}\right)\right)} \]

      24. lower-*.f64 75.8%

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(z3 \cdot \frac{1}{2}\right)}\right)} \]

   5. Applied rewrites 75.8%

      \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)}} \]

   7. Applied rewrites 96.4%

      \[\leadsto \sqrt{\left(\frac{1}{2} - \color{blue}{\mathsf{304\_z0z1z2z3z4}\left(\frac{1}{2}, 1, \left(\cos z1 \cdot \cos z0\right), \sin z1, \sin \left(-z0\right)\right)}\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)} \]
   8. Step-by-step derivation

      1. lift-304-z0z1z2z3z4 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \left(1 \cdot \left(\cos z1 \cdot \cos z0\right) - \sin z1 \cdot \sin \left(-z0\right)\right)}\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \color{blue}{\left(1 \cdot \left(\cos z1 \cdot \cos z0\right) - \sin z1 \cdot \sin \left(-z0\right)\right) \cdot \frac{1}{2}}\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)} \]

      3. *-lft-identity N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \left(\color{blue}{\cos z1 \cdot \cos z0} - \sin z1 \cdot \sin \left(-z0\right)\right) \cdot \frac{1}{2}\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \left(\color{blue}{\cos z1 \cdot \cos z0} - \sin z1 \cdot \sin \left(-z0\right)\right) \cdot \frac{1}{2}\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)} \]

      5. lift-cos.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \left(\color{blue}{\cos z1} \cdot \cos z0 - \sin z1 \cdot \sin \left(-z0\right)\right) \cdot \frac{1}{2}\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)} \]

      6. lift-cos.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \left(\cos z1 \cdot \color{blue}{\cos z0} - \sin z1 \cdot \sin \left(-z0\right)\right) \cdot \frac{1}{2}\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)} \]

      7. cos-neg-rev N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \left(\cos z1 \cdot \color{blue}{\cos \left(\mathsf{neg}\left(z0\right)\right)} - \sin z1 \cdot \sin \left(-z0\right)\right) \cdot \frac{1}{2}\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)} \]

      8. lift-neg.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \left(\cos z1 \cdot \cos \color{blue}{\left(-z0\right)} - \sin z1 \cdot \sin \left(-z0\right)\right) \cdot \frac{1}{2}\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)} \]

      9. lift-sin.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \left(\cos z1 \cdot \cos \left(-z0\right) - \color{blue}{\sin z1} \cdot \sin \left(-z0\right)\right) \cdot \frac{1}{2}\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)} \]

      10. lift-sin.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \left(\cos z1 \cdot \cos \left(-z0\right) - \sin z1 \cdot \color{blue}{\sin \left(-z0\right)}\right) \cdot \frac{1}{2}\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)} \]

      11. cos-sum-rev N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \color{blue}{\cos \left(z1 + \left(-z0\right)\right)} \cdot \frac{1}{2}\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)} \]

      12. lift-neg.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \cos \left(z1 + \color{blue}{\left(\mathsf{neg}\left(z0\right)\right)}\right) \cdot \frac{1}{2}\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)} \]

      13. sub-flip N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \cos \color{blue}{\left(z1 - z0\right)} \cdot \frac{1}{2}\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)} \]

      14. lift--.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \cos \color{blue}{\left(z1 - z0\right)} \cdot \frac{1}{2}\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)} \]

      15. lift-cos.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \color{blue}{\cos \left(z1 - z0\right)} \cdot \frac{1}{2}\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)} \]

      16. lift-*.f64 75.8%

          \[\leadsto \sqrt{\left(\frac{1}{2} - \color{blue}{\cos \left(z1 - z0\right) \cdot \frac{1}{2}}\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)} \]

   9. Applied rewrites 75.8%

      \[\leadsto \sqrt{\color{blue}{\left(\frac{1}{2} - \cos \left(z1 - z0\right) \cdot \frac{1}{2}\right)} + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)} \]

   if 2.8e14 < z0

   1. Initial program 60.5%

      \[\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

   3. Applied rewrites 57.8%

      \[\leadsto \sqrt{\color{blue}{\frac{1}{2} - \left(\cos \left(\left(z0 - z1\right) \cdot 1\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(z2 - z3\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)}} \]

   5. Applied rewrites 57.8%

      \[\leadsto \sqrt{\frac{1}{2} - \color{blue}{\left(\cos \left(z1 - z0\right) \cdot \frac{1}{2} + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\color{blue}{\cos \left(z1 - z0\right) \cdot \frac{1}{2}} + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\color{blue}{\frac{1}{2} \cdot \cos \left(z1 - z0\right)} + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      3. lift-cos.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\frac{1}{2} \cdot \color{blue}{\cos \left(z1 - z0\right)} + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      4. lift--.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\frac{1}{2} \cdot \cos \color{blue}{\left(z1 - z0\right)} + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      5. cos-diff N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\frac{1}{2} \cdot \color{blue}{\left(\cos z1 \cdot \cos z0 + \sin z1 \cdot \sin z0\right)} + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      6. lift-cos.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\frac{1}{2} \cdot \left(\color{blue}{\cos z1} \cdot \cos z0 + \sin z1 \cdot \sin z0\right) + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      7. lift-cos.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\frac{1}{2} \cdot \left(\cos z1 \cdot \color{blue}{\cos z0} + \sin z1 \cdot \sin z0\right) + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\frac{1}{2} \cdot \left(\color{blue}{\cos z1 \cdot \cos z0} + \sin z1 \cdot \sin z0\right) + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      9. *-lft-identity N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\frac{1}{2} \cdot \left(\color{blue}{1 \cdot \left(\cos z1 \cdot \cos z0\right)} + \sin z1 \cdot \sin z0\right) + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      10. add-flip N/A

          \[\leadsto \sqrt{\frac{1}{2} - \left(\frac{1}{2} \cdot \color{blue}{\left(1 \cdot \left(\cos z1 \cdot \cos z0\right) - \left(\mathsf{neg}\left(\sin z1 \cdot \sin z0\right)\right)\right)} + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      11. *-lft-identity N/A

          \[\leadsto \sqrt{\frac{1}{2} - \left(\frac{1}{2} \cdot \left(1 \cdot \color{blue}{\left(1 \cdot \left(\cos z1 \cdot \cos z0\right)\right)} - \left(\mathsf{neg}\left(\sin z1 \cdot \sin z0\right)\right)\right) + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      12. *-commutative N/A

          \[\leadsto \sqrt{\frac{1}{2} - \left(\frac{1}{2} \cdot \left(\color{blue}{\left(1 \cdot \left(\cos z1 \cdot \cos z0\right)\right) \cdot 1} - \left(\mathsf{neg}\left(\sin z1 \cdot \sin z0\right)\right)\right) + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      13. lift-sin.f64 N/A

          \[\leadsto \sqrt{\frac{1}{2} - \left(\frac{1}{2} \cdot \left(\left(1 \cdot \left(\cos z1 \cdot \cos z0\right)\right) \cdot 1 - \left(\mathsf{neg}\left(\color{blue}{\sin z1} \cdot \sin z0\right)\right)\right) + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      14. distribute-rgt-neg-out N/A

          \[\leadsto \sqrt{\frac{1}{2} - \left(\frac{1}{2} \cdot \left(\left(1 \cdot \left(\cos z1 \cdot \cos z0\right)\right) \cdot 1 - \color{blue}{\sin z1 \cdot \left(\mathsf{neg}\left(\sin z0\right)\right)}\right) + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      15. sin-neg N/A

          \[\leadsto \sqrt{\frac{1}{2} - \left(\frac{1}{2} \cdot \left(\left(1 \cdot \left(\cos z1 \cdot \cos z0\right)\right) \cdot 1 - \sin z1 \cdot \color{blue}{\sin \left(\mathsf{neg}\left(z0\right)\right)}\right) + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      16. lift-neg.f64 N/A

          \[\leadsto \sqrt{\frac{1}{2} - \left(\frac{1}{2} \cdot \left(\left(1 \cdot \left(\cos z1 \cdot \cos z0\right)\right) \cdot 1 - \sin z1 \cdot \sin \color{blue}{\left(-z0\right)}\right) + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      17. lift-sin.f64 N/A

          \[\leadsto \sqrt{\frac{1}{2} - \left(\frac{1}{2} \cdot \left(\left(1 \cdot \left(\cos z1 \cdot \cos z0\right)\right) \cdot 1 - \sin z1 \cdot \color{blue}{\sin \left(-z0\right)}\right) + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      18. lower-304-z0z1z2z3z4 N/A

          \[\leadsto \sqrt{\frac{1}{2} - \left(\color{blue}{\mathsf{304\_z0z1z2z3z4}\left(\frac{1}{2}, \left(1 \cdot \left(\cos z1 \cdot \cos z0\right)\right), 1, \sin z1, \sin \left(-z0\right)\right)} + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

   7. Applied rewrites 73.4%

      \[\leadsto \sqrt{\frac{1}{2} - \left(\color{blue}{\mathsf{304\_z0z1z2z3z4}\left(\frac{1}{2}, \left(\cos z1 \cdot \cos z0\right), 1, \sin z1, \left(-\sin z0\right)\right)} + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 85.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(z1 - z0\right)\\ t_1 := \cos z1 \cdot \cos z0\\ t_2 := t\_0 \cdot \frac{1}{2}\\ \mathbf{if}\;z3 \leq \frac{-1770887431076117}{73786976294838206464}:\\ \;\;\;\;\sqrt{\frac{1}{2} - \left(1 - \frac{\left(\left(\frac{1}{2} - \left(\cos z2 \cdot \cos z3 + \sin z2 \cdot \sin z3\right) \cdot \frac{1}{2}\right) \cdot \cos z1\right) \cdot \cos z0}{t\_2}\right) \cdot t\_2}\\ \mathbf{elif}\;z3 \leq 12500000:\\ \;\;\;\;\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \left(t\_1 + \sin z1 \cdot \sin z0\right)\right) + \mathsf{304\_z0z1z2z3z4}\left(\left(\sin \left(\left(z2 - z3\right) \cdot \frac{1}{2}\right) \cdot t\_1\right), \sin \left(z2 \cdot \frac{1}{2}\right), \cos \left(\frac{-1}{2} \cdot z3\right), \sin \left(z3 \cdot \frac{1}{2}\right), \cos \left(\frac{-1}{2} \cdot z2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{2} - \left(t\_1 \cdot \left(\mathsf{304\_z0z1z2z3z4}\left(\frac{1}{2}, \cos z3, \cos z2, \sin z3, \sin \left(-z2\right)\right) - \frac{1}{2}\right) - \frac{-1}{2} \cdot t\_0\right)}\\ \end{array} \]

FPCore

(FPCore (z0 z1 z2 z3)
  :precision binary64
  (let* ((t_0 (cos (- z1 z0)))
       (t_1 (* (cos z1) (cos z0)))
       (t_2 (* t_0 1/2)))
  (if (<= z3 -1770887431076117/73786976294838206464)
    (sqrt
     (-
      1/2
      (*
       (-
        1
        (/
         (*
          (*
           (-
            1/2
            (* (+ (* (cos z2) (cos z3)) (* (sin z2) (sin z3))) 1/2))
           (cos z1))
          (cos z0))
         t_2))
       t_2)))
    (if (<= z3 12500000)
      (sqrt
       (+
        (- 1/2 (* 1/2 (+ t_1 (* (sin z1) (sin z0)))))
        (304-z0z1z2z3z4
         (* (sin (* (- z2 z3) 1/2)) t_1)
         (sin (* z2 1/2))
         (cos (* -1/2 z3))
         (sin (* z3 1/2))
         (cos (* -1/2 z2)))))
      (sqrt
       (-
        1/2
        (-
         (*
          t_1
          (-
           (304-z0z1z2z3z4
            1/2
            (cos z3)
            (cos z2)
            (sin z3)
            (sin (- z2)))
           1/2))
         (* -1/2 t_0))))))))

Derivation

1. Split input into 3 regimes

2. if z3 < -2.4000000000000001e-5

   1. Initial program 60.5%

      \[\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

   3. Applied rewrites 57.8%

      \[\leadsto \sqrt{\color{blue}{\frac{1}{2} - \left(\cos \left(\left(z0 - z1\right) \cdot 1\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(z2 - z3\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)}} \]

   5. Applied rewrites 57.8%

      \[\leadsto \sqrt{\frac{1}{2} - \color{blue}{\left(1 - \frac{\left(\left(\frac{1}{2} - \cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2}\right) \cdot \cos z1\right) \cdot \cos z0}{\cos \left(z1 - z0\right) \cdot \frac{1}{2}}\right) \cdot \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2}\right)}} \]
   6. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(1 - \frac{\left(\left(\frac{1}{2} - \color{blue}{\cos \left(\left(z2 - z3\right) \cdot 1\right)} \cdot \frac{1}{2}\right) \cdot \cos z1\right) \cdot \cos z0}{\cos \left(z1 - z0\right) \cdot \frac{1}{2}}\right) \cdot \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(1 - \frac{\left(\left(\frac{1}{2} - \cos \color{blue}{\left(\left(z2 - z3\right) \cdot 1\right)} \cdot \frac{1}{2}\right) \cdot \cos z1\right) \cdot \cos z0}{\cos \left(z1 - z0\right) \cdot \frac{1}{2}}\right) \cdot \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2}\right)} \]

      3. *-rgt-identity N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(1 - \frac{\left(\left(\frac{1}{2} - \cos \color{blue}{\left(z2 - z3\right)} \cdot \frac{1}{2}\right) \cdot \cos z1\right) \cdot \cos z0}{\cos \left(z1 - z0\right) \cdot \frac{1}{2}}\right) \cdot \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2}\right)} \]

      4. lift--.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(1 - \frac{\left(\left(\frac{1}{2} - \cos \color{blue}{\left(z2 - z3\right)} \cdot \frac{1}{2}\right) \cdot \cos z1\right) \cdot \cos z0}{\cos \left(z1 - z0\right) \cdot \frac{1}{2}}\right) \cdot \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2}\right)} \]

      5. cos-diff N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(1 - \frac{\left(\left(\frac{1}{2} - \color{blue}{\left(\cos z2 \cdot \cos z3 + \sin z2 \cdot \sin z3\right)} \cdot \frac{1}{2}\right) \cdot \cos z1\right) \cdot \cos z0}{\cos \left(z1 - z0\right) \cdot \frac{1}{2}}\right) \cdot \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2}\right)} \]

      6. lower-+.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(1 - \frac{\left(\left(\frac{1}{2} - \color{blue}{\left(\cos z2 \cdot \cos z3 + \sin z2 \cdot \sin z3\right)} \cdot \frac{1}{2}\right) \cdot \cos z1\right) \cdot \cos z0}{\cos \left(z1 - z0\right) \cdot \frac{1}{2}}\right) \cdot \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2}\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(1 - \frac{\left(\left(\frac{1}{2} - \left(\color{blue}{\cos z2 \cdot \cos z3} + \sin z2 \cdot \sin z3\right) \cdot \frac{1}{2}\right) \cdot \cos z1\right) \cdot \cos z0}{\cos \left(z1 - z0\right) \cdot \frac{1}{2}}\right) \cdot \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2}\right)} \]

      8. lower-cos.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(1 - \frac{\left(\left(\frac{1}{2} - \left(\color{blue}{\cos z2} \cdot \cos z3 + \sin z2 \cdot \sin z3\right) \cdot \frac{1}{2}\right) \cdot \cos z1\right) \cdot \cos z0}{\cos \left(z1 - z0\right) \cdot \frac{1}{2}}\right) \cdot \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2}\right)} \]

      9. lower-cos.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(1 - \frac{\left(\left(\frac{1}{2} - \left(\cos z2 \cdot \color{blue}{\cos z3} + \sin z2 \cdot \sin z3\right) \cdot \frac{1}{2}\right) \cdot \cos z1\right) \cdot \cos z0}{\cos \left(z1 - z0\right) \cdot \frac{1}{2}}\right) \cdot \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2}\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \sqrt{\frac{1}{2} - \left(1 - \frac{\left(\left(\frac{1}{2} - \left(\cos z2 \cdot \cos z3 + \color{blue}{\sin z2 \cdot \sin z3}\right) \cdot \frac{1}{2}\right) \cdot \cos z1\right) \cdot \cos z0}{\cos \left(z1 - z0\right) \cdot \frac{1}{2}}\right) \cdot \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2}\right)} \]

      11. lower-sin.f64 N/A

          \[\leadsto \sqrt{\frac{1}{2} - \left(1 - \frac{\left(\left(\frac{1}{2} - \left(\cos z2 \cdot \cos z3 + \color{blue}{\sin z2} \cdot \sin z3\right) \cdot \frac{1}{2}\right) \cdot \cos z1\right) \cdot \cos z0}{\cos \left(z1 - z0\right) \cdot \frac{1}{2}}\right) \cdot \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2}\right)} \]

      12. lower-sin.f64 73.1%

          \[\leadsto \sqrt{\frac{1}{2} - \left(1 - \frac{\left(\left(\frac{1}{2} - \left(\cos z2 \cdot \cos z3 + \sin z2 \cdot \color{blue}{\sin z3}\right) \cdot \frac{1}{2}\right) \cdot \cos z1\right) \cdot \cos z0}{\cos \left(z1 - z0\right) \cdot \frac{1}{2}}\right) \cdot \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2}\right)} \]

   7. Applied rewrites 73.1%

      \[\leadsto \sqrt{\frac{1}{2} - \left(1 - \frac{\left(\left(\frac{1}{2} - \color{blue}{\left(\cos z2 \cdot \cos z3 + \sin z2 \cdot \sin z3\right)} \cdot \frac{1}{2}\right) \cdot \cos z1\right) \cdot \cos z0}{\cos \left(z1 - z0\right) \cdot \frac{1}{2}}\right) \cdot \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2}\right)} \]

   if -2.4000000000000001e-5 < z3 < 1.25e7

   1. Initial program 60.5%

      \[\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot \frac{z0 - z1}{2}\right)}\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      2. cos-neg-rev N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(\mathsf{neg}\left(2 \cdot \frac{z0 - z1}{2}\right)\right)}\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      3. sin-+PI/2-rev N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      4. lower-sin.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      5. lower-+.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \sin \color{blue}{\left(\left(\mathsf{neg}\left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      6. lower-neg.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \sin \left(\color{blue}{\left(-2 \cdot \frac{z0 - z1}{2}\right)} + \frac{\mathsf{PI}\left(\right)}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \sin \left(\left(-\color{blue}{2 \cdot \frac{z0 - z1}{2}}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      8. *-commutative N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \sin \left(\left(-\color{blue}{\frac{z0 - z1}{2} \cdot 2}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      9. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \sin \left(\left(-\color{blue}{\frac{z0 - z1}{2}} \cdot 2\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      10. mult-flip N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \sin \left(\left(-\color{blue}{\left(\left(z0 - z1\right) \cdot \frac{1}{2}\right)} \cdot 2\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      11. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \sin \left(\left(-\left(\left(z0 - z1\right) \cdot \color{blue}{\frac{1}{2}}\right) \cdot 2\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      12. associate-*l* N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \sin \left(\left(-\color{blue}{\left(z0 - z1\right) \cdot \left(\frac{1}{2} \cdot 2\right)}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      13. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \sin \left(\left(-\left(z0 - z1\right) \cdot \color{blue}{1}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      14. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \sin \left(\left(-\left(z0 - z1\right) \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{2}\right)}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      15. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \sin \left(\left(-\left(z0 - z1\right) \cdot \color{blue}{1}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      16. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \sin \left(\left(-\color{blue}{\left(z0 - z1\right) \cdot 1}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      17. mult-flip N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \sin \left(\left(-\left(z0 - z1\right) \cdot 1\right) + \color{blue}{\mathsf{PI}\left(\right) \cdot \frac{1}{2}}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      18. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \sin \left(\left(-\left(z0 - z1\right) \cdot 1\right) + \mathsf{PI}\left(\right) \cdot \color{blue}{\frac{1}{2}}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      19. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \sin \left(\left(-\left(z0 - z1\right) \cdot 1\right) + \color{blue}{\mathsf{PI}\left(\right) \cdot \frac{1}{2}}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      20. lower-PI.f64 31.0%

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \sin \left(\left(-\left(z0 - z1\right) \cdot 1\right) + \color{blue}{\pi} \cdot \frac{1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

   3. Applied rewrites 31.0%

      \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\sin \left(\left(-\left(z0 - z1\right) \cdot 1\right) + \pi \cdot \frac{1}{2}\right)}\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \sin \left(\left(-\left(z0 - z1\right) \cdot 1\right) + \pi \cdot \frac{1}{2}\right)\right) + \color{blue}{\left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)}} \]

      2. lift-sin.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \sin \left(\left(-\left(z0 - z1\right) \cdot 1\right) + \pi \cdot \frac{1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \color{blue}{\sin \left(\frac{z2 - z3}{2}\right)}} \]

      3. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \sin \left(\left(-\left(z0 - z1\right) \cdot 1\right) + \pi \cdot \frac{1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \color{blue}{\left(\frac{z2 - z3}{2}\right)}} \]

      4. mult-flip N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \sin \left(\left(-\left(z0 - z1\right) \cdot 1\right) + \pi \cdot \frac{1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \color{blue}{\left(\left(z2 - z3\right) \cdot \frac{1}{2}\right)}} \]

      5. metadata-eval N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \sin \left(\left(-\left(z0 - z1\right) \cdot 1\right) + \pi \cdot \frac{1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\left(z2 - z3\right) \cdot \color{blue}{\frac{1}{2}}\right)} \]

      6. *-commutative N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \sin \left(\left(-\left(z0 - z1\right) \cdot 1\right) + \pi \cdot \frac{1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \color{blue}{\left(\frac{1}{2} \cdot \left(z2 - z3\right)\right)}} \]

      7. lift--.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \sin \left(\left(-\left(z0 - z1\right) \cdot 1\right) + \pi \cdot \frac{1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \color{blue}{\left(z2 - z3\right)}\right)} \]

      8. distribute-rgt-out-- N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \sin \left(\left(-\left(z0 - z1\right) \cdot 1\right) + \pi \cdot \frac{1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \color{blue}{\left(z2 \cdot \frac{1}{2} - z3 \cdot \frac{1}{2}\right)}} \]

      9. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \sin \left(\left(-\left(z0 - z1\right) \cdot 1\right) + \pi \cdot \frac{1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\color{blue}{z2 \cdot \frac{1}{2}} - z3 \cdot \frac{1}{2}\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \sin \left(\left(-\left(z0 - z1\right) \cdot 1\right) + \pi \cdot \frac{1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(z2 \cdot \frac{1}{2} - \color{blue}{z3 \cdot \frac{1}{2}}\right)} \]

      11. sin-diff-rev N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \sin \left(\left(-\left(z0 - z1\right) \cdot 1\right) + \pi \cdot \frac{1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \color{blue}{\left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)}} \]

      12. lift-sin.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \sin \left(\left(-\left(z0 - z1\right) \cdot 1\right) + \pi \cdot \frac{1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \left(\color{blue}{\sin \left(z2 \cdot \frac{1}{2}\right)} \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)} \]

      13. lift-cos.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \sin \left(\left(-\left(z0 - z1\right) \cdot 1\right) + \pi \cdot \frac{1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(z3 \cdot \frac{1}{2}\right)} - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)} \]

      14. lift-cos.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \sin \left(\left(-\left(z0 - z1\right) \cdot 1\right) + \pi \cdot \frac{1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(z2 \cdot \frac{1}{2}\right)} \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)} \]

      15. lift-sin.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \sin \left(\left(-\left(z0 - z1\right) \cdot 1\right) + \pi \cdot \frac{1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(z3 \cdot \frac{1}{2}\right)}\right)} \]

      16. *-commutative N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \sin \left(\left(-\left(z0 - z1\right) \cdot 1\right) + \pi \cdot \frac{1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \color{blue}{\sin \left(z3 \cdot \frac{1}{2}\right) \cdot \cos \left(z2 \cdot \frac{1}{2}\right)}\right)} \]

   5. Applied rewrites 30.5%

      \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \sin \left(\left(-\left(z0 - z1\right) \cdot 1\right) + \pi \cdot \frac{1}{2}\right)\right) + \color{blue}{\mathsf{304\_z0z1z2z3z4}\left(\left(\sin \left(\left(z2 - z3\right) \cdot \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right), \sin \left(z2 \cdot \frac{1}{2}\right), \cos \left(\frac{-1}{2} \cdot z3\right), \sin \left(z3 \cdot \frac{1}{2}\right), \cos \left(\frac{-1}{2} \cdot z2\right)\right)}} \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\sin \left(\left(-\left(z0 - z1\right) \cdot 1\right) + \pi \cdot \frac{1}{2}\right)}\right) + \mathsf{304\_z0z1z2z3z4}\left(\left(\sin \left(\left(z2 - z3\right) \cdot \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right), \sin \left(z2 \cdot \frac{1}{2}\right), \cos \left(\frac{-1}{2} \cdot z3\right), \sin \left(z3 \cdot \frac{1}{2}\right), \cos \left(\frac{-1}{2} \cdot z2\right)\right)} \]

      2. lift-+.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \sin \color{blue}{\left(\left(-\left(z0 - z1\right) \cdot 1\right) + \pi \cdot \frac{1}{2}\right)}\right) + \mathsf{304\_z0z1z2z3z4}\left(\left(\sin \left(\left(z2 - z3\right) \cdot \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right), \sin \left(z2 \cdot \frac{1}{2}\right), \cos \left(\frac{-1}{2} \cdot z3\right), \sin \left(z3 \cdot \frac{1}{2}\right), \cos \left(\frac{-1}{2} \cdot z2\right)\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \sin \left(\left(-\left(z0 - z1\right) \cdot 1\right) + \color{blue}{\pi \cdot \frac{1}{2}}\right)\right) + \mathsf{304\_z0z1z2z3z4}\left(\left(\sin \left(\left(z2 - z3\right) \cdot \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right), \sin \left(z2 \cdot \frac{1}{2}\right), \cos \left(\frac{-1}{2} \cdot z3\right), \sin \left(z3 \cdot \frac{1}{2}\right), \cos \left(\frac{-1}{2} \cdot z2\right)\right)} \]

      4. lift-neg.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \sin \left(\color{blue}{\left(\mathsf{neg}\left(\left(z0 - z1\right) \cdot 1\right)\right)} + \pi \cdot \frac{1}{2}\right)\right) + \mathsf{304\_z0z1z2z3z4}\left(\left(\sin \left(\left(z2 - z3\right) \cdot \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right), \sin \left(z2 \cdot \frac{1}{2}\right), \cos \left(\frac{-1}{2} \cdot z3\right), \sin \left(z3 \cdot \frac{1}{2}\right), \cos \left(\frac{-1}{2} \cdot z2\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \sin \left(\left(\mathsf{neg}\left(\color{blue}{\left(z0 - z1\right) \cdot 1}\right)\right) + \pi \cdot \frac{1}{2}\right)\right) + \mathsf{304\_z0z1z2z3z4}\left(\left(\sin \left(\left(z2 - z3\right) \cdot \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right), \sin \left(z2 \cdot \frac{1}{2}\right), \cos \left(\frac{-1}{2} \cdot z3\right), \sin \left(z3 \cdot \frac{1}{2}\right), \cos \left(\frac{-1}{2} \cdot z2\right)\right)} \]

      6. *-rgt-identity N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \sin \left(\left(\mathsf{neg}\left(\color{blue}{\left(z0 - z1\right)}\right)\right) + \pi \cdot \frac{1}{2}\right)\right) + \mathsf{304\_z0z1z2z3z4}\left(\left(\sin \left(\left(z2 - z3\right) \cdot \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right), \sin \left(z2 \cdot \frac{1}{2}\right), \cos \left(\frac{-1}{2} \cdot z3\right), \sin \left(z3 \cdot \frac{1}{2}\right), \cos \left(\frac{-1}{2} \cdot z2\right)\right)} \]

      7. lift--.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \sin \left(\left(\mathsf{neg}\left(\color{blue}{\left(z0 - z1\right)}\right)\right) + \pi \cdot \frac{1}{2}\right)\right) + \mathsf{304\_z0z1z2z3z4}\left(\left(\sin \left(\left(z2 - z3\right) \cdot \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right), \sin \left(z2 \cdot \frac{1}{2}\right), \cos \left(\frac{-1}{2} \cdot z3\right), \sin \left(z3 \cdot \frac{1}{2}\right), \cos \left(\frac{-1}{2} \cdot z2\right)\right)} \]

      8. sub-negate-rev N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \sin \left(\color{blue}{\left(z1 - z0\right)} + \pi \cdot \frac{1}{2}\right)\right) + \mathsf{304\_z0z1z2z3z4}\left(\left(\sin \left(\left(z2 - z3\right) \cdot \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right), \sin \left(z2 \cdot \frac{1}{2}\right), \cos \left(\frac{-1}{2} \cdot z3\right), \sin \left(z3 \cdot \frac{1}{2}\right), \cos \left(\frac{-1}{2} \cdot z2\right)\right)} \]

      9. lift--.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \sin \left(\color{blue}{\left(z1 - z0\right)} + \pi \cdot \frac{1}{2}\right)\right) + \mathsf{304\_z0z1z2z3z4}\left(\left(\sin \left(\left(z2 - z3\right) \cdot \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right), \sin \left(z2 \cdot \frac{1}{2}\right), \cos \left(\frac{-1}{2} \cdot z3\right), \sin \left(z3 \cdot \frac{1}{2}\right), \cos \left(\frac{-1}{2} \cdot z2\right)\right)} \]

      10. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \sin \left(\left(z1 - z0\right) + \pi \cdot \color{blue}{\frac{1}{2}}\right)\right) + \mathsf{304\_z0z1z2z3z4}\left(\left(\sin \left(\left(z2 - z3\right) \cdot \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right), \sin \left(z2 \cdot \frac{1}{2}\right), \cos \left(\frac{-1}{2} \cdot z3\right), \sin \left(z3 \cdot \frac{1}{2}\right), \cos \left(\frac{-1}{2} \cdot z2\right)\right)} \]

      11. mult-flip-rev N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \sin \left(\left(z1 - z0\right) + \color{blue}{\frac{\pi}{2}}\right)\right) + \mathsf{304\_z0z1z2z3z4}\left(\left(\sin \left(\left(z2 - z3\right) \cdot \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right), \sin \left(z2 \cdot \frac{1}{2}\right), \cos \left(\frac{-1}{2} \cdot z3\right), \sin \left(z3 \cdot \frac{1}{2}\right), \cos \left(\frac{-1}{2} \cdot z2\right)\right)} \]

      12. lift-PI.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \sin \left(\left(z1 - z0\right) + \frac{\color{blue}{\mathsf{PI}\left(\right)}}{2}\right)\right) + \mathsf{304\_z0z1z2z3z4}\left(\left(\sin \left(\left(z2 - z3\right) \cdot \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right), \sin \left(z2 \cdot \frac{1}{2}\right), \cos \left(\frac{-1}{2} \cdot z3\right), \sin \left(z3 \cdot \frac{1}{2}\right), \cos \left(\frac{-1}{2} \cdot z2\right)\right)} \]

      13. sin-+PI/2-rev N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(z1 - z0\right)}\right) + \mathsf{304\_z0z1z2z3z4}\left(\left(\sin \left(\left(z2 - z3\right) \cdot \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right), \sin \left(z2 \cdot \frac{1}{2}\right), \cos \left(\frac{-1}{2} \cdot z3\right), \sin \left(z3 \cdot \frac{1}{2}\right), \cos \left(\frac{-1}{2} \cdot z2\right)\right)} \]

      14. lift--.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \color{blue}{\left(z1 - z0\right)}\right) + \mathsf{304\_z0z1z2z3z4}\left(\left(\sin \left(\left(z2 - z3\right) \cdot \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right), \sin \left(z2 \cdot \frac{1}{2}\right), \cos \left(\frac{-1}{2} \cdot z3\right), \sin \left(z3 \cdot \frac{1}{2}\right), \cos \left(\frac{-1}{2} \cdot z2\right)\right)} \]

      15. cos-diff N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\left(\cos z1 \cdot \cos z0 + \sin z1 \cdot \sin z0\right)}\right) + \mathsf{304\_z0z1z2z3z4}\left(\left(\sin \left(\left(z2 - z3\right) \cdot \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right), \sin \left(z2 \cdot \frac{1}{2}\right), \cos \left(\frac{-1}{2} \cdot z3\right), \sin \left(z3 \cdot \frac{1}{2}\right), \cos \left(\frac{-1}{2} \cdot z2\right)\right)} \]

      16. lift-cos.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \left(\color{blue}{\cos z1} \cdot \cos z0 + \sin z1 \cdot \sin z0\right)\right) + \mathsf{304\_z0z1z2z3z4}\left(\left(\sin \left(\left(z2 - z3\right) \cdot \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right), \sin \left(z2 \cdot \frac{1}{2}\right), \cos \left(\frac{-1}{2} \cdot z3\right), \sin \left(z3 \cdot \frac{1}{2}\right), \cos \left(\frac{-1}{2} \cdot z2\right)\right)} \]

      17. lift-cos.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \left(\cos z1 \cdot \color{blue}{\cos z0} + \sin z1 \cdot \sin z0\right)\right) + \mathsf{304\_z0z1z2z3z4}\left(\left(\sin \left(\left(z2 - z3\right) \cdot \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right), \sin \left(z2 \cdot \frac{1}{2}\right), \cos \left(\frac{-1}{2} \cdot z3\right), \sin \left(z3 \cdot \frac{1}{2}\right), \cos \left(\frac{-1}{2} \cdot z2\right)\right)} \]

      18. lift-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \left(\color{blue}{\cos z1 \cdot \cos z0} + \sin z1 \cdot \sin z0\right)\right) + \mathsf{304\_z0z1z2z3z4}\left(\left(\sin \left(\left(z2 - z3\right) \cdot \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right), \sin \left(z2 \cdot \frac{1}{2}\right), \cos \left(\frac{-1}{2} \cdot z3\right), \sin \left(z3 \cdot \frac{1}{2}\right), \cos \left(\frac{-1}{2} \cdot z2\right)\right)} \]

      19. lower-+.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\left(\cos z1 \cdot \cos z0 + \sin z1 \cdot \sin z0\right)}\right) + \mathsf{304\_z0z1z2z3z4}\left(\left(\sin \left(\left(z2 - z3\right) \cdot \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right), \sin \left(z2 \cdot \frac{1}{2}\right), \cos \left(\frac{-1}{2} \cdot z3\right), \sin \left(z3 \cdot \frac{1}{2}\right), \cos \left(\frac{-1}{2} \cdot z2\right)\right)} \]

      20. lift-sin.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \left(\cos z1 \cdot \cos z0 + \color{blue}{\sin z1} \cdot \sin z0\right)\right) + \mathsf{304\_z0z1z2z3z4}\left(\left(\sin \left(\left(z2 - z3\right) \cdot \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right), \sin \left(z2 \cdot \frac{1}{2}\right), \cos \left(\frac{-1}{2} \cdot z3\right), \sin \left(z3 \cdot \frac{1}{2}\right), \cos \left(\frac{-1}{2} \cdot z2\right)\right)} \]

      21. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \left(\cos z1 \cdot \cos z0 + \color{blue}{\sin z1 \cdot \sin z0}\right)\right) + \mathsf{304\_z0z1z2z3z4}\left(\left(\sin \left(\left(z2 - z3\right) \cdot \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right), \sin \left(z2 \cdot \frac{1}{2}\right), \cos \left(\frac{-1}{2} \cdot z3\right), \sin \left(z3 \cdot \frac{1}{2}\right), \cos \left(\frac{-1}{2} \cdot z2\right)\right)} \]

      22. lower-sin.f64 75.2%

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \left(\cos z1 \cdot \cos z0 + \sin z1 \cdot \color{blue}{\sin z0}\right)\right) + \mathsf{304\_z0z1z2z3z4}\left(\left(\sin \left(\left(z2 - z3\right) \cdot \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right), \sin \left(z2 \cdot \frac{1}{2}\right), \cos \left(\frac{-1}{2} \cdot z3\right), \sin \left(z3 \cdot \frac{1}{2}\right), \cos \left(\frac{-1}{2} \cdot z2\right)\right)} \]

   7. Applied rewrites 75.2%

      \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\left(\cos z1 \cdot \cos z0 + \sin z1 \cdot \sin z0\right)}\right) + \mathsf{304\_z0z1z2z3z4}\left(\left(\sin \left(\left(z2 - z3\right) \cdot \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right), \sin \left(z2 \cdot \frac{1}{2}\right), \cos \left(\frac{-1}{2} \cdot z3\right), \sin \left(z3 \cdot \frac{1}{2}\right), \cos \left(\frac{-1}{2} \cdot z2\right)\right)} \]

   if 1.25e7 < z3

   1. Initial program 60.5%

      \[\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

   3. Applied rewrites 57.8%

      \[\leadsto \sqrt{\color{blue}{\frac{1}{2} - \left(\cos \left(\left(z0 - z1\right) \cdot 1\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(z2 - z3\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)}} \]

   5. Applied rewrites 57.8%

      \[\leadsto \sqrt{\frac{1}{2} - \color{blue}{\left(1 - \frac{\left(\left(\frac{1}{2} - \cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2}\right) \cdot \cos z1\right) \cdot \cos z0}{\cos \left(z1 - z0\right) \cdot \frac{1}{2}}\right) \cdot \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2}\right)}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \color{blue}{\left(1 - \frac{\left(\left(\frac{1}{2} - \cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2}\right) \cdot \cos z1\right) \cdot \cos z0}{\cos \left(z1 - z0\right) \cdot \frac{1}{2}}\right) \cdot \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2}\right)}} \]

      2. lift--.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \color{blue}{\left(1 - \frac{\left(\left(\frac{1}{2} - \cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2}\right) \cdot \cos z1\right) \cdot \cos z0}{\cos \left(z1 - z0\right) \cdot \frac{1}{2}}\right)} \cdot \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2}\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(1 - \color{blue}{\frac{\left(\left(\frac{1}{2} - \cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2}\right) \cdot \cos z1\right) \cdot \cos z0}{\cos \left(z1 - z0\right) \cdot \frac{1}{2}}}\right) \cdot \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2}\right)} \]

      4. sub-to-mult-rev N/A

         \[\leadsto \sqrt{\frac{1}{2} - \color{blue}{\left(\cos \left(z1 - z0\right) \cdot \frac{1}{2} - \left(\left(\frac{1}{2} - \cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2}\right) \cdot \cos z1\right) \cdot \cos z0\right)}} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2} - \color{blue}{\left(\left(\frac{1}{2} - \cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2}\right) \cdot \cos z1\right) \cdot \cos z0}\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2} - \color{blue}{\left(\left(\frac{1}{2} - \cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2}\right) \cdot \cos z1\right)} \cdot \cos z0\right)} \]

      7. associate-*l* N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2} - \color{blue}{\left(\frac{1}{2} - \cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)}\right)} \]

   7. Applied rewrites 57.8%

      \[\leadsto \sqrt{\frac{1}{2} - \color{blue}{\left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\cos \left(z3 - z2\right) \cdot \frac{1}{2} - \frac{1}{2}\right) - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\color{blue}{\cos \left(z3 - z2\right) \cdot \frac{1}{2}} - \frac{1}{2}\right) - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\color{blue}{\frac{1}{2} \cdot \cos \left(z3 - z2\right)} - \frac{1}{2}\right) - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)} \]

      3. lift-cos.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\frac{1}{2} \cdot \color{blue}{\cos \left(z3 - z2\right)} - \frac{1}{2}\right) - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)} \]

      4. lift--.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\frac{1}{2} \cdot \cos \color{blue}{\left(z3 - z2\right)} - \frac{1}{2}\right) - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)} \]

      5. sub-flip N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\frac{1}{2} \cdot \cos \color{blue}{\left(z3 + \left(\mathsf{neg}\left(z2\right)\right)\right)} - \frac{1}{2}\right) - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)} \]

      6. cos-sum N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(\cos z3 \cdot \cos \left(\mathsf{neg}\left(z2\right)\right) - \sin z3 \cdot \sin \left(\mathsf{neg}\left(z2\right)\right)\right)} - \frac{1}{2}\right) - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)} \]

      7. cos-neg-rev N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\frac{1}{2} \cdot \left(\cos z3 \cdot \color{blue}{\cos z2} - \sin z3 \cdot \sin \left(\mathsf{neg}\left(z2\right)\right)\right) - \frac{1}{2}\right) - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)} \]

      8. lower-304-z0z1z2z3z4 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\color{blue}{\mathsf{304\_z0z1z2z3z4}\left(\frac{1}{2}, \cos z3, \cos z2, \sin z3, \sin \left(\mathsf{neg}\left(z2\right)\right)\right)} - \frac{1}{2}\right) - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)} \]

      9. lower-cos.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\mathsf{304\_z0z1z2z3z4}\left(\frac{1}{2}, \color{blue}{\cos z3}, \cos z2, \sin z3, \sin \left(\mathsf{neg}\left(z2\right)\right)\right) - \frac{1}{2}\right) - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)} \]

      10. lower-cos.f64 N/A

          \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\mathsf{304\_z0z1z2z3z4}\left(\frac{1}{2}, \cos z3, \color{blue}{\cos z2}, \sin z3, \sin \left(\mathsf{neg}\left(z2\right)\right)\right) - \frac{1}{2}\right) - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)} \]

      11. lower-sin.f64 N/A

          \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\mathsf{304\_z0z1z2z3z4}\left(\frac{1}{2}, \cos z3, \cos z2, \color{blue}{\sin z3}, \sin \left(\mathsf{neg}\left(z2\right)\right)\right) - \frac{1}{2}\right) - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)} \]

      12. lower-sin.f64 N/A

          \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\mathsf{304\_z0z1z2z3z4}\left(\frac{1}{2}, \cos z3, \cos z2, \sin z3, \color{blue}{\sin \left(\mathsf{neg}\left(z2\right)\right)}\right) - \frac{1}{2}\right) - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)} \]

      13. lower-neg.f64 73.1%

          \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\mathsf{304\_z0z1z2z3z4}\left(\frac{1}{2}, \cos z3, \cos z2, \sin z3, \sin \color{blue}{\left(-z2\right)}\right) - \frac{1}{2}\right) - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)} \]

   9. Applied rewrites 73.1%

      \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\color{blue}{\mathsf{304\_z0z1z2z3z4}\left(\frac{1}{2}, \cos z3, \cos z2, \sin z3, \sin \left(-z2\right)\right)} - \frac{1}{2}\right) - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 85.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(z1 - z0\right)\\ t_1 := t\_0 \cdot \frac{1}{2}\\ t_2 := \sin \left(\left(z2 - z3\right) \cdot \frac{1}{2}\right)\\ \mathbf{if}\;z3 \leq \frac{-1770887431076117}{73786976294838206464}:\\ \;\;\;\;\sqrt{\frac{1}{2} - \left(1 - \frac{\left(\left(\frac{1}{2} - \left(\cos z2 \cdot \cos z3 + \sin z2 \cdot \sin z3\right) \cdot \frac{1}{2}\right) \cdot \cos z1\right) \cdot \cos z0}{t\_1}\right) \cdot t\_1}\\ \mathbf{elif}\;z3 \leq 46000000000:\\ \;\;\;\;\sqrt{\left(\frac{1}{2} - \mathsf{304\_z0z1z2z3z4}\left(\frac{1}{2}, \cos z1, \cos z0, \sin z1, \sin \left(-z0\right)\right)\right) + \left(\left(t\_2 \cdot \cos z0\right) \cdot \cos z1\right) \cdot t\_2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\mathsf{304\_z0z1z2z3z4}\left(\frac{1}{2}, \cos z3, \cos z2, \sin z3, \sin \left(-z2\right)\right) - \frac{1}{2}\right) - \frac{-1}{2} \cdot t\_0\right)}\\ \end{array} \]

FPCore

(FPCore (z0 z1 z2 z3)
  :precision binary64
  (let* ((t_0 (cos (- z1 z0)))
       (t_1 (* t_0 1/2))
       (t_2 (sin (* (- z2 z3) 1/2))))
  (if (<= z3 -1770887431076117/73786976294838206464)
    (sqrt
     (-
      1/2
      (*
       (-
        1
        (/
         (*
          (*
           (-
            1/2
            (* (+ (* (cos z2) (cos z3)) (* (sin z2) (sin z3))) 1/2))
           (cos z1))
          (cos z0))
         t_1))
       t_1)))
    (if (<= z3 46000000000)
      (sqrt
       (+
        (-
         1/2
         (304-z0z1z2z3z4 1/2 (cos z1) (cos z0) (sin z1) (sin (- z0))))
        (* (* (* t_2 (cos z0)) (cos z1)) t_2)))
      (sqrt
       (-
        1/2
        (-
         (*
          (* (cos z1) (cos z0))
          (-
           (304-z0z1z2z3z4
            1/2
            (cos z3)
            (cos z2)
            (sin z3)
            (sin (- z2)))
           1/2))
         (* -1/2 t_0))))))))

Derivation

1. Split input into 3 regimes

2. if z3 < -2.4000000000000001e-5

   1. Initial program 60.5%

      \[\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

   3. Applied rewrites 57.8%

      \[\leadsto \sqrt{\color{blue}{\frac{1}{2} - \left(\cos \left(\left(z0 - z1\right) \cdot 1\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(z2 - z3\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)}} \]

   5. Applied rewrites 57.8%

      \[\leadsto \sqrt{\frac{1}{2} - \color{blue}{\left(1 - \frac{\left(\left(\frac{1}{2} - \cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2}\right) \cdot \cos z1\right) \cdot \cos z0}{\cos \left(z1 - z0\right) \cdot \frac{1}{2}}\right) \cdot \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2}\right)}} \]
   6. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(1 - \frac{\left(\left(\frac{1}{2} - \color{blue}{\cos \left(\left(z2 - z3\right) \cdot 1\right)} \cdot \frac{1}{2}\right) \cdot \cos z1\right) \cdot \cos z0}{\cos \left(z1 - z0\right) \cdot \frac{1}{2}}\right) \cdot \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(1 - \frac{\left(\left(\frac{1}{2} - \cos \color{blue}{\left(\left(z2 - z3\right) \cdot 1\right)} \cdot \frac{1}{2}\right) \cdot \cos z1\right) \cdot \cos z0}{\cos \left(z1 - z0\right) \cdot \frac{1}{2}}\right) \cdot \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2}\right)} \]

      3. *-rgt-identity N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(1 - \frac{\left(\left(\frac{1}{2} - \cos \color{blue}{\left(z2 - z3\right)} \cdot \frac{1}{2}\right) \cdot \cos z1\right) \cdot \cos z0}{\cos \left(z1 - z0\right) \cdot \frac{1}{2}}\right) \cdot \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2}\right)} \]

      4. lift--.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(1 - \frac{\left(\left(\frac{1}{2} - \cos \color{blue}{\left(z2 - z3\right)} \cdot \frac{1}{2}\right) \cdot \cos z1\right) \cdot \cos z0}{\cos \left(z1 - z0\right) \cdot \frac{1}{2}}\right) \cdot \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2}\right)} \]

      5. cos-diff N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(1 - \frac{\left(\left(\frac{1}{2} - \color{blue}{\left(\cos z2 \cdot \cos z3 + \sin z2 \cdot \sin z3\right)} \cdot \frac{1}{2}\right) \cdot \cos z1\right) \cdot \cos z0}{\cos \left(z1 - z0\right) \cdot \frac{1}{2}}\right) \cdot \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2}\right)} \]

      6. lower-+.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(1 - \frac{\left(\left(\frac{1}{2} - \color{blue}{\left(\cos z2 \cdot \cos z3 + \sin z2 \cdot \sin z3\right)} \cdot \frac{1}{2}\right) \cdot \cos z1\right) \cdot \cos z0}{\cos \left(z1 - z0\right) \cdot \frac{1}{2}}\right) \cdot \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2}\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(1 - \frac{\left(\left(\frac{1}{2} - \left(\color{blue}{\cos z2 \cdot \cos z3} + \sin z2 \cdot \sin z3\right) \cdot \frac{1}{2}\right) \cdot \cos z1\right) \cdot \cos z0}{\cos \left(z1 - z0\right) \cdot \frac{1}{2}}\right) \cdot \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2}\right)} \]

      8. lower-cos.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(1 - \frac{\left(\left(\frac{1}{2} - \left(\color{blue}{\cos z2} \cdot \cos z3 + \sin z2 \cdot \sin z3\right) \cdot \frac{1}{2}\right) \cdot \cos z1\right) \cdot \cos z0}{\cos \left(z1 - z0\right) \cdot \frac{1}{2}}\right) \cdot \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2}\right)} \]

      9. lower-cos.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(1 - \frac{\left(\left(\frac{1}{2} - \left(\cos z2 \cdot \color{blue}{\cos z3} + \sin z2 \cdot \sin z3\right) \cdot \frac{1}{2}\right) \cdot \cos z1\right) \cdot \cos z0}{\cos \left(z1 - z0\right) \cdot \frac{1}{2}}\right) \cdot \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2}\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \sqrt{\frac{1}{2} - \left(1 - \frac{\left(\left(\frac{1}{2} - \left(\cos z2 \cdot \cos z3 + \color{blue}{\sin z2 \cdot \sin z3}\right) \cdot \frac{1}{2}\right) \cdot \cos z1\right) \cdot \cos z0}{\cos \left(z1 - z0\right) \cdot \frac{1}{2}}\right) \cdot \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2}\right)} \]

      11. lower-sin.f64 N/A

          \[\leadsto \sqrt{\frac{1}{2} - \left(1 - \frac{\left(\left(\frac{1}{2} - \left(\cos z2 \cdot \cos z3 + \color{blue}{\sin z2} \cdot \sin z3\right) \cdot \frac{1}{2}\right) \cdot \cos z1\right) \cdot \cos z0}{\cos \left(z1 - z0\right) \cdot \frac{1}{2}}\right) \cdot \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2}\right)} \]

      12. lower-sin.f64 73.1%

          \[\leadsto \sqrt{\frac{1}{2} - \left(1 - \frac{\left(\left(\frac{1}{2} - \left(\cos z2 \cdot \cos z3 + \sin z2 \cdot \color{blue}{\sin z3}\right) \cdot \frac{1}{2}\right) \cdot \cos z1\right) \cdot \cos z0}{\cos \left(z1 - z0\right) \cdot \frac{1}{2}}\right) \cdot \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2}\right)} \]

   7. Applied rewrites 73.1%

      \[\leadsto \sqrt{\frac{1}{2} - \left(1 - \frac{\left(\left(\frac{1}{2} - \color{blue}{\left(\cos z2 \cdot \cos z3 + \sin z2 \cdot \sin z3\right)} \cdot \frac{1}{2}\right) \cdot \cos z1\right) \cdot \cos z0}{\cos \left(z1 - z0\right) \cdot \frac{1}{2}}\right) \cdot \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2}\right)} \]

   if -2.4000000000000001e-5 < z3 < 4.6e10

   1. Initial program 60.5%

      \[\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \color{blue}{\sin \left(\frac{z2 - z3}{2}\right)}\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \color{blue}{\left(\frac{z2 - z3}{2}\right)}\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      3. lift--.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \left(\frac{\color{blue}{z2 - z3}}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      4. div-sub N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \color{blue}{\left(\frac{z2}{2} - \frac{z3}{2}\right)}\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      5. sin-diff N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \color{blue}{\left(\sin \left(\frac{z2}{2}\right) \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)}\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      6. lower--.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \color{blue}{\left(\sin \left(\frac{z2}{2}\right) \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)}\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\color{blue}{\sin \left(\frac{z2}{2}\right) \cdot \cos \left(\frac{z3}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      8. lower-sin.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\color{blue}{\sin \left(\frac{z2}{2}\right)} \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      9. mult-flip N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      10. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      12. lower-cos.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{z3}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      13. mult-flip N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(z3 \cdot \frac{1}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      14. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      15. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(z3 \cdot \frac{1}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      16. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      17. lower-cos.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{z2}{2}\right)} \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      18. mult-flip N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      19. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      20. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      21. lower-sin.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{z3}{2}\right)}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      22. mult-flip N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(z3 \cdot \frac{1}{2}\right)}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      23. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      24. lower-*.f64 59.7%

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(z3 \cdot \frac{1}{2}\right)}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

   3. Applied rewrites 59.7%

      \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \color{blue}{\left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)}\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{z2 - z3}{2}\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{z2 - z3}{2}\right)}} \]

      3. lift--.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{z2 - z3}}{2}\right)} \]

      4. div-sub N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{z2}{2} - \frac{z3}{2}\right)}} \]

      5. sin-diff N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{z2}{2}\right) \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)}} \]

      6. lower--.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{z2}{2}\right) \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{z2}{2}\right) \cdot \cos \left(\frac{z3}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      8. lower-sin.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{z2}{2}\right)} \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      9. mult-flip N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      10. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      12. lower-cos.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{z3}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      13. mult-flip N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(z3 \cdot \frac{1}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      14. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      15. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(z3 \cdot \frac{1}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      16. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)}\right)} \]

      17. lower-cos.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{z2}{2}\right)} \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      18. mult-flip N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      19. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      20. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      21. lower-sin.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{z3}{2}\right)}\right)} \]

      22. mult-flip N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(z3 \cdot \frac{1}{2}\right)}\right)} \]

      23. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \color{blue}{\frac{1}{2}}\right)\right)} \]

      24. lower-*.f64 75.8%

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(z3 \cdot \frac{1}{2}\right)}\right)} \]

   5. Applied rewrites 75.8%

      \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)}} \]

   7. Applied rewrites 96.4%

      \[\leadsto \sqrt{\left(\frac{1}{2} - \color{blue}{\mathsf{304\_z0z1z2z3z4}\left(\frac{1}{2}, \cos z1, \cos z0, \sin z1, \sin \left(-z0\right)\right)}\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)} \]

   9. Applied rewrites 76.1%

      \[\leadsto \sqrt{\left(\frac{1}{2} - \mathsf{304\_z0z1z2z3z4}\left(\frac{1}{2}, \cos z1, \cos z0, \sin z1, \sin \left(-z0\right)\right)\right) + \color{blue}{\left(\left(\sin \left(\left(z2 - z3\right) \cdot \frac{1}{2}\right) \cdot \cos z0\right) \cdot \cos z1\right) \cdot \sin \left(\left(z2 - z3\right) \cdot \frac{1}{2}\right)}} \]

   if 4.6e10 < z3

   1. Initial program 60.5%

      \[\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

   3. Applied rewrites 57.8%

      \[\leadsto \sqrt{\color{blue}{\frac{1}{2} - \left(\cos \left(\left(z0 - z1\right) \cdot 1\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(z2 - z3\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)}} \]

   5. Applied rewrites 57.8%

      \[\leadsto \sqrt{\frac{1}{2} - \color{blue}{\left(1 - \frac{\left(\left(\frac{1}{2} - \cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2}\right) \cdot \cos z1\right) \cdot \cos z0}{\cos \left(z1 - z0\right) \cdot \frac{1}{2}}\right) \cdot \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2}\right)}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \color{blue}{\left(1 - \frac{\left(\left(\frac{1}{2} - \cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2}\right) \cdot \cos z1\right) \cdot \cos z0}{\cos \left(z1 - z0\right) \cdot \frac{1}{2}}\right) \cdot \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2}\right)}} \]

      2. lift--.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \color{blue}{\left(1 - \frac{\left(\left(\frac{1}{2} - \cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2}\right) \cdot \cos z1\right) \cdot \cos z0}{\cos \left(z1 - z0\right) \cdot \frac{1}{2}}\right)} \cdot \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2}\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(1 - \color{blue}{\frac{\left(\left(\frac{1}{2} - \cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2}\right) \cdot \cos z1\right) \cdot \cos z0}{\cos \left(z1 - z0\right) \cdot \frac{1}{2}}}\right) \cdot \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2}\right)} \]

      4. sub-to-mult-rev N/A

         \[\leadsto \sqrt{\frac{1}{2} - \color{blue}{\left(\cos \left(z1 - z0\right) \cdot \frac{1}{2} - \left(\left(\frac{1}{2} - \cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2}\right) \cdot \cos z1\right) \cdot \cos z0\right)}} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2} - \color{blue}{\left(\left(\frac{1}{2} - \cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2}\right) \cdot \cos z1\right) \cdot \cos z0}\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2} - \color{blue}{\left(\left(\frac{1}{2} - \cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2}\right) \cdot \cos z1\right)} \cdot \cos z0\right)} \]

      7. associate-*l* N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2} - \color{blue}{\left(\frac{1}{2} - \cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)}\right)} \]

   7. Applied rewrites 57.8%

      \[\leadsto \sqrt{\frac{1}{2} - \color{blue}{\left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\cos \left(z3 - z2\right) \cdot \frac{1}{2} - \frac{1}{2}\right) - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\color{blue}{\cos \left(z3 - z2\right) \cdot \frac{1}{2}} - \frac{1}{2}\right) - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\color{blue}{\frac{1}{2} \cdot \cos \left(z3 - z2\right)} - \frac{1}{2}\right) - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)} \]

      3. lift-cos.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\frac{1}{2} \cdot \color{blue}{\cos \left(z3 - z2\right)} - \frac{1}{2}\right) - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)} \]

      4. lift--.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\frac{1}{2} \cdot \cos \color{blue}{\left(z3 - z2\right)} - \frac{1}{2}\right) - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)} \]

      5. sub-flip N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\frac{1}{2} \cdot \cos \color{blue}{\left(z3 + \left(\mathsf{neg}\left(z2\right)\right)\right)} - \frac{1}{2}\right) - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)} \]

      6. cos-sum N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(\cos z3 \cdot \cos \left(\mathsf{neg}\left(z2\right)\right) - \sin z3 \cdot \sin \left(\mathsf{neg}\left(z2\right)\right)\right)} - \frac{1}{2}\right) - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)} \]

      7. cos-neg-rev N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\frac{1}{2} \cdot \left(\cos z3 \cdot \color{blue}{\cos z2} - \sin z3 \cdot \sin \left(\mathsf{neg}\left(z2\right)\right)\right) - \frac{1}{2}\right) - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)} \]

      8. lower-304-z0z1z2z3z4 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\color{blue}{\mathsf{304\_z0z1z2z3z4}\left(\frac{1}{2}, \cos z3, \cos z2, \sin z3, \sin \left(\mathsf{neg}\left(z2\right)\right)\right)} - \frac{1}{2}\right) - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)} \]

      9. lower-cos.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\mathsf{304\_z0z1z2z3z4}\left(\frac{1}{2}, \color{blue}{\cos z3}, \cos z2, \sin z3, \sin \left(\mathsf{neg}\left(z2\right)\right)\right) - \frac{1}{2}\right) - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)} \]

      10. lower-cos.f64 N/A

          \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\mathsf{304\_z0z1z2z3z4}\left(\frac{1}{2}, \cos z3, \color{blue}{\cos z2}, \sin z3, \sin \left(\mathsf{neg}\left(z2\right)\right)\right) - \frac{1}{2}\right) - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)} \]

      11. lower-sin.f64 N/A

          \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\mathsf{304\_z0z1z2z3z4}\left(\frac{1}{2}, \cos z3, \cos z2, \color{blue}{\sin z3}, \sin \left(\mathsf{neg}\left(z2\right)\right)\right) - \frac{1}{2}\right) - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)} \]

      12. lower-sin.f64 N/A

          \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\mathsf{304\_z0z1z2z3z4}\left(\frac{1}{2}, \cos z3, \cos z2, \sin z3, \color{blue}{\sin \left(\mathsf{neg}\left(z2\right)\right)}\right) - \frac{1}{2}\right) - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)} \]

      13. lower-neg.f64 73.1%

          \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\mathsf{304\_z0z1z2z3z4}\left(\frac{1}{2}, \cos z3, \cos z2, \sin z3, \sin \color{blue}{\left(-z2\right)}\right) - \frac{1}{2}\right) - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)} \]

   9. Applied rewrites 73.1%

      \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\color{blue}{\mathsf{304\_z0z1z2z3z4}\left(\frac{1}{2}, \cos z3, \cos z2, \sin z3, \sin \left(-z2\right)\right)} - \frac{1}{2}\right) - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 85.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_0 := \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\mathsf{304\_z0z1z2z3z4}\left(\frac{1}{2}, \cos z3, \cos z2, \sin z3, \sin \left(-z2\right)\right) - \frac{1}{2}\right) - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)}\\ t_1 := \sin \left(\left(z2 - z3\right) \cdot \frac{1}{2}\right)\\ \mathbf{if}\;z3 \leq \frac{-1770887431076117}{73786976294838206464}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z3 \leq 46000000000:\\ \;\;\;\;\sqrt{\left(\frac{1}{2} - \mathsf{304\_z0z1z2z3z4}\left(\frac{1}{2}, \cos z1, \cos z0, \sin z1, \sin \left(-z0\right)\right)\right) + \left(\left(t\_1 \cdot \cos z0\right) \cdot \cos z1\right) \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (z0 z1 z2 z3)
  :precision binary64
  (let* ((t_0
        (sqrt
         (-
          1/2
          (-
           (*
            (* (cos z1) (cos z0))
            (-
             (304-z0z1z2z3z4
              1/2
              (cos z3)
              (cos z2)
              (sin z3)
              (sin (- z2)))
             1/2))
           (* -1/2 (cos (- z1 z0)))))))
       (t_1 (sin (* (- z2 z3) 1/2))))
  (if (<= z3 -1770887431076117/73786976294838206464)
    t_0
    (if (<= z3 46000000000)
      (sqrt
       (+
        (-
         1/2
         (304-z0z1z2z3z4 1/2 (cos z1) (cos z0) (sin z1) (sin (- z0))))
        (* (* (* t_1 (cos z0)) (cos z1)) t_1)))
      t_0))))

Derivation

1. Split input into 2 regimes

2. if z3 < -2.4000000000000001e-5 or 4.6e10 < z3

   1. Initial program 60.5%

      \[\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

   3. Applied rewrites 57.8%

      \[\leadsto \sqrt{\color{blue}{\frac{1}{2} - \left(\cos \left(\left(z0 - z1\right) \cdot 1\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(z2 - z3\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)}} \]

   5. Applied rewrites 57.8%

      \[\leadsto \sqrt{\frac{1}{2} - \color{blue}{\left(1 - \frac{\left(\left(\frac{1}{2} - \cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2}\right) \cdot \cos z1\right) \cdot \cos z0}{\cos \left(z1 - z0\right) \cdot \frac{1}{2}}\right) \cdot \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2}\right)}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \color{blue}{\left(1 - \frac{\left(\left(\frac{1}{2} - \cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2}\right) \cdot \cos z1\right) \cdot \cos z0}{\cos \left(z1 - z0\right) \cdot \frac{1}{2}}\right) \cdot \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2}\right)}} \]

      2. lift--.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \color{blue}{\left(1 - \frac{\left(\left(\frac{1}{2} - \cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2}\right) \cdot \cos z1\right) \cdot \cos z0}{\cos \left(z1 - z0\right) \cdot \frac{1}{2}}\right)} \cdot \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2}\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(1 - \color{blue}{\frac{\left(\left(\frac{1}{2} - \cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2}\right) \cdot \cos z1\right) \cdot \cos z0}{\cos \left(z1 - z0\right) \cdot \frac{1}{2}}}\right) \cdot \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2}\right)} \]

      4. sub-to-mult-rev N/A

         \[\leadsto \sqrt{\frac{1}{2} - \color{blue}{\left(\cos \left(z1 - z0\right) \cdot \frac{1}{2} - \left(\left(\frac{1}{2} - \cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2}\right) \cdot \cos z1\right) \cdot \cos z0\right)}} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2} - \color{blue}{\left(\left(\frac{1}{2} - \cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2}\right) \cdot \cos z1\right) \cdot \cos z0}\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2} - \color{blue}{\left(\left(\frac{1}{2} - \cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2}\right) \cdot \cos z1\right)} \cdot \cos z0\right)} \]

      7. associate-*l* N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2} - \color{blue}{\left(\frac{1}{2} - \cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)}\right)} \]

   7. Applied rewrites 57.8%

      \[\leadsto \sqrt{\frac{1}{2} - \color{blue}{\left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\cos \left(z3 - z2\right) \cdot \frac{1}{2} - \frac{1}{2}\right) - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\color{blue}{\cos \left(z3 - z2\right) \cdot \frac{1}{2}} - \frac{1}{2}\right) - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\color{blue}{\frac{1}{2} \cdot \cos \left(z3 - z2\right)} - \frac{1}{2}\right) - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)} \]

      3. lift-cos.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\frac{1}{2} \cdot \color{blue}{\cos \left(z3 - z2\right)} - \frac{1}{2}\right) - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)} \]

      4. lift--.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\frac{1}{2} \cdot \cos \color{blue}{\left(z3 - z2\right)} - \frac{1}{2}\right) - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)} \]

      5. sub-flip N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\frac{1}{2} \cdot \cos \color{blue}{\left(z3 + \left(\mathsf{neg}\left(z2\right)\right)\right)} - \frac{1}{2}\right) - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)} \]

      6. cos-sum N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(\cos z3 \cdot \cos \left(\mathsf{neg}\left(z2\right)\right) - \sin z3 \cdot \sin \left(\mathsf{neg}\left(z2\right)\right)\right)} - \frac{1}{2}\right) - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)} \]

      7. cos-neg-rev N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\frac{1}{2} \cdot \left(\cos z3 \cdot \color{blue}{\cos z2} - \sin z3 \cdot \sin \left(\mathsf{neg}\left(z2\right)\right)\right) - \frac{1}{2}\right) - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)} \]

      8. lower-304-z0z1z2z3z4 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\color{blue}{\mathsf{304\_z0z1z2z3z4}\left(\frac{1}{2}, \cos z3, \cos z2, \sin z3, \sin \left(\mathsf{neg}\left(z2\right)\right)\right)} - \frac{1}{2}\right) - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)} \]

      9. lower-cos.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\mathsf{304\_z0z1z2z3z4}\left(\frac{1}{2}, \color{blue}{\cos z3}, \cos z2, \sin z3, \sin \left(\mathsf{neg}\left(z2\right)\right)\right) - \frac{1}{2}\right) - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)} \]

      10. lower-cos.f64 N/A

          \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\mathsf{304\_z0z1z2z3z4}\left(\frac{1}{2}, \cos z3, \color{blue}{\cos z2}, \sin z3, \sin \left(\mathsf{neg}\left(z2\right)\right)\right) - \frac{1}{2}\right) - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)} \]

      11. lower-sin.f64 N/A

          \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\mathsf{304\_z0z1z2z3z4}\left(\frac{1}{2}, \cos z3, \cos z2, \color{blue}{\sin z3}, \sin \left(\mathsf{neg}\left(z2\right)\right)\right) - \frac{1}{2}\right) - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)} \]

      12. lower-sin.f64 N/A

          \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\mathsf{304\_z0z1z2z3z4}\left(\frac{1}{2}, \cos z3, \cos z2, \sin z3, \color{blue}{\sin \left(\mathsf{neg}\left(z2\right)\right)}\right) - \frac{1}{2}\right) - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)} \]

      13. lower-neg.f64 73.1%

          \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\mathsf{304\_z0z1z2z3z4}\left(\frac{1}{2}, \cos z3, \cos z2, \sin z3, \sin \color{blue}{\left(-z2\right)}\right) - \frac{1}{2}\right) - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)} \]

   9. Applied rewrites 73.1%

      \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\color{blue}{\mathsf{304\_z0z1z2z3z4}\left(\frac{1}{2}, \cos z3, \cos z2, \sin z3, \sin \left(-z2\right)\right)} - \frac{1}{2}\right) - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)} \]

   if -2.4000000000000001e-5 < z3 < 4.6e10

   1. Initial program 60.5%

      \[\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \color{blue}{\sin \left(\frac{z2 - z3}{2}\right)}\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \color{blue}{\left(\frac{z2 - z3}{2}\right)}\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      3. lift--.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \left(\frac{\color{blue}{z2 - z3}}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      4. div-sub N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \color{blue}{\left(\frac{z2}{2} - \frac{z3}{2}\right)}\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      5. sin-diff N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \color{blue}{\left(\sin \left(\frac{z2}{2}\right) \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)}\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      6. lower--.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \color{blue}{\left(\sin \left(\frac{z2}{2}\right) \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)}\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\color{blue}{\sin \left(\frac{z2}{2}\right) \cdot \cos \left(\frac{z3}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      8. lower-sin.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\color{blue}{\sin \left(\frac{z2}{2}\right)} \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      9. mult-flip N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      10. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      12. lower-cos.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{z3}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      13. mult-flip N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(z3 \cdot \frac{1}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      14. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      15. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(z3 \cdot \frac{1}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      16. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      17. lower-cos.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{z2}{2}\right)} \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      18. mult-flip N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      19. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      20. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      21. lower-sin.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{z3}{2}\right)}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      22. mult-flip N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(z3 \cdot \frac{1}{2}\right)}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      23. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      24. lower-*.f64 59.7%

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(z3 \cdot \frac{1}{2}\right)}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

   3. Applied rewrites 59.7%

      \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \color{blue}{\left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)}\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{z2 - z3}{2}\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{z2 - z3}{2}\right)}} \]

      3. lift--.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{z2 - z3}}{2}\right)} \]

      4. div-sub N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{z2}{2} - \frac{z3}{2}\right)}} \]

      5. sin-diff N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{z2}{2}\right) \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)}} \]

      6. lower--.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{z2}{2}\right) \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{z2}{2}\right) \cdot \cos \left(\frac{z3}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      8. lower-sin.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{z2}{2}\right)} \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      9. mult-flip N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      10. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      12. lower-cos.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{z3}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      13. mult-flip N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(z3 \cdot \frac{1}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      14. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      15. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(z3 \cdot \frac{1}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      16. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)}\right)} \]

      17. lower-cos.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{z2}{2}\right)} \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      18. mult-flip N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      19. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      20. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      21. lower-sin.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{z3}{2}\right)}\right)} \]

      22. mult-flip N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(z3 \cdot \frac{1}{2}\right)}\right)} \]

      23. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \color{blue}{\frac{1}{2}}\right)\right)} \]

      24. lower-*.f64 75.8%

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(z3 \cdot \frac{1}{2}\right)}\right)} \]

   5. Applied rewrites 75.8%

      \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)}} \]

   7. Applied rewrites 96.4%

      \[\leadsto \sqrt{\left(\frac{1}{2} - \color{blue}{\mathsf{304\_z0z1z2z3z4}\left(\frac{1}{2}, \cos z1, \cos z0, \sin z1, \sin \left(-z0\right)\right)}\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)} \]

   9. Applied rewrites 76.1%

      \[\leadsto \sqrt{\left(\frac{1}{2} - \mathsf{304\_z0z1z2z3z4}\left(\frac{1}{2}, \cos z1, \cos z0, \sin z1, \sin \left(-z0\right)\right)\right) + \color{blue}{\left(\left(\sin \left(\left(z2 - z3\right) \cdot \frac{1}{2}\right) \cdot \cos z0\right) \cdot \cos z1\right) \cdot \sin \left(\left(z2 - z3\right) \cdot \frac{1}{2}\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 83.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_0 := \cos z1 \cdot \cos z0\\ \mathbf{if}\;z0 \leq \frac{-944473296573929}{1180591620717411303424}:\\ \;\;\;\;\sqrt{\left(\frac{1}{2} - \mathsf{304\_z0z1z2z3z4}\left(\frac{1}{2}, \cos z1, \cos z0, \sin z1, \sin \left(-z0\right)\right)\right) + \cos z0 \cdot \left(\cos z1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(z2 - z3\right) \cdot \frac{1}{2}\right)\right)\right)\right)}\\ \mathbf{elif}\;z0 \leq 280000000000000:\\ \;\;\;\;\sqrt{\frac{1}{2} - \left(t\_0 \cdot \left(\mathsf{304\_z0z1z2z3z4}\left(\frac{1}{2}, \cos z3, \cos z2, \sin z3, \sin \left(-z2\right)\right) - \frac{1}{2}\right) - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{2} - \left(\mathsf{304\_z0z1z2z3z4}\left(\frac{1}{2}, t\_0, 1, \sin z1, \left(-\sin z0\right)\right) + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot t\_0\right)}\\ \end{array} \]

FPCore

(FPCore (z0 z1 z2 z3)
  :precision binary64
  (let* ((t_0 (* (cos z1) (cos z0))))
  (if (<= z0 -944473296573929/1180591620717411303424)
    (sqrt
     (+
      (-
       1/2
       (304-z0z1z2z3z4 1/2 (cos z1) (cos z0) (sin z1) (sin (- z0))))
      (*
       (cos z0)
       (* (cos z1) (- 1/2 (* 1/2 (cos (* 2 (* (- z2 z3) 1/2)))))))))
    (if (<= z0 280000000000000)
      (sqrt
       (-
        1/2
        (-
         (*
          t_0
          (-
           (304-z0z1z2z3z4
            1/2
            (cos z3)
            (cos z2)
            (sin z3)
            (sin (- z2)))
           1/2))
         (* -1/2 (cos (- z1 z0))))))
      (sqrt
       (-
        1/2
        (+
         (304-z0z1z2z3z4 1/2 t_0 1 (sin z1) (- (sin z0)))
         (* (- (* (cos (* (- z2 z3) 1)) 1/2) 1/2) t_0))))))))

Derivation

1. Split input into 3 regimes

2. if z0 < -7.9999999999999996e-7

   1. Initial program 60.5%

      \[\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \color{blue}{\sin \left(\frac{z2 - z3}{2}\right)}\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \color{blue}{\left(\frac{z2 - z3}{2}\right)}\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      3. lift--.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \left(\frac{\color{blue}{z2 - z3}}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      4. div-sub N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \color{blue}{\left(\frac{z2}{2} - \frac{z3}{2}\right)}\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      5. sin-diff N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \color{blue}{\left(\sin \left(\frac{z2}{2}\right) \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)}\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      6. lower--.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \color{blue}{\left(\sin \left(\frac{z2}{2}\right) \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)}\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\color{blue}{\sin \left(\frac{z2}{2}\right) \cdot \cos \left(\frac{z3}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      8. lower-sin.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\color{blue}{\sin \left(\frac{z2}{2}\right)} \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      9. mult-flip N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      10. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      12. lower-cos.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{z3}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      13. mult-flip N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(z3 \cdot \frac{1}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      14. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      15. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(z3 \cdot \frac{1}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      16. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      17. lower-cos.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{z2}{2}\right)} \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      18. mult-flip N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      19. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      20. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      21. lower-sin.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{z3}{2}\right)}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      22. mult-flip N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(z3 \cdot \frac{1}{2}\right)}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      23. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      24. lower-*.f64 59.7%

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(z3 \cdot \frac{1}{2}\right)}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

   3. Applied rewrites 59.7%

      \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \color{blue}{\left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)}\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{z2 - z3}{2}\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{z2 - z3}{2}\right)}} \]

      3. lift--.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{z2 - z3}}{2}\right)} \]

      4. div-sub N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{z2}{2} - \frac{z3}{2}\right)}} \]

      5. sin-diff N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{z2}{2}\right) \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)}} \]

      6. lower--.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{z2}{2}\right) \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{z2}{2}\right) \cdot \cos \left(\frac{z3}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      8. lower-sin.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{z2}{2}\right)} \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      9. mult-flip N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      10. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      12. lower-cos.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{z3}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      13. mult-flip N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(z3 \cdot \frac{1}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      14. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      15. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(z3 \cdot \frac{1}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      16. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)}\right)} \]

      17. lower-cos.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{z2}{2}\right)} \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      18. mult-flip N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      19. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      20. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      21. lower-sin.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{z3}{2}\right)}\right)} \]

      22. mult-flip N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(z3 \cdot \frac{1}{2}\right)}\right)} \]

      23. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \color{blue}{\frac{1}{2}}\right)\right)} \]

      24. lower-*.f64 75.8%

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(z3 \cdot \frac{1}{2}\right)}\right)} \]

   5. Applied rewrites 75.8%

      \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)}} \]

   7. Applied rewrites 96.4%

      \[\leadsto \sqrt{\left(\frac{1}{2} - \color{blue}{\mathsf{304\_z0z1z2z3z4}\left(\frac{1}{2}, \cos z1, \cos z0, \sin z1, \sin \left(-z0\right)\right)}\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)} \]

   9. Applied rewrites 73.4%

      \[\leadsto \sqrt{\left(\frac{1}{2} - \mathsf{304\_z0z1z2z3z4}\left(\frac{1}{2}, \cos z1, \cos z0, \sin z1, \sin \left(-z0\right)\right)\right) + \color{blue}{\cos z0 \cdot \left(\cos z1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(z2 - z3\right) \cdot \frac{1}{2}\right)\right)\right)\right)}} \]

   if -7.9999999999999996e-7 < z0 < 2.8e14

   1. Initial program 60.5%

      \[\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

   3. Applied rewrites 57.8%

      \[\leadsto \sqrt{\color{blue}{\frac{1}{2} - \left(\cos \left(\left(z0 - z1\right) \cdot 1\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(z2 - z3\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)}} \]

   5. Applied rewrites 57.8%

      \[\leadsto \sqrt{\frac{1}{2} - \color{blue}{\left(1 - \frac{\left(\left(\frac{1}{2} - \cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2}\right) \cdot \cos z1\right) \cdot \cos z0}{\cos \left(z1 - z0\right) \cdot \frac{1}{2}}\right) \cdot \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2}\right)}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \color{blue}{\left(1 - \frac{\left(\left(\frac{1}{2} - \cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2}\right) \cdot \cos z1\right) \cdot \cos z0}{\cos \left(z1 - z0\right) \cdot \frac{1}{2}}\right) \cdot \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2}\right)}} \]

      2. lift--.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \color{blue}{\left(1 - \frac{\left(\left(\frac{1}{2} - \cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2}\right) \cdot \cos z1\right) \cdot \cos z0}{\cos \left(z1 - z0\right) \cdot \frac{1}{2}}\right)} \cdot \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2}\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(1 - \color{blue}{\frac{\left(\left(\frac{1}{2} - \cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2}\right) \cdot \cos z1\right) \cdot \cos z0}{\cos \left(z1 - z0\right) \cdot \frac{1}{2}}}\right) \cdot \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2}\right)} \]

      4. sub-to-mult-rev N/A

         \[\leadsto \sqrt{\frac{1}{2} - \color{blue}{\left(\cos \left(z1 - z0\right) \cdot \frac{1}{2} - \left(\left(\frac{1}{2} - \cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2}\right) \cdot \cos z1\right) \cdot \cos z0\right)}} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2} - \color{blue}{\left(\left(\frac{1}{2} - \cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2}\right) \cdot \cos z1\right) \cdot \cos z0}\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2} - \color{blue}{\left(\left(\frac{1}{2} - \cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2}\right) \cdot \cos z1\right)} \cdot \cos z0\right)} \]

      7. associate-*l* N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2} - \color{blue}{\left(\frac{1}{2} - \cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)}\right)} \]

   7. Applied rewrites 57.8%

      \[\leadsto \sqrt{\frac{1}{2} - \color{blue}{\left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\cos \left(z3 - z2\right) \cdot \frac{1}{2} - \frac{1}{2}\right) - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\color{blue}{\cos \left(z3 - z2\right) \cdot \frac{1}{2}} - \frac{1}{2}\right) - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\color{blue}{\frac{1}{2} \cdot \cos \left(z3 - z2\right)} - \frac{1}{2}\right) - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)} \]

      3. lift-cos.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\frac{1}{2} \cdot \color{blue}{\cos \left(z3 - z2\right)} - \frac{1}{2}\right) - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)} \]

      4. lift--.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\frac{1}{2} \cdot \cos \color{blue}{\left(z3 - z2\right)} - \frac{1}{2}\right) - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)} \]

      5. sub-flip N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\frac{1}{2} \cdot \cos \color{blue}{\left(z3 + \left(\mathsf{neg}\left(z2\right)\right)\right)} - \frac{1}{2}\right) - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)} \]

      6. cos-sum N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(\cos z3 \cdot \cos \left(\mathsf{neg}\left(z2\right)\right) - \sin z3 \cdot \sin \left(\mathsf{neg}\left(z2\right)\right)\right)} - \frac{1}{2}\right) - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)} \]

      7. cos-neg-rev N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\frac{1}{2} \cdot \left(\cos z3 \cdot \color{blue}{\cos z2} - \sin z3 \cdot \sin \left(\mathsf{neg}\left(z2\right)\right)\right) - \frac{1}{2}\right) - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)} \]

      8. lower-304-z0z1z2z3z4 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\color{blue}{\mathsf{304\_z0z1z2z3z4}\left(\frac{1}{2}, \cos z3, \cos z2, \sin z3, \sin \left(\mathsf{neg}\left(z2\right)\right)\right)} - \frac{1}{2}\right) - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)} \]

      9. lower-cos.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\mathsf{304\_z0z1z2z3z4}\left(\frac{1}{2}, \color{blue}{\cos z3}, \cos z2, \sin z3, \sin \left(\mathsf{neg}\left(z2\right)\right)\right) - \frac{1}{2}\right) - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)} \]

      10. lower-cos.f64 N/A

          \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\mathsf{304\_z0z1z2z3z4}\left(\frac{1}{2}, \cos z3, \color{blue}{\cos z2}, \sin z3, \sin \left(\mathsf{neg}\left(z2\right)\right)\right) - \frac{1}{2}\right) - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)} \]

      11. lower-sin.f64 N/A

          \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\mathsf{304\_z0z1z2z3z4}\left(\frac{1}{2}, \cos z3, \cos z2, \color{blue}{\sin z3}, \sin \left(\mathsf{neg}\left(z2\right)\right)\right) - \frac{1}{2}\right) - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)} \]

      12. lower-sin.f64 N/A

          \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\mathsf{304\_z0z1z2z3z4}\left(\frac{1}{2}, \cos z3, \cos z2, \sin z3, \color{blue}{\sin \left(\mathsf{neg}\left(z2\right)\right)}\right) - \frac{1}{2}\right) - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)} \]

      13. lower-neg.f64 73.1%

          \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\mathsf{304\_z0z1z2z3z4}\left(\frac{1}{2}, \cos z3, \cos z2, \sin z3, \sin \color{blue}{\left(-z2\right)}\right) - \frac{1}{2}\right) - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)} \]

   9. Applied rewrites 73.1%

      \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\color{blue}{\mathsf{304\_z0z1z2z3z4}\left(\frac{1}{2}, \cos z3, \cos z2, \sin z3, \sin \left(-z2\right)\right)} - \frac{1}{2}\right) - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)} \]

   if 2.8e14 < z0

   1. Initial program 60.5%

      \[\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

   3. Applied rewrites 57.8%

      \[\leadsto \sqrt{\color{blue}{\frac{1}{2} - \left(\cos \left(\left(z0 - z1\right) \cdot 1\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(z2 - z3\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)}} \]

   5. Applied rewrites 57.8%

      \[\leadsto \sqrt{\frac{1}{2} - \color{blue}{\left(\cos \left(z1 - z0\right) \cdot \frac{1}{2} + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\color{blue}{\cos \left(z1 - z0\right) \cdot \frac{1}{2}} + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\color{blue}{\frac{1}{2} \cdot \cos \left(z1 - z0\right)} + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      3. lift-cos.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\frac{1}{2} \cdot \color{blue}{\cos \left(z1 - z0\right)} + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      4. lift--.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\frac{1}{2} \cdot \cos \color{blue}{\left(z1 - z0\right)} + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      5. cos-diff N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\frac{1}{2} \cdot \color{blue}{\left(\cos z1 \cdot \cos z0 + \sin z1 \cdot \sin z0\right)} + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      6. lift-cos.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\frac{1}{2} \cdot \left(\color{blue}{\cos z1} \cdot \cos z0 + \sin z1 \cdot \sin z0\right) + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      7. lift-cos.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\frac{1}{2} \cdot \left(\cos z1 \cdot \color{blue}{\cos z0} + \sin z1 \cdot \sin z0\right) + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\frac{1}{2} \cdot \left(\color{blue}{\cos z1 \cdot \cos z0} + \sin z1 \cdot \sin z0\right) + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      9. *-lft-identity N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\frac{1}{2} \cdot \left(\color{blue}{1 \cdot \left(\cos z1 \cdot \cos z0\right)} + \sin z1 \cdot \sin z0\right) + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      10. add-flip N/A

          \[\leadsto \sqrt{\frac{1}{2} - \left(\frac{1}{2} \cdot \color{blue}{\left(1 \cdot \left(\cos z1 \cdot \cos z0\right) - \left(\mathsf{neg}\left(\sin z1 \cdot \sin z0\right)\right)\right)} + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      11. *-lft-identity N/A

          \[\leadsto \sqrt{\frac{1}{2} - \left(\frac{1}{2} \cdot \left(1 \cdot \color{blue}{\left(1 \cdot \left(\cos z1 \cdot \cos z0\right)\right)} - \left(\mathsf{neg}\left(\sin z1 \cdot \sin z0\right)\right)\right) + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      12. *-commutative N/A

          \[\leadsto \sqrt{\frac{1}{2} - \left(\frac{1}{2} \cdot \left(\color{blue}{\left(1 \cdot \left(\cos z1 \cdot \cos z0\right)\right) \cdot 1} - \left(\mathsf{neg}\left(\sin z1 \cdot \sin z0\right)\right)\right) + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      13. lift-sin.f64 N/A

          \[\leadsto \sqrt{\frac{1}{2} - \left(\frac{1}{2} \cdot \left(\left(1 \cdot \left(\cos z1 \cdot \cos z0\right)\right) \cdot 1 - \left(\mathsf{neg}\left(\color{blue}{\sin z1} \cdot \sin z0\right)\right)\right) + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      14. distribute-rgt-neg-out N/A

          \[\leadsto \sqrt{\frac{1}{2} - \left(\frac{1}{2} \cdot \left(\left(1 \cdot \left(\cos z1 \cdot \cos z0\right)\right) \cdot 1 - \color{blue}{\sin z1 \cdot \left(\mathsf{neg}\left(\sin z0\right)\right)}\right) + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      15. sin-neg N/A

          \[\leadsto \sqrt{\frac{1}{2} - \left(\frac{1}{2} \cdot \left(\left(1 \cdot \left(\cos z1 \cdot \cos z0\right)\right) \cdot 1 - \sin z1 \cdot \color{blue}{\sin \left(\mathsf{neg}\left(z0\right)\right)}\right) + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      16. lift-neg.f64 N/A

          \[\leadsto \sqrt{\frac{1}{2} - \left(\frac{1}{2} \cdot \left(\left(1 \cdot \left(\cos z1 \cdot \cos z0\right)\right) \cdot 1 - \sin z1 \cdot \sin \color{blue}{\left(-z0\right)}\right) + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      17. lift-sin.f64 N/A

          \[\leadsto \sqrt{\frac{1}{2} - \left(\frac{1}{2} \cdot \left(\left(1 \cdot \left(\cos z1 \cdot \cos z0\right)\right) \cdot 1 - \sin z1 \cdot \color{blue}{\sin \left(-z0\right)}\right) + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      18. lower-304-z0z1z2z3z4 N/A

          \[\leadsto \sqrt{\frac{1}{2} - \left(\color{blue}{\mathsf{304\_z0z1z2z3z4}\left(\frac{1}{2}, \left(1 \cdot \left(\cos z1 \cdot \cos z0\right)\right), 1, \sin z1, \sin \left(-z0\right)\right)} + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

   7. Applied rewrites 73.4%

      \[\leadsto \sqrt{\frac{1}{2} - \left(\color{blue}{\mathsf{304\_z0z1z2z3z4}\left(\frac{1}{2}, \left(\cos z1 \cdot \cos z0\right), 1, \sin z1, \left(-\sin z0\right)\right)} + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 83.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \cos z1 \cdot \cos z0\\ \mathbf{if}\;z0 \leq \frac{-944473296573929}{1180591620717411303424}:\\ \;\;\;\;\sqrt{\left(\frac{1}{2} - \mathsf{304\_z0z1z2z3z4}\left(\frac{1}{2}, \cos z1, \cos z0, \sin z1, \sin \left(-z0\right)\right)\right) + \cos z0 \cdot \left(\cos z1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(z2 - z3\right) \cdot \frac{1}{2}\right)\right)\right)\right)}\\ \mathbf{elif}\;z0 \leq 280000000000000:\\ \;\;\;\;\sqrt{\frac{1}{2} - \left(t\_0 \cdot \left(\mathsf{304\_z0z1z2z3z4}\left(\frac{1}{2}, \cos z3, \cos z2, \sin z3, \sin \left(-z2\right)\right) - \frac{1}{2}\right) - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{2} - \left(\left(t\_0 + \sin z1 \cdot \sin z0\right) \cdot \frac{1}{2} + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot t\_0\right)}\\ \end{array} \]

FPCore

(FPCore (z0 z1 z2 z3)
  :precision binary64
  (let* ((t_0 (* (cos z1) (cos z0))))
  (if (<= z0 -944473296573929/1180591620717411303424)
    (sqrt
     (+
      (-
       1/2
       (304-z0z1z2z3z4 1/2 (cos z1) (cos z0) (sin z1) (sin (- z0))))
      (*
       (cos z0)
       (* (cos z1) (- 1/2 (* 1/2 (cos (* 2 (* (- z2 z3) 1/2)))))))))
    (if (<= z0 280000000000000)
      (sqrt
       (-
        1/2
        (-
         (*
          t_0
          (-
           (304-z0z1z2z3z4
            1/2
            (cos z3)
            (cos z2)
            (sin z3)
            (sin (- z2)))
           1/2))
         (* -1/2 (cos (- z1 z0))))))
      (sqrt
       (-
        1/2
        (+
         (* (+ t_0 (* (sin z1) (sin z0))) 1/2)
         (* (- (* (cos (* (- z2 z3) 1)) 1/2) 1/2) t_0))))))))

Derivation

1. Split input into 3 regimes

2. if z0 < -7.9999999999999996e-7

   1. Initial program 60.5%

      \[\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \color{blue}{\sin \left(\frac{z2 - z3}{2}\right)}\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \color{blue}{\left(\frac{z2 - z3}{2}\right)}\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      3. lift--.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \left(\frac{\color{blue}{z2 - z3}}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      4. div-sub N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \color{blue}{\left(\frac{z2}{2} - \frac{z3}{2}\right)}\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      5. sin-diff N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \color{blue}{\left(\sin \left(\frac{z2}{2}\right) \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)}\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      6. lower--.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \color{blue}{\left(\sin \left(\frac{z2}{2}\right) \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)}\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\color{blue}{\sin \left(\frac{z2}{2}\right) \cdot \cos \left(\frac{z3}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      8. lower-sin.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\color{blue}{\sin \left(\frac{z2}{2}\right)} \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      9. mult-flip N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      10. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      12. lower-cos.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{z3}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      13. mult-flip N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(z3 \cdot \frac{1}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      14. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      15. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(z3 \cdot \frac{1}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      16. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      17. lower-cos.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{z2}{2}\right)} \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      18. mult-flip N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      19. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      20. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{z3}{2}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      21. lower-sin.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{z3}{2}\right)}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      22. mult-flip N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(z3 \cdot \frac{1}{2}\right)}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      23. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      24. lower-*.f64 59.7%

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(z3 \cdot \frac{1}{2}\right)}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

   3. Applied rewrites 59.7%

      \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \color{blue}{\left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)}\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{z2 - z3}{2}\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{z2 - z3}{2}\right)}} \]

      3. lift--.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{z2 - z3}}{2}\right)} \]

      4. div-sub N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{z2}{2} - \frac{z3}{2}\right)}} \]

      5. sin-diff N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{z2}{2}\right) \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)}} \]

      6. lower--.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{z2}{2}\right) \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{z2}{2}\right) \cdot \cos \left(\frac{z3}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      8. lower-sin.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{z2}{2}\right)} \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      9. mult-flip N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      10. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{z3}{2}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      12. lower-cos.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \color{blue}{\cos \left(\frac{z3}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      13. mult-flip N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(z3 \cdot \frac{1}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      14. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \color{blue}{\frac{1}{2}}\right) - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      15. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \color{blue}{\left(z3 \cdot \frac{1}{2}\right)} - \cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      16. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{z2}{2}\right) \cdot \sin \left(\frac{z3}{2}\right)}\right)} \]

      17. lower-cos.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \color{blue}{\cos \left(\frac{z2}{2}\right)} \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      18. mult-flip N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      19. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      20. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \color{blue}{\left(z2 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{z3}{2}\right)\right)} \]

      21. lower-sin.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{z3}{2}\right)}\right)} \]

      22. mult-flip N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(z3 \cdot \frac{1}{2}\right)}\right)} \]

      23. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \color{blue}{\frac{1}{2}}\right)\right)} \]

      24. lower-*.f64 75.8%

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \color{blue}{\left(z3 \cdot \frac{1}{2}\right)}\right)} \]

   5. Applied rewrites 75.8%

      \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)}} \]

   7. Applied rewrites 96.4%

      \[\leadsto \sqrt{\left(\frac{1}{2} - \color{blue}{\mathsf{304\_z0z1z2z3z4}\left(\frac{1}{2}, \cos z1, \cos z0, \sin z1, \sin \left(-z0\right)\right)}\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(z2 \cdot \frac{1}{2}\right) \cdot \cos \left(z3 \cdot \frac{1}{2}\right) - \cos \left(z2 \cdot \frac{1}{2}\right) \cdot \sin \left(z3 \cdot \frac{1}{2}\right)\right)} \]

   9. Applied rewrites 73.4%

      \[\leadsto \sqrt{\left(\frac{1}{2} - \mathsf{304\_z0z1z2z3z4}\left(\frac{1}{2}, \cos z1, \cos z0, \sin z1, \sin \left(-z0\right)\right)\right) + \color{blue}{\cos z0 \cdot \left(\cos z1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(z2 - z3\right) \cdot \frac{1}{2}\right)\right)\right)\right)}} \]

   if -7.9999999999999996e-7 < z0 < 2.8e14

   1. Initial program 60.5%

      \[\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

   3. Applied rewrites 57.8%

      \[\leadsto \sqrt{\color{blue}{\frac{1}{2} - \left(\cos \left(\left(z0 - z1\right) \cdot 1\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(z2 - z3\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)}} \]

   5. Applied rewrites 57.8%

      \[\leadsto \sqrt{\frac{1}{2} - \color{blue}{\left(1 - \frac{\left(\left(\frac{1}{2} - \cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2}\right) \cdot \cos z1\right) \cdot \cos z0}{\cos \left(z1 - z0\right) \cdot \frac{1}{2}}\right) \cdot \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2}\right)}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \color{blue}{\left(1 - \frac{\left(\left(\frac{1}{2} - \cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2}\right) \cdot \cos z1\right) \cdot \cos z0}{\cos \left(z1 - z0\right) \cdot \frac{1}{2}}\right) \cdot \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2}\right)}} \]

      2. lift--.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \color{blue}{\left(1 - \frac{\left(\left(\frac{1}{2} - \cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2}\right) \cdot \cos z1\right) \cdot \cos z0}{\cos \left(z1 - z0\right) \cdot \frac{1}{2}}\right)} \cdot \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2}\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(1 - \color{blue}{\frac{\left(\left(\frac{1}{2} - \cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2}\right) \cdot \cos z1\right) \cdot \cos z0}{\cos \left(z1 - z0\right) \cdot \frac{1}{2}}}\right) \cdot \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2}\right)} \]

      4. sub-to-mult-rev N/A

         \[\leadsto \sqrt{\frac{1}{2} - \color{blue}{\left(\cos \left(z1 - z0\right) \cdot \frac{1}{2} - \left(\left(\frac{1}{2} - \cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2}\right) \cdot \cos z1\right) \cdot \cos z0\right)}} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2} - \color{blue}{\left(\left(\frac{1}{2} - \cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2}\right) \cdot \cos z1\right) \cdot \cos z0}\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2} - \color{blue}{\left(\left(\frac{1}{2} - \cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2}\right) \cdot \cos z1\right)} \cdot \cos z0\right)} \]

      7. associate-*l* N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2} - \color{blue}{\left(\frac{1}{2} - \cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)}\right)} \]

   7. Applied rewrites 57.8%

      \[\leadsto \sqrt{\frac{1}{2} - \color{blue}{\left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\cos \left(z3 - z2\right) \cdot \frac{1}{2} - \frac{1}{2}\right) - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\color{blue}{\cos \left(z3 - z2\right) \cdot \frac{1}{2}} - \frac{1}{2}\right) - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\color{blue}{\frac{1}{2} \cdot \cos \left(z3 - z2\right)} - \frac{1}{2}\right) - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)} \]

      3. lift-cos.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\frac{1}{2} \cdot \color{blue}{\cos \left(z3 - z2\right)} - \frac{1}{2}\right) - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)} \]

      4. lift--.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\frac{1}{2} \cdot \cos \color{blue}{\left(z3 - z2\right)} - \frac{1}{2}\right) - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)} \]

      5. sub-flip N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\frac{1}{2} \cdot \cos \color{blue}{\left(z3 + \left(\mathsf{neg}\left(z2\right)\right)\right)} - \frac{1}{2}\right) - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)} \]

      6. cos-sum N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(\cos z3 \cdot \cos \left(\mathsf{neg}\left(z2\right)\right) - \sin z3 \cdot \sin \left(\mathsf{neg}\left(z2\right)\right)\right)} - \frac{1}{2}\right) - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)} \]

      7. cos-neg-rev N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\frac{1}{2} \cdot \left(\cos z3 \cdot \color{blue}{\cos z2} - \sin z3 \cdot \sin \left(\mathsf{neg}\left(z2\right)\right)\right) - \frac{1}{2}\right) - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)} \]

      8. lower-304-z0z1z2z3z4 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\color{blue}{\mathsf{304\_z0z1z2z3z4}\left(\frac{1}{2}, \cos z3, \cos z2, \sin z3, \sin \left(\mathsf{neg}\left(z2\right)\right)\right)} - \frac{1}{2}\right) - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)} \]

      9. lower-cos.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\mathsf{304\_z0z1z2z3z4}\left(\frac{1}{2}, \color{blue}{\cos z3}, \cos z2, \sin z3, \sin \left(\mathsf{neg}\left(z2\right)\right)\right) - \frac{1}{2}\right) - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)} \]

      10. lower-cos.f64 N/A

          \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\mathsf{304\_z0z1z2z3z4}\left(\frac{1}{2}, \cos z3, \color{blue}{\cos z2}, \sin z3, \sin \left(\mathsf{neg}\left(z2\right)\right)\right) - \frac{1}{2}\right) - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)} \]

      11. lower-sin.f64 N/A

          \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\mathsf{304\_z0z1z2z3z4}\left(\frac{1}{2}, \cos z3, \cos z2, \color{blue}{\sin z3}, \sin \left(\mathsf{neg}\left(z2\right)\right)\right) - \frac{1}{2}\right) - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)} \]

      12. lower-sin.f64 N/A

          \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\mathsf{304\_z0z1z2z3z4}\left(\frac{1}{2}, \cos z3, \cos z2, \sin z3, \color{blue}{\sin \left(\mathsf{neg}\left(z2\right)\right)}\right) - \frac{1}{2}\right) - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)} \]

      13. lower-neg.f64 73.1%

          \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\mathsf{304\_z0z1z2z3z4}\left(\frac{1}{2}, \cos z3, \cos z2, \sin z3, \sin \color{blue}{\left(-z2\right)}\right) - \frac{1}{2}\right) - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)} \]

   9. Applied rewrites 73.1%

      \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\color{blue}{\mathsf{304\_z0z1z2z3z4}\left(\frac{1}{2}, \cos z3, \cos z2, \sin z3, \sin \left(-z2\right)\right)} - \frac{1}{2}\right) - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)} \]

   if 2.8e14 < z0

   1. Initial program 60.5%

      \[\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

   3. Applied rewrites 57.8%

      \[\leadsto \sqrt{\color{blue}{\frac{1}{2} - \left(\cos \left(\left(z0 - z1\right) \cdot 1\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(z2 - z3\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)}} \]

   5. Applied rewrites 57.8%

      \[\leadsto \sqrt{\frac{1}{2} - \color{blue}{\left(\cos \left(z1 - z0\right) \cdot \frac{1}{2} + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)}} \]
   6. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\color{blue}{\cos \left(z1 - z0\right)} \cdot \frac{1}{2} + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      2. lift--.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\cos \color{blue}{\left(z1 - z0\right)} \cdot \frac{1}{2} + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      3. cos-diff N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\color{blue}{\left(\cos z1 \cdot \cos z0 + \sin z1 \cdot \sin z0\right)} \cdot \frac{1}{2} + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      4. lift-cos.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\color{blue}{\cos z1} \cdot \cos z0 + \sin z1 \cdot \sin z0\right) \cdot \frac{1}{2} + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      5. lift-cos.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \color{blue}{\cos z0} + \sin z1 \cdot \sin z0\right) \cdot \frac{1}{2} + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\color{blue}{\cos z1 \cdot \cos z0} + \sin z1 \cdot \sin z0\right) \cdot \frac{1}{2} + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      7. *-lft-identity N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\color{blue}{1 \cdot \left(\cos z1 \cdot \cos z0\right)} + \sin z1 \cdot \sin z0\right) \cdot \frac{1}{2} + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      8. lower-+.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\color{blue}{\left(1 \cdot \left(\cos z1 \cdot \cos z0\right) + \sin z1 \cdot \sin z0\right)} \cdot \frac{1}{2} + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      9. *-lft-identity N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\color{blue}{\cos z1 \cdot \cos z0} + \sin z1 \cdot \sin z0\right) \cdot \frac{1}{2} + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      10. lift-sin.f64 N/A

          \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0 + \color{blue}{\sin z1} \cdot \sin z0\right) \cdot \frac{1}{2} + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0 + \color{blue}{\sin z1 \cdot \sin z0}\right) \cdot \frac{1}{2} + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      12. lower-sin.f64 73.4%

          \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0 + \sin z1 \cdot \color{blue}{\sin z0}\right) \cdot \frac{1}{2} + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

   7. Applied rewrites 73.4%

      \[\leadsto \sqrt{\frac{1}{2} - \left(\color{blue}{\left(\cos z1 \cdot \cos z0 + \sin z1 \cdot \sin z0\right)} \cdot \frac{1}{2} + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 83.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \cos z1 \cdot \cos z0\\ \mathbf{if}\;z0 \leq \frac{-944473296573929}{1180591620717411303424}:\\ \;\;\;\;\sqrt{\frac{1}{2} - \left(t\_0 \cdot \left(\cos \left(z3 - z2\right) \cdot \frac{1}{2} - \frac{1}{2}\right) - \mathsf{304\_z0z1z2z3z4}\left(\frac{1}{2}, \sin z1, \left(-\sin z0\right), \cos z1, \cos z0\right)\right)}\\ \mathbf{elif}\;z0 \leq 280000000000000:\\ \;\;\;\;\sqrt{\frac{1}{2} - \left(t\_0 \cdot \left(\mathsf{304\_z0z1z2z3z4}\left(\frac{1}{2}, \cos z3, \cos z2, \sin z3, \sin \left(-z2\right)\right) - \frac{1}{2}\right) - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{2} - \left(\left(t\_0 + \sin z1 \cdot \sin z0\right) \cdot \frac{1}{2} + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot t\_0\right)}\\ \end{array} \]

FPCore

(FPCore (z0 z1 z2 z3)
  :precision binary64
  (let* ((t_0 (* (cos z1) (cos z0))))
  (if (<= z0 -944473296573929/1180591620717411303424)
    (sqrt
     (-
      1/2
      (-
       (* t_0 (- (* (cos (- z3 z2)) 1/2) 1/2))
       (304-z0z1z2z3z4 1/2 (sin z1) (- (sin z0)) (cos z1) (cos z0)))))
    (if (<= z0 280000000000000)
      (sqrt
       (-
        1/2
        (-
         (*
          t_0
          (-
           (304-z0z1z2z3z4
            1/2
            (cos z3)
            (cos z2)
            (sin z3)
            (sin (- z2)))
           1/2))
         (* -1/2 (cos (- z1 z0))))))
      (sqrt
       (-
        1/2
        (+
         (* (+ t_0 (* (sin z1) (sin z0))) 1/2)
         (* (- (* (cos (* (- z2 z3) 1)) 1/2) 1/2) t_0))))))))

Derivation

1. Split input into 3 regimes

2. if z0 < -7.9999999999999996e-7

   1. Initial program 60.5%

      \[\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

   3. Applied rewrites 57.8%

      \[\leadsto \sqrt{\color{blue}{\frac{1}{2} - \left(\cos \left(\left(z0 - z1\right) \cdot 1\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(z2 - z3\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)}} \]

   5. Applied rewrites 57.8%

      \[\leadsto \sqrt{\frac{1}{2} - \color{blue}{\left(1 - \frac{\left(\left(\frac{1}{2} - \cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2}\right) \cdot \cos z1\right) \cdot \cos z0}{\cos \left(z1 - z0\right) \cdot \frac{1}{2}}\right) \cdot \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2}\right)}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \color{blue}{\left(1 - \frac{\left(\left(\frac{1}{2} - \cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2}\right) \cdot \cos z1\right) \cdot \cos z0}{\cos \left(z1 - z0\right) \cdot \frac{1}{2}}\right) \cdot \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2}\right)}} \]

      2. lift--.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \color{blue}{\left(1 - \frac{\left(\left(\frac{1}{2} - \cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2}\right) \cdot \cos z1\right) \cdot \cos z0}{\cos \left(z1 - z0\right) \cdot \frac{1}{2}}\right)} \cdot \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2}\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(1 - \color{blue}{\frac{\left(\left(\frac{1}{2} - \cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2}\right) \cdot \cos z1\right) \cdot \cos z0}{\cos \left(z1 - z0\right) \cdot \frac{1}{2}}}\right) \cdot \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2}\right)} \]

      4. sub-to-mult-rev N/A

         \[\leadsto \sqrt{\frac{1}{2} - \color{blue}{\left(\cos \left(z1 - z0\right) \cdot \frac{1}{2} - \left(\left(\frac{1}{2} - \cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2}\right) \cdot \cos z1\right) \cdot \cos z0\right)}} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2} - \color{blue}{\left(\left(\frac{1}{2} - \cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2}\right) \cdot \cos z1\right) \cdot \cos z0}\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2} - \color{blue}{\left(\left(\frac{1}{2} - \cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2}\right) \cdot \cos z1\right)} \cdot \cos z0\right)} \]

      7. associate-*l* N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2} - \color{blue}{\left(\frac{1}{2} - \cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)}\right)} \]

   7. Applied rewrites 57.8%

      \[\leadsto \sqrt{\frac{1}{2} - \color{blue}{\left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\cos \left(z3 - z2\right) \cdot \frac{1}{2} - \frac{1}{2}\right) - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\cos \left(z3 - z2\right) \cdot \frac{1}{2} - \frac{1}{2}\right) - \color{blue}{\frac{-1}{2} \cdot \cos \left(z1 - z0\right)}\right)} \]

      2. metadata-eval N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\cos \left(z3 - z2\right) \cdot \frac{1}{2} - \frac{1}{2}\right) - \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \cdot \cos \left(z1 - z0\right)\right)} \]

      3. distribute-lft-neg-out N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\cos \left(z3 - z2\right) \cdot \frac{1}{2} - \frac{1}{2}\right) - \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot \cos \left(z1 - z0\right)\right)\right)}\right)} \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\cos \left(z3 - z2\right) \cdot \frac{1}{2} - \frac{1}{2}\right) - \color{blue}{\frac{1}{2} \cdot \left(\mathsf{neg}\left(\cos \left(z1 - z0\right)\right)\right)}\right)} \]

      5. lift-cos.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\cos \left(z3 - z2\right) \cdot \frac{1}{2} - \frac{1}{2}\right) - \frac{1}{2} \cdot \left(\mathsf{neg}\left(\color{blue}{\cos \left(z1 - z0\right)}\right)\right)\right)} \]

      6. lift--.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\cos \left(z3 - z2\right) \cdot \frac{1}{2} - \frac{1}{2}\right) - \frac{1}{2} \cdot \left(\mathsf{neg}\left(\cos \color{blue}{\left(z1 - z0\right)}\right)\right)\right)} \]

      7. sub-flip N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\cos \left(z3 - z2\right) \cdot \frac{1}{2} - \frac{1}{2}\right) - \frac{1}{2} \cdot \left(\mathsf{neg}\left(\cos \color{blue}{\left(z1 + \left(\mathsf{neg}\left(z0\right)\right)\right)}\right)\right)\right)} \]

      8. lift-neg.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\cos \left(z3 - z2\right) \cdot \frac{1}{2} - \frac{1}{2}\right) - \frac{1}{2} \cdot \left(\mathsf{neg}\left(\cos \left(z1 + \color{blue}{\left(-z0\right)}\right)\right)\right)\right)} \]

      9. cos-sum-rev N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\cos \left(z3 - z2\right) \cdot \frac{1}{2} - \frac{1}{2}\right) - \frac{1}{2} \cdot \left(\mathsf{neg}\left(\color{blue}{\left(\cos z1 \cdot \cos \left(-z0\right) - \sin z1 \cdot \sin \left(-z0\right)\right)}\right)\right)\right)} \]

      10. lift-cos.f64 N/A

          \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\cos \left(z3 - z2\right) \cdot \frac{1}{2} - \frac{1}{2}\right) - \frac{1}{2} \cdot \left(\mathsf{neg}\left(\left(\color{blue}{\cos z1} \cdot \cos \left(-z0\right) - \sin z1 \cdot \sin \left(-z0\right)\right)\right)\right)\right)} \]

      11. lift-neg.f64 N/A

          \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\cos \left(z3 - z2\right) \cdot \frac{1}{2} - \frac{1}{2}\right) - \frac{1}{2} \cdot \left(\mathsf{neg}\left(\left(\cos z1 \cdot \cos \color{blue}{\left(\mathsf{neg}\left(z0\right)\right)} - \sin z1 \cdot \sin \left(-z0\right)\right)\right)\right)\right)} \]

      12. cos-neg-rev N/A

          \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\cos \left(z3 - z2\right) \cdot \frac{1}{2} - \frac{1}{2}\right) - \frac{1}{2} \cdot \left(\mathsf{neg}\left(\left(\cos z1 \cdot \color{blue}{\cos z0} - \sin z1 \cdot \sin \left(-z0\right)\right)\right)\right)\right)} \]

      13. lift-cos.f64 N/A

          \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\cos \left(z3 - z2\right) \cdot \frac{1}{2} - \frac{1}{2}\right) - \frac{1}{2} \cdot \left(\mathsf{neg}\left(\left(\cos z1 \cdot \color{blue}{\cos z0} - \sin z1 \cdot \sin \left(-z0\right)\right)\right)\right)\right)} \]

      14. lift-sin.f64 N/A

          \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\cos \left(z3 - z2\right) \cdot \frac{1}{2} - \frac{1}{2}\right) - \frac{1}{2} \cdot \left(\mathsf{neg}\left(\left(\cos z1 \cdot \cos z0 - \color{blue}{\sin z1} \cdot \sin \left(-z0\right)\right)\right)\right)\right)} \]

      15. lift-sin.f64 N/A

          \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\cos \left(z3 - z2\right) \cdot \frac{1}{2} - \frac{1}{2}\right) - \frac{1}{2} \cdot \left(\mathsf{neg}\left(\left(\cos z1 \cdot \cos z0 - \sin z1 \cdot \color{blue}{\sin \left(-z0\right)}\right)\right)\right)\right)} \]

      16. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\cos \left(z3 - z2\right) \cdot \frac{1}{2} - \frac{1}{2}\right) - \frac{1}{2} \cdot \left(\mathsf{neg}\left(\left(\color{blue}{\cos z1 \cdot \cos z0} - \sin z1 \cdot \sin \left(-z0\right)\right)\right)\right)\right)} \]

      17. sub-negate-rev N/A

          \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\cos \left(z3 - z2\right) \cdot \frac{1}{2} - \frac{1}{2}\right) - \frac{1}{2} \cdot \color{blue}{\left(\sin z1 \cdot \sin \left(-z0\right) - \cos z1 \cdot \cos z0\right)}\right)} \]

      18. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\cos \left(z3 - z2\right) \cdot \frac{1}{2} - \frac{1}{2}\right) - \frac{1}{2} \cdot \left(\sin z1 \cdot \sin \left(-z0\right) - \color{blue}{\cos z1 \cdot \cos z0}\right)\right)} \]

   9. Applied rewrites 73.4%

      \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\cos \left(z3 - z2\right) \cdot \frac{1}{2} - \frac{1}{2}\right) - \color{blue}{\mathsf{304\_z0z1z2z3z4}\left(\frac{1}{2}, \sin z1, \left(-\sin z0\right), \cos z1, \cos z0\right)}\right)} \]

   if -7.9999999999999996e-7 < z0 < 2.8e14

   1. Initial program 60.5%

      \[\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

   3. Applied rewrites 57.8%

      \[\leadsto \sqrt{\color{blue}{\frac{1}{2} - \left(\cos \left(\left(z0 - z1\right) \cdot 1\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(z2 - z3\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)}} \]

   5. Applied rewrites 57.8%

      \[\leadsto \sqrt{\frac{1}{2} - \color{blue}{\left(1 - \frac{\left(\left(\frac{1}{2} - \cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2}\right) \cdot \cos z1\right) \cdot \cos z0}{\cos \left(z1 - z0\right) \cdot \frac{1}{2}}\right) \cdot \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2}\right)}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \color{blue}{\left(1 - \frac{\left(\left(\frac{1}{2} - \cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2}\right) \cdot \cos z1\right) \cdot \cos z0}{\cos \left(z1 - z0\right) \cdot \frac{1}{2}}\right) \cdot \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2}\right)}} \]

      2. lift--.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \color{blue}{\left(1 - \frac{\left(\left(\frac{1}{2} - \cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2}\right) \cdot \cos z1\right) \cdot \cos z0}{\cos \left(z1 - z0\right) \cdot \frac{1}{2}}\right)} \cdot \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2}\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(1 - \color{blue}{\frac{\left(\left(\frac{1}{2} - \cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2}\right) \cdot \cos z1\right) \cdot \cos z0}{\cos \left(z1 - z0\right) \cdot \frac{1}{2}}}\right) \cdot \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2}\right)} \]

      4. sub-to-mult-rev N/A

         \[\leadsto \sqrt{\frac{1}{2} - \color{blue}{\left(\cos \left(z1 - z0\right) \cdot \frac{1}{2} - \left(\left(\frac{1}{2} - \cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2}\right) \cdot \cos z1\right) \cdot \cos z0\right)}} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2} - \color{blue}{\left(\left(\frac{1}{2} - \cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2}\right) \cdot \cos z1\right) \cdot \cos z0}\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2} - \color{blue}{\left(\left(\frac{1}{2} - \cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2}\right) \cdot \cos z1\right)} \cdot \cos z0\right)} \]

      7. associate-*l* N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2} - \color{blue}{\left(\frac{1}{2} - \cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)}\right)} \]

   7. Applied rewrites 57.8%

      \[\leadsto \sqrt{\frac{1}{2} - \color{blue}{\left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\cos \left(z3 - z2\right) \cdot \frac{1}{2} - \frac{1}{2}\right) - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\color{blue}{\cos \left(z3 - z2\right) \cdot \frac{1}{2}} - \frac{1}{2}\right) - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\color{blue}{\frac{1}{2} \cdot \cos \left(z3 - z2\right)} - \frac{1}{2}\right) - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)} \]

      3. lift-cos.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\frac{1}{2} \cdot \color{blue}{\cos \left(z3 - z2\right)} - \frac{1}{2}\right) - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)} \]

      4. lift--.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\frac{1}{2} \cdot \cos \color{blue}{\left(z3 - z2\right)} - \frac{1}{2}\right) - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)} \]

      5. sub-flip N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\frac{1}{2} \cdot \cos \color{blue}{\left(z3 + \left(\mathsf{neg}\left(z2\right)\right)\right)} - \frac{1}{2}\right) - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)} \]

      6. cos-sum N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(\cos z3 \cdot \cos \left(\mathsf{neg}\left(z2\right)\right) - \sin z3 \cdot \sin \left(\mathsf{neg}\left(z2\right)\right)\right)} - \frac{1}{2}\right) - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)} \]

      7. cos-neg-rev N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\frac{1}{2} \cdot \left(\cos z3 \cdot \color{blue}{\cos z2} - \sin z3 \cdot \sin \left(\mathsf{neg}\left(z2\right)\right)\right) - \frac{1}{2}\right) - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)} \]

      8. lower-304-z0z1z2z3z4 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\color{blue}{\mathsf{304\_z0z1z2z3z4}\left(\frac{1}{2}, \cos z3, \cos z2, \sin z3, \sin \left(\mathsf{neg}\left(z2\right)\right)\right)} - \frac{1}{2}\right) - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)} \]

      9. lower-cos.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\mathsf{304\_z0z1z2z3z4}\left(\frac{1}{2}, \color{blue}{\cos z3}, \cos z2, \sin z3, \sin \left(\mathsf{neg}\left(z2\right)\right)\right) - \frac{1}{2}\right) - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)} \]

      10. lower-cos.f64 N/A

          \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\mathsf{304\_z0z1z2z3z4}\left(\frac{1}{2}, \cos z3, \color{blue}{\cos z2}, \sin z3, \sin \left(\mathsf{neg}\left(z2\right)\right)\right) - \frac{1}{2}\right) - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)} \]

      11. lower-sin.f64 N/A

          \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\mathsf{304\_z0z1z2z3z4}\left(\frac{1}{2}, \cos z3, \cos z2, \color{blue}{\sin z3}, \sin \left(\mathsf{neg}\left(z2\right)\right)\right) - \frac{1}{2}\right) - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)} \]

      12. lower-sin.f64 N/A

          \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\mathsf{304\_z0z1z2z3z4}\left(\frac{1}{2}, \cos z3, \cos z2, \sin z3, \color{blue}{\sin \left(\mathsf{neg}\left(z2\right)\right)}\right) - \frac{1}{2}\right) - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)} \]

      13. lower-neg.f64 73.1%

          \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\mathsf{304\_z0z1z2z3z4}\left(\frac{1}{2}, \cos z3, \cos z2, \sin z3, \sin \color{blue}{\left(-z2\right)}\right) - \frac{1}{2}\right) - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)} \]

   9. Applied rewrites 73.1%

      \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\color{blue}{\mathsf{304\_z0z1z2z3z4}\left(\frac{1}{2}, \cos z3, \cos z2, \sin z3, \sin \left(-z2\right)\right)} - \frac{1}{2}\right) - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)} \]

   if 2.8e14 < z0

   1. Initial program 60.5%

      \[\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

   3. Applied rewrites 57.8%

      \[\leadsto \sqrt{\color{blue}{\frac{1}{2} - \left(\cos \left(\left(z0 - z1\right) \cdot 1\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(z2 - z3\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)}} \]

   5. Applied rewrites 57.8%

      \[\leadsto \sqrt{\frac{1}{2} - \color{blue}{\left(\cos \left(z1 - z0\right) \cdot \frac{1}{2} + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)}} \]
   6. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\color{blue}{\cos \left(z1 - z0\right)} \cdot \frac{1}{2} + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      2. lift--.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\cos \color{blue}{\left(z1 - z0\right)} \cdot \frac{1}{2} + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      3. cos-diff N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\color{blue}{\left(\cos z1 \cdot \cos z0 + \sin z1 \cdot \sin z0\right)} \cdot \frac{1}{2} + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      4. lift-cos.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\color{blue}{\cos z1} \cdot \cos z0 + \sin z1 \cdot \sin z0\right) \cdot \frac{1}{2} + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      5. lift-cos.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \color{blue}{\cos z0} + \sin z1 \cdot \sin z0\right) \cdot \frac{1}{2} + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\color{blue}{\cos z1 \cdot \cos z0} + \sin z1 \cdot \sin z0\right) \cdot \frac{1}{2} + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      7. *-lft-identity N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\color{blue}{1 \cdot \left(\cos z1 \cdot \cos z0\right)} + \sin z1 \cdot \sin z0\right) \cdot \frac{1}{2} + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      8. lower-+.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\color{blue}{\left(1 \cdot \left(\cos z1 \cdot \cos z0\right) + \sin z1 \cdot \sin z0\right)} \cdot \frac{1}{2} + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      9. *-lft-identity N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\color{blue}{\cos z1 \cdot \cos z0} + \sin z1 \cdot \sin z0\right) \cdot \frac{1}{2} + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      10. lift-sin.f64 N/A

          \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0 + \color{blue}{\sin z1} \cdot \sin z0\right) \cdot \frac{1}{2} + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0 + \color{blue}{\sin z1 \cdot \sin z0}\right) \cdot \frac{1}{2} + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      12. lower-sin.f64 73.4%

          \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0 + \sin z1 \cdot \color{blue}{\sin z0}\right) \cdot \frac{1}{2} + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

   7. Applied rewrites 73.4%

      \[\leadsto \sqrt{\frac{1}{2} - \left(\color{blue}{\left(\cos z1 \cdot \cos z0 + \sin z1 \cdot \sin z0\right)} \cdot \frac{1}{2} + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 14: 83.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \cos z1 \cdot \cos z0\\ t_1 := \sqrt{\frac{1}{2} - \left(t\_0 \cdot \left(\cos \left(z3 - z2\right) \cdot \frac{1}{2} - \frac{1}{2}\right) - \mathsf{304\_z0z1z2z3z4}\left(\frac{1}{2}, \sin z1, \left(-\sin z0\right), \cos z1, \cos z0\right)\right)}\\ \mathbf{if}\;z0 \leq \frac{-944473296573929}{1180591620717411303424}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z0 \leq 280000000000000:\\ \;\;\;\;\sqrt{\frac{1}{2} - \left(t\_0 \cdot \left(\mathsf{304\_z0z1z2z3z4}\left(\frac{1}{2}, \cos z3, \cos z2, \sin z3, \sin \left(-z2\right)\right) - \frac{1}{2}\right) - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (z0 z1 z2 z3)
  :precision binary64
  (let* ((t_0 (* (cos z1) (cos z0)))
       (t_1
        (sqrt
         (-
          1/2
          (-
           (* t_0 (- (* (cos (- z3 z2)) 1/2) 1/2))
           (304-z0z1z2z3z4
            1/2
            (sin z1)
            (- (sin z0))
            (cos z1)
            (cos z0)))))))
  (if (<= z0 -944473296573929/1180591620717411303424)
    t_1
    (if (<= z0 280000000000000)
      (sqrt
       (-
        1/2
        (-
         (*
          t_0
          (-
           (304-z0z1z2z3z4
            1/2
            (cos z3)
            (cos z2)
            (sin z3)
            (sin (- z2)))
           1/2))
         (* -1/2 (cos (- z1 z0))))))
      t_1))))

Derivation

1. Split input into 2 regimes

2. if z0 < -7.9999999999999996e-7 or 2.8e14 < z0

   1. Initial program 60.5%

      \[\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

   3. Applied rewrites 57.8%

      \[\leadsto \sqrt{\color{blue}{\frac{1}{2} - \left(\cos \left(\left(z0 - z1\right) \cdot 1\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(z2 - z3\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)}} \]

   5. Applied rewrites 57.8%

      \[\leadsto \sqrt{\frac{1}{2} - \color{blue}{\left(1 - \frac{\left(\left(\frac{1}{2} - \cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2}\right) \cdot \cos z1\right) \cdot \cos z0}{\cos \left(z1 - z0\right) \cdot \frac{1}{2}}\right) \cdot \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2}\right)}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \color{blue}{\left(1 - \frac{\left(\left(\frac{1}{2} - \cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2}\right) \cdot \cos z1\right) \cdot \cos z0}{\cos \left(z1 - z0\right) \cdot \frac{1}{2}}\right) \cdot \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2}\right)}} \]

      2. lift--.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \color{blue}{\left(1 - \frac{\left(\left(\frac{1}{2} - \cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2}\right) \cdot \cos z1\right) \cdot \cos z0}{\cos \left(z1 - z0\right) \cdot \frac{1}{2}}\right)} \cdot \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2}\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(1 - \color{blue}{\frac{\left(\left(\frac{1}{2} - \cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2}\right) \cdot \cos z1\right) \cdot \cos z0}{\cos \left(z1 - z0\right) \cdot \frac{1}{2}}}\right) \cdot \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2}\right)} \]

      4. sub-to-mult-rev N/A

         \[\leadsto \sqrt{\frac{1}{2} - \color{blue}{\left(\cos \left(z1 - z0\right) \cdot \frac{1}{2} - \left(\left(\frac{1}{2} - \cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2}\right) \cdot \cos z1\right) \cdot \cos z0\right)}} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2} - \color{blue}{\left(\left(\frac{1}{2} - \cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2}\right) \cdot \cos z1\right) \cdot \cos z0}\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2} - \color{blue}{\left(\left(\frac{1}{2} - \cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2}\right) \cdot \cos z1\right)} \cdot \cos z0\right)} \]

      7. associate-*l* N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2} - \color{blue}{\left(\frac{1}{2} - \cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)}\right)} \]

   7. Applied rewrites 57.8%

      \[\leadsto \sqrt{\frac{1}{2} - \color{blue}{\left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\cos \left(z3 - z2\right) \cdot \frac{1}{2} - \frac{1}{2}\right) - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\cos \left(z3 - z2\right) \cdot \frac{1}{2} - \frac{1}{2}\right) - \color{blue}{\frac{-1}{2} \cdot \cos \left(z1 - z0\right)}\right)} \]

      2. metadata-eval N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\cos \left(z3 - z2\right) \cdot \frac{1}{2} - \frac{1}{2}\right) - \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \cdot \cos \left(z1 - z0\right)\right)} \]

      3. distribute-lft-neg-out N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\cos \left(z3 - z2\right) \cdot \frac{1}{2} - \frac{1}{2}\right) - \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot \cos \left(z1 - z0\right)\right)\right)}\right)} \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\cos \left(z3 - z2\right) \cdot \frac{1}{2} - \frac{1}{2}\right) - \color{blue}{\frac{1}{2} \cdot \left(\mathsf{neg}\left(\cos \left(z1 - z0\right)\right)\right)}\right)} \]

      5. lift-cos.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\cos \left(z3 - z2\right) \cdot \frac{1}{2} - \frac{1}{2}\right) - \frac{1}{2} \cdot \left(\mathsf{neg}\left(\color{blue}{\cos \left(z1 - z0\right)}\right)\right)\right)} \]

      6. lift--.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\cos \left(z3 - z2\right) \cdot \frac{1}{2} - \frac{1}{2}\right) - \frac{1}{2} \cdot \left(\mathsf{neg}\left(\cos \color{blue}{\left(z1 - z0\right)}\right)\right)\right)} \]

      7. sub-flip N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\cos \left(z3 - z2\right) \cdot \frac{1}{2} - \frac{1}{2}\right) - \frac{1}{2} \cdot \left(\mathsf{neg}\left(\cos \color{blue}{\left(z1 + \left(\mathsf{neg}\left(z0\right)\right)\right)}\right)\right)\right)} \]

      8. lift-neg.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\cos \left(z3 - z2\right) \cdot \frac{1}{2} - \frac{1}{2}\right) - \frac{1}{2} \cdot \left(\mathsf{neg}\left(\cos \left(z1 + \color{blue}{\left(-z0\right)}\right)\right)\right)\right)} \]

      9. cos-sum-rev N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\cos \left(z3 - z2\right) \cdot \frac{1}{2} - \frac{1}{2}\right) - \frac{1}{2} \cdot \left(\mathsf{neg}\left(\color{blue}{\left(\cos z1 \cdot \cos \left(-z0\right) - \sin z1 \cdot \sin \left(-z0\right)\right)}\right)\right)\right)} \]

      10. lift-cos.f64 N/A

          \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\cos \left(z3 - z2\right) \cdot \frac{1}{2} - \frac{1}{2}\right) - \frac{1}{2} \cdot \left(\mathsf{neg}\left(\left(\color{blue}{\cos z1} \cdot \cos \left(-z0\right) - \sin z1 \cdot \sin \left(-z0\right)\right)\right)\right)\right)} \]

      11. lift-neg.f64 N/A

          \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\cos \left(z3 - z2\right) \cdot \frac{1}{2} - \frac{1}{2}\right) - \frac{1}{2} \cdot \left(\mathsf{neg}\left(\left(\cos z1 \cdot \cos \color{blue}{\left(\mathsf{neg}\left(z0\right)\right)} - \sin z1 \cdot \sin \left(-z0\right)\right)\right)\right)\right)} \]

      12. cos-neg-rev N/A

          \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\cos \left(z3 - z2\right) \cdot \frac{1}{2} - \frac{1}{2}\right) - \frac{1}{2} \cdot \left(\mathsf{neg}\left(\left(\cos z1 \cdot \color{blue}{\cos z0} - \sin z1 \cdot \sin \left(-z0\right)\right)\right)\right)\right)} \]

      13. lift-cos.f64 N/A

          \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\cos \left(z3 - z2\right) \cdot \frac{1}{2} - \frac{1}{2}\right) - \frac{1}{2} \cdot \left(\mathsf{neg}\left(\left(\cos z1 \cdot \color{blue}{\cos z0} - \sin z1 \cdot \sin \left(-z0\right)\right)\right)\right)\right)} \]

      14. lift-sin.f64 N/A

          \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\cos \left(z3 - z2\right) \cdot \frac{1}{2} - \frac{1}{2}\right) - \frac{1}{2} \cdot \left(\mathsf{neg}\left(\left(\cos z1 \cdot \cos z0 - \color{blue}{\sin z1} \cdot \sin \left(-z0\right)\right)\right)\right)\right)} \]

      15. lift-sin.f64 N/A

          \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\cos \left(z3 - z2\right) \cdot \frac{1}{2} - \frac{1}{2}\right) - \frac{1}{2} \cdot \left(\mathsf{neg}\left(\left(\cos z1 \cdot \cos z0 - \sin z1 \cdot \color{blue}{\sin \left(-z0\right)}\right)\right)\right)\right)} \]

      16. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\cos \left(z3 - z2\right) \cdot \frac{1}{2} - \frac{1}{2}\right) - \frac{1}{2} \cdot \left(\mathsf{neg}\left(\left(\color{blue}{\cos z1 \cdot \cos z0} - \sin z1 \cdot \sin \left(-z0\right)\right)\right)\right)\right)} \]

      17. sub-negate-rev N/A

          \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\cos \left(z3 - z2\right) \cdot \frac{1}{2} - \frac{1}{2}\right) - \frac{1}{2} \cdot \color{blue}{\left(\sin z1 \cdot \sin \left(-z0\right) - \cos z1 \cdot \cos z0\right)}\right)} \]

      18. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\cos \left(z3 - z2\right) \cdot \frac{1}{2} - \frac{1}{2}\right) - \frac{1}{2} \cdot \left(\sin z1 \cdot \sin \left(-z0\right) - \color{blue}{\cos z1 \cdot \cos z0}\right)\right)} \]

   9. Applied rewrites 73.4%

      \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\cos \left(z3 - z2\right) \cdot \frac{1}{2} - \frac{1}{2}\right) - \color{blue}{\mathsf{304\_z0z1z2z3z4}\left(\frac{1}{2}, \sin z1, \left(-\sin z0\right), \cos z1, \cos z0\right)}\right)} \]

   if -7.9999999999999996e-7 < z0 < 2.8e14

   1. Initial program 60.5%

      \[\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

   3. Applied rewrites 57.8%

      \[\leadsto \sqrt{\color{blue}{\frac{1}{2} - \left(\cos \left(\left(z0 - z1\right) \cdot 1\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(z2 - z3\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)}} \]

   5. Applied rewrites 57.8%

      \[\leadsto \sqrt{\frac{1}{2} - \color{blue}{\left(1 - \frac{\left(\left(\frac{1}{2} - \cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2}\right) \cdot \cos z1\right) \cdot \cos z0}{\cos \left(z1 - z0\right) \cdot \frac{1}{2}}\right) \cdot \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2}\right)}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \color{blue}{\left(1 - \frac{\left(\left(\frac{1}{2} - \cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2}\right) \cdot \cos z1\right) \cdot \cos z0}{\cos \left(z1 - z0\right) \cdot \frac{1}{2}}\right) \cdot \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2}\right)}} \]

      2. lift--.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \color{blue}{\left(1 - \frac{\left(\left(\frac{1}{2} - \cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2}\right) \cdot \cos z1\right) \cdot \cos z0}{\cos \left(z1 - z0\right) \cdot \frac{1}{2}}\right)} \cdot \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2}\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(1 - \color{blue}{\frac{\left(\left(\frac{1}{2} - \cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2}\right) \cdot \cos z1\right) \cdot \cos z0}{\cos \left(z1 - z0\right) \cdot \frac{1}{2}}}\right) \cdot \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2}\right)} \]

      4. sub-to-mult-rev N/A

         \[\leadsto \sqrt{\frac{1}{2} - \color{blue}{\left(\cos \left(z1 - z0\right) \cdot \frac{1}{2} - \left(\left(\frac{1}{2} - \cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2}\right) \cdot \cos z1\right) \cdot \cos z0\right)}} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2} - \color{blue}{\left(\left(\frac{1}{2} - \cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2}\right) \cdot \cos z1\right) \cdot \cos z0}\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2} - \color{blue}{\left(\left(\frac{1}{2} - \cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2}\right) \cdot \cos z1\right)} \cdot \cos z0\right)} \]

      7. associate-*l* N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2} - \color{blue}{\left(\frac{1}{2} - \cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)}\right)} \]

   7. Applied rewrites 57.8%

      \[\leadsto \sqrt{\frac{1}{2} - \color{blue}{\left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\cos \left(z3 - z2\right) \cdot \frac{1}{2} - \frac{1}{2}\right) - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\color{blue}{\cos \left(z3 - z2\right) \cdot \frac{1}{2}} - \frac{1}{2}\right) - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\color{blue}{\frac{1}{2} \cdot \cos \left(z3 - z2\right)} - \frac{1}{2}\right) - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)} \]

      3. lift-cos.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\frac{1}{2} \cdot \color{blue}{\cos \left(z3 - z2\right)} - \frac{1}{2}\right) - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)} \]

      4. lift--.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\frac{1}{2} \cdot \cos \color{blue}{\left(z3 - z2\right)} - \frac{1}{2}\right) - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)} \]

      5. sub-flip N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\frac{1}{2} \cdot \cos \color{blue}{\left(z3 + \left(\mathsf{neg}\left(z2\right)\right)\right)} - \frac{1}{2}\right) - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)} \]

      6. cos-sum N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\frac{1}{2} \cdot \color{blue}{\left(\cos z3 \cdot \cos \left(\mathsf{neg}\left(z2\right)\right) - \sin z3 \cdot \sin \left(\mathsf{neg}\left(z2\right)\right)\right)} - \frac{1}{2}\right) - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)} \]

      7. cos-neg-rev N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\frac{1}{2} \cdot \left(\cos z3 \cdot \color{blue}{\cos z2} - \sin z3 \cdot \sin \left(\mathsf{neg}\left(z2\right)\right)\right) - \frac{1}{2}\right) - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)} \]

      8. lower-304-z0z1z2z3z4 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\color{blue}{\mathsf{304\_z0z1z2z3z4}\left(\frac{1}{2}, \cos z3, \cos z2, \sin z3, \sin \left(\mathsf{neg}\left(z2\right)\right)\right)} - \frac{1}{2}\right) - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)} \]

      9. lower-cos.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\mathsf{304\_z0z1z2z3z4}\left(\frac{1}{2}, \color{blue}{\cos z3}, \cos z2, \sin z3, \sin \left(\mathsf{neg}\left(z2\right)\right)\right) - \frac{1}{2}\right) - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)} \]

      10. lower-cos.f64 N/A

          \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\mathsf{304\_z0z1z2z3z4}\left(\frac{1}{2}, \cos z3, \color{blue}{\cos z2}, \sin z3, \sin \left(\mathsf{neg}\left(z2\right)\right)\right) - \frac{1}{2}\right) - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)} \]

      11. lower-sin.f64 N/A

          \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\mathsf{304\_z0z1z2z3z4}\left(\frac{1}{2}, \cos z3, \cos z2, \color{blue}{\sin z3}, \sin \left(\mathsf{neg}\left(z2\right)\right)\right) - \frac{1}{2}\right) - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)} \]

      12. lower-sin.f64 N/A

          \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\mathsf{304\_z0z1z2z3z4}\left(\frac{1}{2}, \cos z3, \cos z2, \sin z3, \color{blue}{\sin \left(\mathsf{neg}\left(z2\right)\right)}\right) - \frac{1}{2}\right) - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)} \]

      13. lower-neg.f64 73.1%

          \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\mathsf{304\_z0z1z2z3z4}\left(\frac{1}{2}, \cos z3, \cos z2, \sin z3, \sin \color{blue}{\left(-z2\right)}\right) - \frac{1}{2}\right) - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)} \]

   9. Applied rewrites 73.1%

      \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\color{blue}{\mathsf{304\_z0z1z2z3z4}\left(\frac{1}{2}, \cos z3, \cos z2, \sin z3, \sin \left(-z2\right)\right)} - \frac{1}{2}\right) - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 83.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \cos z1 \cdot \cos z0\\ t_1 := \sqrt{\frac{1}{2} - \left(t\_0 \cdot \left(\cos \left(z3 - z2\right) \cdot \frac{1}{2} - \frac{1}{2}\right) - \mathsf{304\_z0z1z2z3z4}\left(\frac{1}{2}, \sin z1, \left(-\sin z0\right), \cos z1, \cos z0\right)\right)}\\ \mathbf{if}\;z0 \leq \frac{-944473296573929}{1180591620717411303424}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z0 \leq 280000000000000:\\ \;\;\;\;\sqrt{\frac{1}{2} - \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2} + \left(\left(\cos z2 \cdot \cos z3 + \sin z2 \cdot \sin z3\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (z0 z1 z2 z3)
  :precision binary64
  (let* ((t_0 (* (cos z1) (cos z0)))
       (t_1
        (sqrt
         (-
          1/2
          (-
           (* t_0 (- (* (cos (- z3 z2)) 1/2) 1/2))
           (304-z0z1z2z3z4
            1/2
            (sin z1)
            (- (sin z0))
            (cos z1)
            (cos z0)))))))
  (if (<= z0 -944473296573929/1180591620717411303424)
    t_1
    (if (<= z0 280000000000000)
      (sqrt
       (-
        1/2
        (+
         (* (cos (- z1 z0)) 1/2)
         (*
          (-
           (* (+ (* (cos z2) (cos z3)) (* (sin z2) (sin z3))) 1/2)
           1/2)
          t_0))))
      t_1))))

Derivation

1. Split input into 2 regimes

2. if z0 < -7.9999999999999996e-7 or 2.8e14 < z0

   1. Initial program 60.5%

      \[\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

   3. Applied rewrites 57.8%

      \[\leadsto \sqrt{\color{blue}{\frac{1}{2} - \left(\cos \left(\left(z0 - z1\right) \cdot 1\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(z2 - z3\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)}} \]

   5. Applied rewrites 57.8%

      \[\leadsto \sqrt{\frac{1}{2} - \color{blue}{\left(1 - \frac{\left(\left(\frac{1}{2} - \cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2}\right) \cdot \cos z1\right) \cdot \cos z0}{\cos \left(z1 - z0\right) \cdot \frac{1}{2}}\right) \cdot \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2}\right)}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \color{blue}{\left(1 - \frac{\left(\left(\frac{1}{2} - \cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2}\right) \cdot \cos z1\right) \cdot \cos z0}{\cos \left(z1 - z0\right) \cdot \frac{1}{2}}\right) \cdot \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2}\right)}} \]

      2. lift--.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \color{blue}{\left(1 - \frac{\left(\left(\frac{1}{2} - \cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2}\right) \cdot \cos z1\right) \cdot \cos z0}{\cos \left(z1 - z0\right) \cdot \frac{1}{2}}\right)} \cdot \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2}\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(1 - \color{blue}{\frac{\left(\left(\frac{1}{2} - \cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2}\right) \cdot \cos z1\right) \cdot \cos z0}{\cos \left(z1 - z0\right) \cdot \frac{1}{2}}}\right) \cdot \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2}\right)} \]

      4. sub-to-mult-rev N/A

         \[\leadsto \sqrt{\frac{1}{2} - \color{blue}{\left(\cos \left(z1 - z0\right) \cdot \frac{1}{2} - \left(\left(\frac{1}{2} - \cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2}\right) \cdot \cos z1\right) \cdot \cos z0\right)}} \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2} - \color{blue}{\left(\left(\frac{1}{2} - \cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2}\right) \cdot \cos z1\right) \cdot \cos z0}\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2} - \color{blue}{\left(\left(\frac{1}{2} - \cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2}\right) \cdot \cos z1\right)} \cdot \cos z0\right)} \]

      7. associate-*l* N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2} - \color{blue}{\left(\frac{1}{2} - \cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)}\right)} \]

   7. Applied rewrites 57.8%

      \[\leadsto \sqrt{\frac{1}{2} - \color{blue}{\left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\cos \left(z3 - z2\right) \cdot \frac{1}{2} - \frac{1}{2}\right) - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\cos \left(z3 - z2\right) \cdot \frac{1}{2} - \frac{1}{2}\right) - \color{blue}{\frac{-1}{2} \cdot \cos \left(z1 - z0\right)}\right)} \]

      2. metadata-eval N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\cos \left(z3 - z2\right) \cdot \frac{1}{2} - \frac{1}{2}\right) - \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \cdot \cos \left(z1 - z0\right)\right)} \]

      3. distribute-lft-neg-out N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\cos \left(z3 - z2\right) \cdot \frac{1}{2} - \frac{1}{2}\right) - \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2} \cdot \cos \left(z1 - z0\right)\right)\right)}\right)} \]

      4. distribute-rgt-neg-in N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\cos \left(z3 - z2\right) \cdot \frac{1}{2} - \frac{1}{2}\right) - \color{blue}{\frac{1}{2} \cdot \left(\mathsf{neg}\left(\cos \left(z1 - z0\right)\right)\right)}\right)} \]

      5. lift-cos.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\cos \left(z3 - z2\right) \cdot \frac{1}{2} - \frac{1}{2}\right) - \frac{1}{2} \cdot \left(\mathsf{neg}\left(\color{blue}{\cos \left(z1 - z0\right)}\right)\right)\right)} \]

      6. lift--.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\cos \left(z3 - z2\right) \cdot \frac{1}{2} - \frac{1}{2}\right) - \frac{1}{2} \cdot \left(\mathsf{neg}\left(\cos \color{blue}{\left(z1 - z0\right)}\right)\right)\right)} \]

      7. sub-flip N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\cos \left(z3 - z2\right) \cdot \frac{1}{2} - \frac{1}{2}\right) - \frac{1}{2} \cdot \left(\mathsf{neg}\left(\cos \color{blue}{\left(z1 + \left(\mathsf{neg}\left(z0\right)\right)\right)}\right)\right)\right)} \]

      8. lift-neg.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\cos \left(z3 - z2\right) \cdot \frac{1}{2} - \frac{1}{2}\right) - \frac{1}{2} \cdot \left(\mathsf{neg}\left(\cos \left(z1 + \color{blue}{\left(-z0\right)}\right)\right)\right)\right)} \]

      9. cos-sum-rev N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\cos \left(z3 - z2\right) \cdot \frac{1}{2} - \frac{1}{2}\right) - \frac{1}{2} \cdot \left(\mathsf{neg}\left(\color{blue}{\left(\cos z1 \cdot \cos \left(-z0\right) - \sin z1 \cdot \sin \left(-z0\right)\right)}\right)\right)\right)} \]

      10. lift-cos.f64 N/A

          \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\cos \left(z3 - z2\right) \cdot \frac{1}{2} - \frac{1}{2}\right) - \frac{1}{2} \cdot \left(\mathsf{neg}\left(\left(\color{blue}{\cos z1} \cdot \cos \left(-z0\right) - \sin z1 \cdot \sin \left(-z0\right)\right)\right)\right)\right)} \]

      11. lift-neg.f64 N/A

          \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\cos \left(z3 - z2\right) \cdot \frac{1}{2} - \frac{1}{2}\right) - \frac{1}{2} \cdot \left(\mathsf{neg}\left(\left(\cos z1 \cdot \cos \color{blue}{\left(\mathsf{neg}\left(z0\right)\right)} - \sin z1 \cdot \sin \left(-z0\right)\right)\right)\right)\right)} \]

      12. cos-neg-rev N/A

          \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\cos \left(z3 - z2\right) \cdot \frac{1}{2} - \frac{1}{2}\right) - \frac{1}{2} \cdot \left(\mathsf{neg}\left(\left(\cos z1 \cdot \color{blue}{\cos z0} - \sin z1 \cdot \sin \left(-z0\right)\right)\right)\right)\right)} \]

      13. lift-cos.f64 N/A

          \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\cos \left(z3 - z2\right) \cdot \frac{1}{2} - \frac{1}{2}\right) - \frac{1}{2} \cdot \left(\mathsf{neg}\left(\left(\cos z1 \cdot \color{blue}{\cos z0} - \sin z1 \cdot \sin \left(-z0\right)\right)\right)\right)\right)} \]

      14. lift-sin.f64 N/A

          \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\cos \left(z3 - z2\right) \cdot \frac{1}{2} - \frac{1}{2}\right) - \frac{1}{2} \cdot \left(\mathsf{neg}\left(\left(\cos z1 \cdot \cos z0 - \color{blue}{\sin z1} \cdot \sin \left(-z0\right)\right)\right)\right)\right)} \]

      15. lift-sin.f64 N/A

          \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\cos \left(z3 - z2\right) \cdot \frac{1}{2} - \frac{1}{2}\right) - \frac{1}{2} \cdot \left(\mathsf{neg}\left(\left(\cos z1 \cdot \cos z0 - \sin z1 \cdot \color{blue}{\sin \left(-z0\right)}\right)\right)\right)\right)} \]

      16. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\cos \left(z3 - z2\right) \cdot \frac{1}{2} - \frac{1}{2}\right) - \frac{1}{2} \cdot \left(\mathsf{neg}\left(\left(\color{blue}{\cos z1 \cdot \cos z0} - \sin z1 \cdot \sin \left(-z0\right)\right)\right)\right)\right)} \]

      17. sub-negate-rev N/A

          \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\cos \left(z3 - z2\right) \cdot \frac{1}{2} - \frac{1}{2}\right) - \frac{1}{2} \cdot \color{blue}{\left(\sin z1 \cdot \sin \left(-z0\right) - \cos z1 \cdot \cos z0\right)}\right)} \]

      18. lift-*.f64 N/A

          \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\cos \left(z3 - z2\right) \cdot \frac{1}{2} - \frac{1}{2}\right) - \frac{1}{2} \cdot \left(\sin z1 \cdot \sin \left(-z0\right) - \color{blue}{\cos z1 \cdot \cos z0}\right)\right)} \]

   9. Applied rewrites 73.4%

      \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\cos \left(z3 - z2\right) \cdot \frac{1}{2} - \frac{1}{2}\right) - \color{blue}{\mathsf{304\_z0z1z2z3z4}\left(\frac{1}{2}, \sin z1, \left(-\sin z0\right), \cos z1, \cos z0\right)}\right)} \]

   if -7.9999999999999996e-7 < z0 < 2.8e14

   1. Initial program 60.5%

      \[\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

   3. Applied rewrites 57.8%

      \[\leadsto \sqrt{\color{blue}{\frac{1}{2} - \left(\cos \left(\left(z0 - z1\right) \cdot 1\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(z2 - z3\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)}} \]

   5. Applied rewrites 57.8%

      \[\leadsto \sqrt{\frac{1}{2} - \color{blue}{\left(\cos \left(z1 - z0\right) \cdot \frac{1}{2} + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)}} \]
   6. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2} + \left(\color{blue}{\cos \left(\left(z2 - z3\right) \cdot 1\right)} \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2} + \left(\cos \color{blue}{\left(\left(z2 - z3\right) \cdot 1\right)} \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      3. *-rgt-identity N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2} + \left(\cos \color{blue}{\left(z2 - z3\right)} \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      4. lift--.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2} + \left(\cos \color{blue}{\left(z2 - z3\right)} \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      5. cos-diff N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2} + \left(\color{blue}{\left(\cos z2 \cdot \cos z3 + \sin z2 \cdot \sin z3\right)} \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      6. lower-+.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2} + \left(\color{blue}{\left(\cos z2 \cdot \cos z3 + \sin z2 \cdot \sin z3\right)} \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2} + \left(\left(\color{blue}{\cos z2 \cdot \cos z3} + \sin z2 \cdot \sin z3\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      8. lower-cos.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2} + \left(\left(\color{blue}{\cos z2} \cdot \cos z3 + \sin z2 \cdot \sin z3\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      9. lower-cos.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2} + \left(\left(\cos z2 \cdot \color{blue}{\cos z3} + \sin z2 \cdot \sin z3\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \sqrt{\frac{1}{2} - \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2} + \left(\left(\cos z2 \cdot \cos z3 + \color{blue}{\sin z2 \cdot \sin z3}\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      11. lower-sin.f64 N/A

          \[\leadsto \sqrt{\frac{1}{2} - \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2} + \left(\left(\cos z2 \cdot \cos z3 + \color{blue}{\sin z2} \cdot \sin z3\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      12. lower-sin.f64 73.1%

          \[\leadsto \sqrt{\frac{1}{2} - \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2} + \left(\left(\cos z2 \cdot \cos z3 + \sin z2 \cdot \color{blue}{\sin z3}\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

   7. Applied rewrites 73.1%

      \[\leadsto \sqrt{\frac{1}{2} - \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2} + \left(\color{blue}{\left(\cos z2 \cdot \cos z3 + \sin z2 \cdot \sin z3\right)} \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 76.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(\frac{z2 - z3}{2}\right)\\ \mathbf{if}\;\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot t\_0\right) \cdot t\_0 \leq \frac{3022314549036573}{151115727451828646838272}:\\ \;\;\;\;\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \sin \left(\left(-\left(z0 - z1\right) \cdot 1\right) + \pi \cdot \frac{1}{2}\right)\right) + \left(\left(\sin \left(z0 + \pi \cdot \frac{1}{2}\right) \cdot \cos z1\right) \cdot t\_0\right) \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{2} - \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2} + \left(\left(\cos z2 \cdot \cos z3 + \sin z2 \cdot \sin z3\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)}\\ \end{array} \]

FPCore

(FPCore (z0 z1 z2 z3)
  :precision binary64
  (let* ((t_0 (sin (/ (- z2 z3) 2))))
  (if (<=
       (+
        (- 1/2 (* 1/2 (cos (* 2 (/ (- z0 z1) 2)))))
        (* (* (* (cos z0) (cos z1)) t_0) t_0))
       3022314549036573/151115727451828646838272)
    (sqrt
     (+
      (- 1/2 (* 1/2 (sin (+ (- (* (- z0 z1) 1)) (* PI 1/2)))))
      (* (* (* (sin (+ z0 (* PI 1/2))) (cos z1)) t_0) t_0)))
    (sqrt
     (-
      1/2
      (+
       (* (cos (- z1 z0)) 1/2)
       (*
        (-
         (* (+ (* (cos z2) (cos z3)) (* (sin z2) (sin z3))) 1/2)
         1/2)
        (* (cos z1) (cos z0)))))))))

C

double code(double z0, double z1, double z2, double z3) {
	double t_0 = sin(((z2 - z3) / 2.0));
	double tmp;
	if (((0.5 - (0.5 * cos((2.0 * ((z0 - z1) / 2.0))))) + (((cos(z0) * cos(z1)) * t_0) * t_0)) <= 2e-8) {
		tmp = sqrt(((0.5 - (0.5 * sin((-((z0 - z1) * 1.0) + (((double) M_PI) * 0.5))))) + (((sin((z0 + (((double) M_PI) * 0.5))) * cos(z1)) * t_0) * t_0)));
	} else {
		tmp = sqrt((0.5 - ((cos((z1 - z0)) * 0.5) + (((((cos(z2) * cos(z3)) + (sin(z2) * sin(z3))) * 0.5) - 0.5) * (cos(z1) * cos(z0))))));
	}
	return tmp;
}

Java

public static double code(double z0, double z1, double z2, double z3) {
	double t_0 = Math.sin(((z2 - z3) / 2.0));
	double tmp;
	if (((0.5 - (0.5 * Math.cos((2.0 * ((z0 - z1) / 2.0))))) + (((Math.cos(z0) * Math.cos(z1)) * t_0) * t_0)) <= 2e-8) {
		tmp = Math.sqrt(((0.5 - (0.5 * Math.sin((-((z0 - z1) * 1.0) + (Math.PI * 0.5))))) + (((Math.sin((z0 + (Math.PI * 0.5))) * Math.cos(z1)) * t_0) * t_0)));
	} else {
		tmp = Math.sqrt((0.5 - ((Math.cos((z1 - z0)) * 0.5) + (((((Math.cos(z2) * Math.cos(z3)) + (Math.sin(z2) * Math.sin(z3))) * 0.5) - 0.5) * (Math.cos(z1) * Math.cos(z0))))));
	}
	return tmp;
}

Python

def code(z0, z1, z2, z3):
	t_0 = math.sin(((z2 - z3) / 2.0))
	tmp = 0
	if ((0.5 - (0.5 * math.cos((2.0 * ((z0 - z1) / 2.0))))) + (((math.cos(z0) * math.cos(z1)) * t_0) * t_0)) <= 2e-8:
		tmp = math.sqrt(((0.5 - (0.5 * math.sin((-((z0 - z1) * 1.0) + (math.pi * 0.5))))) + (((math.sin((z0 + (math.pi * 0.5))) * math.cos(z1)) * t_0) * t_0)))
	else:
		tmp = math.sqrt((0.5 - ((math.cos((z1 - z0)) * 0.5) + (((((math.cos(z2) * math.cos(z3)) + (math.sin(z2) * math.sin(z3))) * 0.5) - 0.5) * (math.cos(z1) * math.cos(z0))))))
	return tmp

Julia

function code(z0, z1, z2, z3)
	t_0 = sin(Float64(Float64(z2 - z3) / 2.0))
	tmp = 0.0
	if (Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(Float64(z0 - z1) / 2.0))))) + Float64(Float64(Float64(cos(z0) * cos(z1)) * t_0) * t_0)) <= 2e-8)
		tmp = sqrt(Float64(Float64(0.5 - Float64(0.5 * sin(Float64(Float64(-Float64(Float64(z0 - z1) * 1.0)) + Float64(pi * 0.5))))) + Float64(Float64(Float64(sin(Float64(z0 + Float64(pi * 0.5))) * cos(z1)) * t_0) * t_0)));
	else
		tmp = sqrt(Float64(0.5 - Float64(Float64(cos(Float64(z1 - z0)) * 0.5) + Float64(Float64(Float64(Float64(Float64(cos(z2) * cos(z3)) + Float64(sin(z2) * sin(z3))) * 0.5) - 0.5) * Float64(cos(z1) * cos(z0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z0, z1, z2, z3)
	t_0 = sin(((z2 - z3) / 2.0));
	tmp = 0.0;
	if (((0.5 - (0.5 * cos((2.0 * ((z0 - z1) / 2.0))))) + (((cos(z0) * cos(z1)) * t_0) * t_0)) <= 2e-8)
		tmp = sqrt(((0.5 - (0.5 * sin((-((z0 - z1) * 1.0) + (pi * 0.5))))) + (((sin((z0 + (pi * 0.5))) * cos(z1)) * t_0) * t_0)));
	else
		tmp = sqrt((0.5 - ((cos((z1 - z0)) * 0.5) + (((((cos(z2) * cos(z3)) + (sin(z2) * sin(z3))) * 0.5) - 0.5) * (cos(z1) * cos(z0))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[z0_, z1_, z2_, z3_] := Block[{t$95$0 = N[Sin[N[(N[(z2 - z3), $MachinePrecision] / 2), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(1/2 - N[(1/2 * N[Cos[N[(2 * N[(N[(z0 - z1), $MachinePrecision] / 2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[Cos[z0], $MachinePrecision] * N[Cos[z1], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 3022314549036573/151115727451828646838272], N[Sqrt[N[(N[(1/2 - N[(1/2 * N[Sin[N[((-N[(N[(z0 - z1), $MachinePrecision] * 1), $MachinePrecision]) + N[(Pi * 1/2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[Sin[N[(z0 + N[(Pi * 1/2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[z1], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(1/2 - N[(N[(N[Cos[N[(z1 - z0), $MachinePrecision]], $MachinePrecision] * 1/2), $MachinePrecision] + N[(N[(N[(N[(N[(N[Cos[z2], $MachinePrecision] * N[Cos[z3], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[z2], $MachinePrecision] * N[Sin[z3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1/2), $MachinePrecision] - 1/2), $MachinePrecision] * N[(N[Cos[z1], $MachinePrecision] * N[Cos[z0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 #s(literal 1/2 binary64) (*.f64 #s(literal 1/2 binary64) (cos.f64 (*.f64 #s(literal 2 binary64) (/.f64 (-.f64 z0 z1) #s(literal 2 binary64)))))) (*.f64 (*.f64 (*.f64 (cos.f64 z0) (cos.f64 z1)) (sin.f64 (/.f64 (-.f64 z2 z3) #s(literal 2 binary64)))) (sin.f64 (/.f64 (-.f64 z2 z3) #s(literal 2 binary64))))) < 2e-8

   1. Initial program 60.5%

      \[\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(2 \cdot \frac{z0 - z1}{2}\right)}\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      2. cos-neg-rev N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(\mathsf{neg}\left(2 \cdot \frac{z0 - z1}{2}\right)\right)}\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      3. sin-+PI/2-rev N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      4. lower-sin.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      5. lower-+.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \sin \color{blue}{\left(\left(\mathsf{neg}\left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      6. lower-neg.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \sin \left(\color{blue}{\left(-2 \cdot \frac{z0 - z1}{2}\right)} + \frac{\mathsf{PI}\left(\right)}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \sin \left(\left(-\color{blue}{2 \cdot \frac{z0 - z1}{2}}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      8. *-commutative N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \sin \left(\left(-\color{blue}{\frac{z0 - z1}{2} \cdot 2}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      9. lift-/.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \sin \left(\left(-\color{blue}{\frac{z0 - z1}{2}} \cdot 2\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      10. mult-flip N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \sin \left(\left(-\color{blue}{\left(\left(z0 - z1\right) \cdot \frac{1}{2}\right)} \cdot 2\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      11. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \sin \left(\left(-\left(\left(z0 - z1\right) \cdot \color{blue}{\frac{1}{2}}\right) \cdot 2\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      12. associate-*l* N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \sin \left(\left(-\color{blue}{\left(z0 - z1\right) \cdot \left(\frac{1}{2} \cdot 2\right)}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      13. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \sin \left(\left(-\left(z0 - z1\right) \cdot \color{blue}{1}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      14. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \sin \left(\left(-\left(z0 - z1\right) \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{2}\right)}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      15. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \sin \left(\left(-\left(z0 - z1\right) \cdot \color{blue}{1}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      16. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \sin \left(\left(-\color{blue}{\left(z0 - z1\right) \cdot 1}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      17. mult-flip N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \sin \left(\left(-\left(z0 - z1\right) \cdot 1\right) + \color{blue}{\mathsf{PI}\left(\right) \cdot \frac{1}{2}}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      18. metadata-eval N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \sin \left(\left(-\left(z0 - z1\right) \cdot 1\right) + \mathsf{PI}\left(\right) \cdot \color{blue}{\frac{1}{2}}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      19. lower-*.f64 N/A

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \sin \left(\left(-\left(z0 - z1\right) \cdot 1\right) + \color{blue}{\mathsf{PI}\left(\right) \cdot \frac{1}{2}}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      20. lower-PI.f64 31.0%

          \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \sin \left(\left(-\left(z0 - z1\right) \cdot 1\right) + \color{blue}{\pi} \cdot \frac{1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

   3. Applied rewrites 31.0%

      \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\sin \left(\left(-\left(z0 - z1\right) \cdot 1\right) + \pi \cdot \frac{1}{2}\right)}\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]
   4. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \sin \left(\left(-\left(z0 - z1\right) \cdot 1\right) + \pi \cdot \frac{1}{2}\right)\right) + \left(\left(\color{blue}{\cos z0} \cdot \cos z1\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      2. sin-+PI/2-rev N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \sin \left(\left(-\left(z0 - z1\right) \cdot 1\right) + \pi \cdot \frac{1}{2}\right)\right) + \left(\left(\color{blue}{\sin \left(z0 + \frac{\mathsf{PI}\left(\right)}{2}\right)} \cdot \cos z1\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      3. lower-sin.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \sin \left(\left(-\left(z0 - z1\right) \cdot 1\right) + \pi \cdot \frac{1}{2}\right)\right) + \left(\left(\color{blue}{\sin \left(z0 + \frac{\mathsf{PI}\left(\right)}{2}\right)} \cdot \cos z1\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      4. lower-+.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \sin \left(\left(-\left(z0 - z1\right) \cdot 1\right) + \pi \cdot \frac{1}{2}\right)\right) + \left(\left(\sin \color{blue}{\left(z0 + \frac{\mathsf{PI}\left(\right)}{2}\right)} \cdot \cos z1\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      5. lift-PI.f64 N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \sin \left(\left(-\left(z0 - z1\right) \cdot 1\right) + \pi \cdot \frac{1}{2}\right)\right) + \left(\left(\sin \left(z0 + \frac{\color{blue}{\pi}}{2}\right) \cdot \cos z1\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      6. mult-flip-rev N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \sin \left(\left(-\left(z0 - z1\right) \cdot 1\right) + \pi \cdot \frac{1}{2}\right)\right) + \left(\left(\sin \left(z0 + \color{blue}{\pi \cdot \frac{1}{2}}\right) \cdot \cos z1\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      7. metadata-eval N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \sin \left(\left(-\left(z0 - z1\right) \cdot 1\right) + \pi \cdot \frac{1}{2}\right)\right) + \left(\left(\sin \left(z0 + \pi \cdot \color{blue}{\frac{1}{2}}\right) \cdot \cos z1\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

      8. lift-*.f64 30.8%

         \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \sin \left(\left(-\left(z0 - z1\right) \cdot 1\right) + \pi \cdot \frac{1}{2}\right)\right) + \left(\left(\sin \left(z0 + \color{blue}{\pi \cdot \frac{1}{2}}\right) \cdot \cos z1\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

   5. Applied rewrites 30.8%

      \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \sin \left(\left(-\left(z0 - z1\right) \cdot 1\right) + \pi \cdot \frac{1}{2}\right)\right) + \left(\left(\color{blue}{\sin \left(z0 + \pi \cdot \frac{1}{2}\right)} \cdot \cos z1\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

   if 2e-8 < (+.f64 (-.f64 #s(literal 1/2 binary64) (*.f64 #s(literal 1/2 binary64) (cos.f64 (*.f64 #s(literal 2 binary64) (/.f64 (-.f64 z0 z1) #s(literal 2 binary64)))))) (*.f64 (*.f64 (*.f64 (cos.f64 z0) (cos.f64 z1)) (sin.f64 (/.f64 (-.f64 z2 z3) #s(literal 2 binary64)))) (sin.f64 (/.f64 (-.f64 z2 z3) #s(literal 2 binary64)))))

   1. Initial program 60.5%

      \[\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

   3. Applied rewrites 57.8%

      \[\leadsto \sqrt{\color{blue}{\frac{1}{2} - \left(\cos \left(\left(z0 - z1\right) \cdot 1\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(z2 - z3\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)}} \]

   5. Applied rewrites 57.8%

      \[\leadsto \sqrt{\frac{1}{2} - \color{blue}{\left(\cos \left(z1 - z0\right) \cdot \frac{1}{2} + \left(\cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)}} \]
   6. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2} + \left(\color{blue}{\cos \left(\left(z2 - z3\right) \cdot 1\right)} \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2} + \left(\cos \color{blue}{\left(\left(z2 - z3\right) \cdot 1\right)} \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      3. *-rgt-identity N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2} + \left(\cos \color{blue}{\left(z2 - z3\right)} \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      4. lift--.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2} + \left(\cos \color{blue}{\left(z2 - z3\right)} \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      5. cos-diff N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2} + \left(\color{blue}{\left(\cos z2 \cdot \cos z3 + \sin z2 \cdot \sin z3\right)} \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      6. lower-+.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2} + \left(\color{blue}{\left(\cos z2 \cdot \cos z3 + \sin z2 \cdot \sin z3\right)} \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2} + \left(\left(\color{blue}{\cos z2 \cdot \cos z3} + \sin z2 \cdot \sin z3\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      8. lower-cos.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2} + \left(\left(\color{blue}{\cos z2} \cdot \cos z3 + \sin z2 \cdot \sin z3\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      9. lower-cos.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2} - \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2} + \left(\left(\cos z2 \cdot \color{blue}{\cos z3} + \sin z2 \cdot \sin z3\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \sqrt{\frac{1}{2} - \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2} + \left(\left(\cos z2 \cdot \cos z3 + \color{blue}{\sin z2 \cdot \sin z3}\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      11. lower-sin.f64 N/A

          \[\leadsto \sqrt{\frac{1}{2} - \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2} + \left(\left(\cos z2 \cdot \cos z3 + \color{blue}{\sin z2} \cdot \sin z3\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

      12. lower-sin.f64 73.1%

          \[\leadsto \sqrt{\frac{1}{2} - \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2} + \left(\left(\cos z2 \cdot \cos z3 + \sin z2 \cdot \color{blue}{\sin z3}\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]

   7. Applied rewrites 73.1%

      \[\leadsto \sqrt{\frac{1}{2} - \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2} + \left(\color{blue}{\left(\cos z2 \cdot \cos z3 + \sin z2 \cdot \sin z3\right)} \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 61.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(\frac{z2 - z3}{2}\right)\\ t_1 := \cos \left(z1 - z0\right)\\ \sqrt{\left(\frac{1}{2} - t\_1 \cdot \frac{1}{2}\right) + \left(\frac{1}{\frac{2}{t\_1 + \cos \left(z1 + z0\right)}} \cdot t\_0\right) \cdot t\_0} \end{array} \]

FPCore

(FPCore (z0 z1 z2 z3)
  :precision binary64
  (let* ((t_0 (sin (/ (- z2 z3) 2))) (t_1 (cos (- z1 z0))))
  (sqrt
   (+
    (- 1/2 (* t_1 1/2))
    (* (* (/ 1 (/ 2 (+ t_1 (cos (+ z1 z0))))) t_0) t_0)))))

C

double code(double z0, double z1, double z2, double z3) {
	double t_0 = sin(((z2 - z3) / 2.0));
	double t_1 = cos((z1 - z0));
	return sqrt(((0.5 - (t_1 * 0.5)) + (((1.0 / (2.0 / (t_1 + cos((z1 + z0))))) * t_0) * t_0)));
}

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(z0, z1, z2, z3)
use fmin_fmax_functions
    real(8), intent (in) :: z0
    real(8), intent (in) :: z1
    real(8), intent (in) :: z2
    real(8), intent (in) :: z3
    real(8) :: t_0
    real(8) :: t_1
    t_0 = sin(((z2 - z3) / 2.0d0))
    t_1 = cos((z1 - z0))
    code = sqrt(((0.5d0 - (t_1 * 0.5d0)) + (((1.0d0 / (2.0d0 / (t_1 + cos((z1 + z0))))) * t_0) * t_0)))
end function

Java

public static double code(double z0, double z1, double z2, double z3) {
	double t_0 = Math.sin(((z2 - z3) / 2.0));
	double t_1 = Math.cos((z1 - z0));
	return Math.sqrt(((0.5 - (t_1 * 0.5)) + (((1.0 / (2.0 / (t_1 + Math.cos((z1 + z0))))) * t_0) * t_0)));
}

Python

def code(z0, z1, z2, z3):
	t_0 = math.sin(((z2 - z3) / 2.0))
	t_1 = math.cos((z1 - z0))
	return math.sqrt(((0.5 - (t_1 * 0.5)) + (((1.0 / (2.0 / (t_1 + math.cos((z1 + z0))))) * t_0) * t_0)))

Julia

function code(z0, z1, z2, z3)
	t_0 = sin(Float64(Float64(z2 - z3) / 2.0))
	t_1 = cos(Float64(z1 - z0))
	return sqrt(Float64(Float64(0.5 - Float64(t_1 * 0.5)) + Float64(Float64(Float64(1.0 / Float64(2.0 / Float64(t_1 + cos(Float64(z1 + z0))))) * t_0) * t_0)))
end

MATLAB

function tmp = code(z0, z1, z2, z3)
	t_0 = sin(((z2 - z3) / 2.0));
	t_1 = cos((z1 - z0));
	tmp = sqrt(((0.5 - (t_1 * 0.5)) + (((1.0 / (2.0 / (t_1 + cos((z1 + z0))))) * t_0) * t_0)));
end

Wolfram

code[z0_, z1_, z2_, z3_] := Block[{t$95$0 = N[Sin[N[(N[(z2 - z3), $MachinePrecision] / 2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(z1 - z0), $MachinePrecision]], $MachinePrecision]}, N[Sqrt[N[(N[(1/2 - N[(t$95$1 * 1/2), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(1 / N[(2 / N[(t$95$1 + N[Cos[N[(z1 + z0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Initial program 60.5%

   \[\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]
2. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\color{blue}{\left(\cos z0 \cdot \cos z1\right)} \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

   2. lift-cos.f64 N/A

      \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\color{blue}{\cos z0} \cdot \cos z1\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

   3. lift-cos.f64 N/A

      \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \color{blue}{\cos z1}\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

   4. cos-mult N/A

      \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\color{blue}{\frac{\cos \left(z0 + z1\right) + \cos \left(z0 - z1\right)}{2}} \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

   5. div-flip N/A

      \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\color{blue}{\frac{1}{\frac{2}{\cos \left(z0 + z1\right) + \cos \left(z0 - z1\right)}}} \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

   6. lower-unsound-/.f64 N/A

      \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\color{blue}{\frac{1}{\frac{2}{\cos \left(z0 + z1\right) + \cos \left(z0 - z1\right)}}} \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

   7. lower-unsound-/.f64 N/A

      \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\frac{1}{\color{blue}{\frac{2}{\cos \left(z0 + z1\right) + \cos \left(z0 - z1\right)}}} \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

   8. +-commutative N/A

      \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\frac{1}{\frac{2}{\color{blue}{\cos \left(z0 - z1\right) + \cos \left(z0 + z1\right)}}} \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

   9. lower-+.f64 N/A

      \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\frac{1}{\frac{2}{\color{blue}{\cos \left(z0 - z1\right) + \cos \left(z0 + z1\right)}}} \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

   10. lift--.f64 N/A

       \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\frac{1}{\frac{2}{\cos \color{blue}{\left(z0 - z1\right)} + \cos \left(z0 + z1\right)}} \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

   11. cos-neg-rev N/A

       \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\frac{1}{\frac{2}{\color{blue}{\cos \left(\mathsf{neg}\left(\left(z0 - z1\right)\right)\right)} + \cos \left(z0 + z1\right)}} \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

   12. lower-cos.f64 N/A

       \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\frac{1}{\frac{2}{\color{blue}{\cos \left(\mathsf{neg}\left(\left(z0 - z1\right)\right)\right)} + \cos \left(z0 + z1\right)}} \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

   13. lift--.f64 N/A

       \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\frac{1}{\frac{2}{\cos \left(\mathsf{neg}\left(\color{blue}{\left(z0 - z1\right)}\right)\right) + \cos \left(z0 + z1\right)}} \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

   14. sub-negate-rev N/A

       \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\frac{1}{\frac{2}{\cos \color{blue}{\left(z1 - z0\right)} + \cos \left(z0 + z1\right)}} \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

   15. lower--.f64 N/A

       \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\frac{1}{\frac{2}{\cos \color{blue}{\left(z1 - z0\right)} + \cos \left(z0 + z1\right)}} \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

   16. lower-cos.f64 N/A

       \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\frac{1}{\frac{2}{\cos \left(z1 - z0\right) + \color{blue}{\cos \left(z0 + z1\right)}}} \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

   17. +-commutative N/A

       \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\frac{1}{\frac{2}{\cos \left(z1 - z0\right) + \cos \color{blue}{\left(z1 + z0\right)}}} \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

   18. lower-+.f64 61.0%

       \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\frac{1}{\frac{2}{\cos \left(z1 - z0\right) + \cos \color{blue}{\left(z1 + z0\right)}}} \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

3. Applied rewrites 61.0%

   \[\leadsto \sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\color{blue}{\frac{1}{\frac{2}{\cos \left(z1 - z0\right) + \cos \left(z1 + z0\right)}}} \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]
4. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \sqrt{\left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)}\right) + \left(\frac{1}{\frac{2}{\cos \left(z1 - z0\right) + \cos \left(z1 + z0\right)}} \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

   2. metadata-eval N/A

      \[\leadsto \sqrt{\left(\frac{1}{2} - \color{blue}{\frac{1}{2}} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\frac{1}{\frac{2}{\cos \left(z1 - z0\right) + \cos \left(z1 + z0\right)}} \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

   3. associate-*l/ N/A

      \[\leadsto \sqrt{\left(\frac{1}{2} - \color{blue}{\frac{1 \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)}{2}}\right) + \left(\frac{1}{\frac{2}{\cos \left(z1 - z0\right) + \cos \left(z1 + z0\right)}} \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

5. Applied rewrites 61.0%

   \[\leadsto \sqrt{\left(\frac{1}{2} - \color{blue}{\cos \left(z1 - z0\right) \cdot \frac{1}{2}}\right) + \left(\frac{1}{\frac{2}{\cos \left(z1 - z0\right) + \cos \left(z1 + z0\right)}} \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]
6. Add Preprocessing

Alternative 18: 60.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(\left(z0 - z1\right) \cdot \frac{1}{2}\right)\\ \sqrt{\left(\left(\frac{1}{2} - \cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2}\right) \cdot \cos z1\right) \cdot \cos z0 - \left(-t\_0\right) \cdot t\_0} \end{array} \]

FPCore

(FPCore (z0 z1 z2 z3)
  :precision binary64
  (let* ((t_0 (sin (* (- z0 z1) 1/2))))
  (sqrt
   (-
    (* (* (- 1/2 (* (cos (* (- z2 z3) 1)) 1/2)) (cos z1)) (cos z0))
    (* (- t_0) t_0)))))

C

double code(double z0, double z1, double z2, double z3) {
	double t_0 = sin(((z0 - z1) * 0.5));
	return sqrt(((((0.5 - (cos(((z2 - z3) * 1.0)) * 0.5)) * cos(z1)) * cos(z0)) - (-t_0 * t_0)));
}

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(z0, z1, z2, z3)
use fmin_fmax_functions
    real(8), intent (in) :: z0
    real(8), intent (in) :: z1
    real(8), intent (in) :: z2
    real(8), intent (in) :: z3
    real(8) :: t_0
    t_0 = sin(((z0 - z1) * 0.5d0))
    code = sqrt(((((0.5d0 - (cos(((z2 - z3) * 1.0d0)) * 0.5d0)) * cos(z1)) * cos(z0)) - (-t_0 * t_0)))
end function

Java

public static double code(double z0, double z1, double z2, double z3) {
	double t_0 = Math.sin(((z0 - z1) * 0.5));
	return Math.sqrt(((((0.5 - (Math.cos(((z2 - z3) * 1.0)) * 0.5)) * Math.cos(z1)) * Math.cos(z0)) - (-t_0 * t_0)));
}

Python

def code(z0, z1, z2, z3):
	t_0 = math.sin(((z0 - z1) * 0.5))
	return math.sqrt(((((0.5 - (math.cos(((z2 - z3) * 1.0)) * 0.5)) * math.cos(z1)) * math.cos(z0)) - (-t_0 * t_0)))

Julia

function code(z0, z1, z2, z3)
	t_0 = sin(Float64(Float64(z0 - z1) * 0.5))
	return sqrt(Float64(Float64(Float64(Float64(0.5 - Float64(cos(Float64(Float64(z2 - z3) * 1.0)) * 0.5)) * cos(z1)) * cos(z0)) - Float64(Float64(-t_0) * t_0)))
end

MATLAB

function tmp = code(z0, z1, z2, z3)
	t_0 = sin(((z0 - z1) * 0.5));
	tmp = sqrt(((((0.5 - (cos(((z2 - z3) * 1.0)) * 0.5)) * cos(z1)) * cos(z0)) - (-t_0 * t_0)));
end

Wolfram

code[z0_, z1_, z2_, z3_] := Block[{t$95$0 = N[Sin[N[(N[(z0 - z1), $MachinePrecision] * 1/2), $MachinePrecision]], $MachinePrecision]}, N[Sqrt[N[(N[(N[(N[(1/2 - N[(N[Cos[N[(N[(z2 - z3), $MachinePrecision] * 1), $MachinePrecision]], $MachinePrecision] * 1/2), $MachinePrecision]), $MachinePrecision] * N[Cos[z1], $MachinePrecision]), $MachinePrecision] * N[Cos[z0], $MachinePrecision]), $MachinePrecision] - N[((-t$95$0) * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Initial program 60.5%

   \[\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

3. Applied rewrites 57.8%

   \[\leadsto \sqrt{\color{blue}{\frac{1}{2} - \left(\cos \left(\left(z0 - z1\right) \cdot 1\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(z2 - z3\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)}} \]

5. Applied rewrites 60.4%

   \[\leadsto \sqrt{\color{blue}{\left(\left(\frac{1}{2} - \cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2}\right) \cdot \cos z1\right) \cdot \cos z0 - \left(-\sin \left(\left(z0 - z1\right) \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\left(z0 - z1\right) \cdot \frac{1}{2}\right)}} \]
6. Add Preprocessing

Alternative 19: 58.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(z1 - z0\right)\\ \sqrt{\frac{1}{2} - \left(\frac{\left(\cos \left(z3 - z2\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos \left(z1 + z0\right) + t\_0\right)}{2} - \frac{-1}{2} \cdot t\_0\right)} \end{array} \]

FPCore

(FPCore (z0 z1 z2 z3)
  :precision binary64
  (let* ((t_0 (cos (- z1 z0))))
  (sqrt
   (-
    1/2
    (-
     (/ (* (- (* (cos (- z3 z2)) 1/2) 1/2) (+ (cos (+ z1 z0)) t_0)) 2)
     (* -1/2 t_0))))))

C

double code(double z0, double z1, double z2, double z3) {
	double t_0 = cos((z1 - z0));
	return sqrt((0.5 - (((((cos((z3 - z2)) * 0.5) - 0.5) * (cos((z1 + z0)) + t_0)) / 2.0) - (-0.5 * t_0))));
}

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(z0, z1, z2, z3)
use fmin_fmax_functions
    real(8), intent (in) :: z0
    real(8), intent (in) :: z1
    real(8), intent (in) :: z2
    real(8), intent (in) :: z3
    real(8) :: t_0
    t_0 = cos((z1 - z0))
    code = sqrt((0.5d0 - (((((cos((z3 - z2)) * 0.5d0) - 0.5d0) * (cos((z1 + z0)) + t_0)) / 2.0d0) - ((-0.5d0) * t_0))))
end function

Java

public static double code(double z0, double z1, double z2, double z3) {
	double t_0 = Math.cos((z1 - z0));
	return Math.sqrt((0.5 - (((((Math.cos((z3 - z2)) * 0.5) - 0.5) * (Math.cos((z1 + z0)) + t_0)) / 2.0) - (-0.5 * t_0))));
}

Python

def code(z0, z1, z2, z3):
	t_0 = math.cos((z1 - z0))
	return math.sqrt((0.5 - (((((math.cos((z3 - z2)) * 0.5) - 0.5) * (math.cos((z1 + z0)) + t_0)) / 2.0) - (-0.5 * t_0))))

Julia

function code(z0, z1, z2, z3)
	t_0 = cos(Float64(z1 - z0))
	return sqrt(Float64(0.5 - Float64(Float64(Float64(Float64(Float64(cos(Float64(z3 - z2)) * 0.5) - 0.5) * Float64(cos(Float64(z1 + z0)) + t_0)) / 2.0) - Float64(-0.5 * t_0))))
end

MATLAB

function tmp = code(z0, z1, z2, z3)
	t_0 = cos((z1 - z0));
	tmp = sqrt((0.5 - (((((cos((z3 - z2)) * 0.5) - 0.5) * (cos((z1 + z0)) + t_0)) / 2.0) - (-0.5 * t_0))));
end

Wolfram

code[z0_, z1_, z2_, z3_] := Block[{t$95$0 = N[Cos[N[(z1 - z0), $MachinePrecision]], $MachinePrecision]}, N[Sqrt[N[(1/2 - N[(N[(N[(N[(N[(N[Cos[N[(z3 - z2), $MachinePrecision]], $MachinePrecision] * 1/2), $MachinePrecision] - 1/2), $MachinePrecision] * N[(N[Cos[N[(z1 + z0), $MachinePrecision]], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision] / 2), $MachinePrecision] - N[(-1/2 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Initial program 60.5%

   \[\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

3. Applied rewrites 57.8%

   \[\leadsto \sqrt{\color{blue}{\frac{1}{2} - \left(\cos \left(\left(z0 - z1\right) \cdot 1\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(z2 - z3\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)}} \]

5. Applied rewrites 57.8%

   \[\leadsto \sqrt{\frac{1}{2} - \color{blue}{\left(1 - \frac{\left(\left(\frac{1}{2} - \cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2}\right) \cdot \cos z1\right) \cdot \cos z0}{\cos \left(z1 - z0\right) \cdot \frac{1}{2}}\right) \cdot \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2}\right)}} \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \sqrt{\frac{1}{2} - \color{blue}{\left(1 - \frac{\left(\left(\frac{1}{2} - \cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2}\right) \cdot \cos z1\right) \cdot \cos z0}{\cos \left(z1 - z0\right) \cdot \frac{1}{2}}\right) \cdot \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2}\right)}} \]

   2. lift--.f64 N/A

      \[\leadsto \sqrt{\frac{1}{2} - \color{blue}{\left(1 - \frac{\left(\left(\frac{1}{2} - \cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2}\right) \cdot \cos z1\right) \cdot \cos z0}{\cos \left(z1 - z0\right) \cdot \frac{1}{2}}\right)} \cdot \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2}\right)} \]

   3. lift-/.f64 N/A

      \[\leadsto \sqrt{\frac{1}{2} - \left(1 - \color{blue}{\frac{\left(\left(\frac{1}{2} - \cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2}\right) \cdot \cos z1\right) \cdot \cos z0}{\cos \left(z1 - z0\right) \cdot \frac{1}{2}}}\right) \cdot \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2}\right)} \]

   4. sub-to-mult-rev N/A

      \[\leadsto \sqrt{\frac{1}{2} - \color{blue}{\left(\cos \left(z1 - z0\right) \cdot \frac{1}{2} - \left(\left(\frac{1}{2} - \cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2}\right) \cdot \cos z1\right) \cdot \cos z0\right)}} \]

   5. lift-*.f64 N/A

      \[\leadsto \sqrt{\frac{1}{2} - \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2} - \color{blue}{\left(\left(\frac{1}{2} - \cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2}\right) \cdot \cos z1\right) \cdot \cos z0}\right)} \]

   6. lift-*.f64 N/A

      \[\leadsto \sqrt{\frac{1}{2} - \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2} - \color{blue}{\left(\left(\frac{1}{2} - \cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2}\right) \cdot \cos z1\right)} \cdot \cos z0\right)} \]

   7. associate-*l* N/A

      \[\leadsto \sqrt{\frac{1}{2} - \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2} - \color{blue}{\left(\frac{1}{2} - \cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)}\right)} \]

7. Applied rewrites 57.8%

   \[\leadsto \sqrt{\frac{1}{2} - \color{blue}{\left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\cos \left(z3 - z2\right) \cdot \frac{1}{2} - \frac{1}{2}\right) - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)}} \]
8. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \color{blue}{\left(\cos \left(z3 - z2\right) \cdot \frac{1}{2} - \frac{1}{2}\right)} - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)} \]

   2. sub-to-mult-rev N/A

      \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \color{blue}{\left(\left(1 - \frac{\frac{1}{2}}{\cos \left(z3 - z2\right) \cdot \frac{1}{2}}\right) \cdot \left(\cos \left(z3 - z2\right) \cdot \frac{1}{2}\right)\right)} - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)} \]

   3. lift-/.f64 N/A

      \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\left(1 - \color{blue}{\frac{\frac{1}{2}}{\cos \left(z3 - z2\right) \cdot \frac{1}{2}}}\right) \cdot \left(\cos \left(z3 - z2\right) \cdot \frac{1}{2}\right)\right) - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)} \]

   4. lift--.f64 N/A

      \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\color{blue}{\left(1 - \frac{\frac{1}{2}}{\cos \left(z3 - z2\right) \cdot \frac{1}{2}}\right)} \cdot \left(\cos \left(z3 - z2\right) \cdot \frac{1}{2}\right)\right) - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)} \]

   5. lift-*.f64 57.7%

      \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \color{blue}{\left(\left(1 - \frac{\frac{1}{2}}{\cos \left(z3 - z2\right) \cdot \frac{1}{2}}\right) \cdot \left(\cos \left(z3 - z2\right) \cdot \frac{1}{2}\right)\right)} - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)} \]

   6. lower-*.f64 N/A

      \[\leadsto \sqrt{\frac{1}{2} - \left(\color{blue}{\left(\cos z1 \cdot \cos z0\right) \cdot \left(\left(1 - \frac{\frac{1}{2}}{\cos \left(z3 - z2\right) \cdot \frac{1}{2}}\right) \cdot \left(\cos \left(z3 - z2\right) \cdot \frac{1}{2}\right)\right)} - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)} \]

   7. *-commutative N/A

      \[\leadsto \sqrt{\frac{1}{2} - \left(\color{blue}{\left(\left(1 - \frac{\frac{1}{2}}{\cos \left(z3 - z2\right) \cdot \frac{1}{2}}\right) \cdot \left(\cos \left(z3 - z2\right) \cdot \frac{1}{2}\right)\right) \cdot \left(\cos z1 \cdot \cos z0\right)} - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)} \]

   8. lift-*.f64 N/A

      \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\left(1 - \frac{\frac{1}{2}}{\cos \left(z3 - z2\right) \cdot \frac{1}{2}}\right) \cdot \left(\cos \left(z3 - z2\right) \cdot \frac{1}{2}\right)\right) \cdot \color{blue}{\left(\cos z1 \cdot \cos z0\right)} - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)} \]

   9. lift-cos.f64 N/A

      \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\left(1 - \frac{\frac{1}{2}}{\cos \left(z3 - z2\right) \cdot \frac{1}{2}}\right) \cdot \left(\cos \left(z3 - z2\right) \cdot \frac{1}{2}\right)\right) \cdot \left(\color{blue}{\cos z1} \cdot \cos z0\right) - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)} \]

   10. lift-cos.f64 N/A

       \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\left(1 - \frac{\frac{1}{2}}{\cos \left(z3 - z2\right) \cdot \frac{1}{2}}\right) \cdot \left(\cos \left(z3 - z2\right) \cdot \frac{1}{2}\right)\right) \cdot \left(\cos z1 \cdot \color{blue}{\cos z0}\right) - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)} \]

   11. cos-mult N/A

       \[\leadsto \sqrt{\frac{1}{2} - \left(\left(\left(1 - \frac{\frac{1}{2}}{\cos \left(z3 - z2\right) \cdot \frac{1}{2}}\right) \cdot \left(\cos \left(z3 - z2\right) \cdot \frac{1}{2}\right)\right) \cdot \color{blue}{\frac{\cos \left(z1 + z0\right) + \cos \left(z1 - z0\right)}{2}} - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)} \]

   12. associate-*r/ N/A

       \[\leadsto \sqrt{\frac{1}{2} - \left(\color{blue}{\frac{\left(\left(1 - \frac{\frac{1}{2}}{\cos \left(z3 - z2\right) \cdot \frac{1}{2}}\right) \cdot \left(\cos \left(z3 - z2\right) \cdot \frac{1}{2}\right)\right) \cdot \left(\cos \left(z1 + z0\right) + \cos \left(z1 - z0\right)\right)}{2}} - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)} \]

   13. lower-/.f64 N/A

       \[\leadsto \sqrt{\frac{1}{2} - \left(\color{blue}{\frac{\left(\left(1 - \frac{\frac{1}{2}}{\cos \left(z3 - z2\right) \cdot \frac{1}{2}}\right) \cdot \left(\cos \left(z3 - z2\right) \cdot \frac{1}{2}\right)\right) \cdot \left(\cos \left(z1 + z0\right) + \cos \left(z1 - z0\right)\right)}{2}} - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)} \]

9. Applied rewrites 58.3%

   \[\leadsto \sqrt{\frac{1}{2} - \left(\color{blue}{\frac{\left(\cos \left(z3 - z2\right) \cdot \frac{1}{2} - \frac{1}{2}\right) \cdot \left(\cos \left(z1 + z0\right) + \cos \left(z1 - z0\right)\right)}{2}} - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)} \]
10. Add Preprocessing

Alternative 20: 57.8% accurate, 1.3× speedup

Math

\[\sqrt{\frac{1}{2} - \left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\cos \left(z3 - z2\right) \cdot \frac{1}{2} - \frac{1}{2}\right) - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)} \]

FPCore

(FPCore (z0 z1 z2 z3)
  :precision binary64
  (sqrt
 (-
  1/2
  (-
   (* (* (cos z1) (cos z0)) (- (* (cos (- z3 z2)) 1/2) 1/2))
   (* -1/2 (cos (- z1 z0)))))))

C

double code(double z0, double z1, double z2, double z3) {
	return sqrt((0.5 - (((cos(z1) * cos(z0)) * ((cos((z3 - z2)) * 0.5) - 0.5)) - (-0.5 * cos((z1 - z0))))));
}

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(z0, z1, z2, z3)
use fmin_fmax_functions
    real(8), intent (in) :: z0
    real(8), intent (in) :: z1
    real(8), intent (in) :: z2
    real(8), intent (in) :: z3
    code = sqrt((0.5d0 - (((cos(z1) * cos(z0)) * ((cos((z3 - z2)) * 0.5d0) - 0.5d0)) - ((-0.5d0) * cos((z1 - z0))))))
end function

Java

public static double code(double z0, double z1, double z2, double z3) {
	return Math.sqrt((0.5 - (((Math.cos(z1) * Math.cos(z0)) * ((Math.cos((z3 - z2)) * 0.5) - 0.5)) - (-0.5 * Math.cos((z1 - z0))))));
}

Python

def code(z0, z1, z2, z3):
	return math.sqrt((0.5 - (((math.cos(z1) * math.cos(z0)) * ((math.cos((z3 - z2)) * 0.5) - 0.5)) - (-0.5 * math.cos((z1 - z0))))))

Julia

function code(z0, z1, z2, z3)
	return sqrt(Float64(0.5 - Float64(Float64(Float64(cos(z1) * cos(z0)) * Float64(Float64(cos(Float64(z3 - z2)) * 0.5) - 0.5)) - Float64(-0.5 * cos(Float64(z1 - z0))))))
end

MATLAB

function tmp = code(z0, z1, z2, z3)
	tmp = sqrt((0.5 - (((cos(z1) * cos(z0)) * ((cos((z3 - z2)) * 0.5) - 0.5)) - (-0.5 * cos((z1 - z0))))));
end

Wolfram

code[z0_, z1_, z2_, z3_] := N[Sqrt[N[(1/2 - N[(N[(N[(N[Cos[z1], $MachinePrecision] * N[Cos[z0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[N[(z3 - z2), $MachinePrecision]], $MachinePrecision] * 1/2), $MachinePrecision] - 1/2), $MachinePrecision]), $MachinePrecision] - N[(-1/2 * N[Cos[N[(z1 - z0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 60.5%

   \[\sqrt{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \frac{z0 - z1}{2}\right)\right) + \left(\left(\cos z0 \cdot \cos z1\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)\right) \cdot \sin \left(\frac{z2 - z3}{2}\right)} \]

3. Applied rewrites 57.8%

   \[\leadsto \sqrt{\color{blue}{\frac{1}{2} - \left(\cos \left(\left(z0 - z1\right) \cdot 1\right) \cdot \frac{1}{2} - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(z2 - z3\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\cos z1 \cdot \cos z0\right)\right)}} \]

5. Applied rewrites 57.8%

   \[\leadsto \sqrt{\frac{1}{2} - \color{blue}{\left(1 - \frac{\left(\left(\frac{1}{2} - \cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2}\right) \cdot \cos z1\right) \cdot \cos z0}{\cos \left(z1 - z0\right) \cdot \frac{1}{2}}\right) \cdot \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2}\right)}} \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \sqrt{\frac{1}{2} - \color{blue}{\left(1 - \frac{\left(\left(\frac{1}{2} - \cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2}\right) \cdot \cos z1\right) \cdot \cos z0}{\cos \left(z1 - z0\right) \cdot \frac{1}{2}}\right) \cdot \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2}\right)}} \]

   2. lift--.f64 N/A

      \[\leadsto \sqrt{\frac{1}{2} - \color{blue}{\left(1 - \frac{\left(\left(\frac{1}{2} - \cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2}\right) \cdot \cos z1\right) \cdot \cos z0}{\cos \left(z1 - z0\right) \cdot \frac{1}{2}}\right)} \cdot \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2}\right)} \]

   3. lift-/.f64 N/A

      \[\leadsto \sqrt{\frac{1}{2} - \left(1 - \color{blue}{\frac{\left(\left(\frac{1}{2} - \cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2}\right) \cdot \cos z1\right) \cdot \cos z0}{\cos \left(z1 - z0\right) \cdot \frac{1}{2}}}\right) \cdot \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2}\right)} \]

   4. sub-to-mult-rev N/A

      \[\leadsto \sqrt{\frac{1}{2} - \color{blue}{\left(\cos \left(z1 - z0\right) \cdot \frac{1}{2} - \left(\left(\frac{1}{2} - \cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2}\right) \cdot \cos z1\right) \cdot \cos z0\right)}} \]

   5. lift-*.f64 N/A

      \[\leadsto \sqrt{\frac{1}{2} - \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2} - \color{blue}{\left(\left(\frac{1}{2} - \cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2}\right) \cdot \cos z1\right) \cdot \cos z0}\right)} \]

   6. lift-*.f64 N/A

      \[\leadsto \sqrt{\frac{1}{2} - \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2} - \color{blue}{\left(\left(\frac{1}{2} - \cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2}\right) \cdot \cos z1\right)} \cdot \cos z0\right)} \]

   7. associate-*l* N/A

      \[\leadsto \sqrt{\frac{1}{2} - \left(\cos \left(z1 - z0\right) \cdot \frac{1}{2} - \color{blue}{\left(\frac{1}{2} - \cos \left(\left(z2 - z3\right) \cdot 1\right) \cdot \frac{1}{2}\right) \cdot \left(\cos z1 \cdot \cos z0\right)}\right)} \]

7. Applied rewrites 57.8%

   \[\leadsto \sqrt{\frac{1}{2} - \color{blue}{\left(\left(\cos z1 \cdot \cos z0\right) \cdot \left(\cos \left(z3 - z2\right) \cdot \frac{1}{2} - \frac{1}{2}\right) - \frac{-1}{2} \cdot \cos \left(z1 - z0\right)\right)}} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025277 -o generate:taylor -o generate:evaluate
(FPCore (z0 z1 z2 z3)
  :name "(sqrt (+ (- 1/2 (* 1/2 (cos (* 2 (/ (- z0 z1) 2))))) (* (* (* (cos z0) (cos z1)) (sin (/ (- z2 z3) 2))) (sin (/ (- z2 z3) 2)))))"
  :precision binary64
  (sqrt (+ (- 1/2 (* 1/2 (cos (* 2 (/ (- z0 z1) 2))))) (* (* (* (cos z0) (cos z1)) (sin (/ (- z2 z3) 2))) (sin (/ (- z2 z3) 2))))))