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

Percentage Accurate: 63.7% → 99.1%

Time: 16.8s

Alternatives: 13

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

double code(double z1, double z0, double z2, double z3) {
	return sqrt(((0.5 * (1.0 + cos((z1 - z0)))) - ((-0.5 * (-1.0 + cos(((z2 - z3) * 1.0)))) * (cos(z1) * cos(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(z1, z0, z2, z3)
use fmin_fmax_functions
    real(8), intent (in) :: z1
    real(8), intent (in) :: z0
    real(8), intent (in) :: z2
    real(8), intent (in) :: z3
    code = sqrt(((0.5d0 * (1.0d0 + cos((z1 - z0)))) - (((-0.5d0) * ((-1.0d0) + cos(((z2 - z3) * 1.0d0)))) * (cos(z1) * cos(z0)))))
end function

Java

public static double code(double z1, double z0, double z2, double z3) {
	return Math.sqrt(((0.5 * (1.0 + Math.cos((z1 - z0)))) - ((-0.5 * (-1.0 + Math.cos(((z2 - z3) * 1.0)))) * (Math.cos(z1) * Math.cos(z0)))));
}

Python

def code(z1, z0, z2, z3):
	return math.sqrt(((0.5 * (1.0 + math.cos((z1 - z0)))) - ((-0.5 * (-1.0 + math.cos(((z2 - z3) * 1.0)))) * (math.cos(z1) * math.cos(z0)))))

Julia

function code(z1, z0, z2, z3)
	return sqrt(Float64(Float64(0.5 * Float64(1.0 + cos(Float64(z1 - z0)))) - Float64(Float64(-0.5 * Float64(-1.0 + cos(Float64(Float64(z2 - z3) * 1.0)))) * Float64(cos(z1) * cos(z0)))))
end

MATLAB

function tmp = code(z1, z0, z2, z3)
	tmp = sqrt(((0.5 * (1.0 + cos((z1 - z0)))) - ((-0.5 * (-1.0 + cos(((z2 - z3) * 1.0)))) * (cos(z1) * cos(z0)))));
end

Wolfram

code[z1_, z0_, z2_, z3_] := N[Sqrt[N[(N[(1/2 * N[(1 + N[Cos[N[(z1 - z0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(-1/2 * N[(-1 + N[Cos[N[(N[(z2 - z3), $MachinePrecision] * 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[z1], $MachinePrecision] * N[Cos[z0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Initial Program: 63.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double z1, double z0, double z2, double z3) {
	return sqrt(((0.5 * (1.0 + cos((z1 - z0)))) - ((-0.5 * (-1.0 + cos(((z2 - z3) * 1.0)))) * (cos(z1) * cos(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(z1, z0, z2, z3)
use fmin_fmax_functions
    real(8), intent (in) :: z1
    real(8), intent (in) :: z0
    real(8), intent (in) :: z2
    real(8), intent (in) :: z3
    code = sqrt(((0.5d0 * (1.0d0 + cos((z1 - z0)))) - (((-0.5d0) * ((-1.0d0) + cos(((z2 - z3) * 1.0d0)))) * (cos(z1) * cos(z0)))))
end function

Java

public static double code(double z1, double z0, double z2, double z3) {
	return Math.sqrt(((0.5 * (1.0 + Math.cos((z1 - z0)))) - ((-0.5 * (-1.0 + Math.cos(((z2 - z3) * 1.0)))) * (Math.cos(z1) * Math.cos(z0)))));
}

Python

def code(z1, z0, z2, z3):
	return math.sqrt(((0.5 * (1.0 + math.cos((z1 - z0)))) - ((-0.5 * (-1.0 + math.cos(((z2 - z3) * 1.0)))) * (math.cos(z1) * math.cos(z0)))))

Julia

function code(z1, z0, z2, z3)
	return sqrt(Float64(Float64(0.5 * Float64(1.0 + cos(Float64(z1 - z0)))) - Float64(Float64(-0.5 * Float64(-1.0 + cos(Float64(Float64(z2 - z3) * 1.0)))) * Float64(cos(z1) * cos(z0)))))
end

MATLAB

function tmp = code(z1, z0, z2, z3)
	tmp = sqrt(((0.5 * (1.0 + cos((z1 - z0)))) - ((-0.5 * (-1.0 + cos(((z2 - z3) * 1.0)))) * (cos(z1) * cos(z0)))));
end

Wolfram

code[z1_, z0_, z2_, z3_] := N[Sqrt[N[(N[(1/2 * N[(1 + N[Cos[N[(z1 - z0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(-1/2 * N[(-1 + N[Cos[N[(N[(z2 - z3), $MachinePrecision] * 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[z1], $MachinePrecision] * N[Cos[z0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Alternative 1: 99.1% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (z1 z0 z2 z3)
  :precision binary64
  (sqrt
 (-
  (* 1/2 (+ (* (cos z0) (cos z1)) (+ (* (sin z0) (sin z1)) 1)))
  (*
   (*
    -1/2
    (+ -1 (* (- (* (tan z3) (tan z2)) -1) (* (cos z2) (cos z3)))))
   (304-z0z1z2z3z4
    (cos z1)
    (sin (- z0))
    (cos (* PI 1/2))
    (cos z0)
    (sin (* PI -1/2)))))))

Derivation

1. Initial program 63.7%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-cos.f64 N/A

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

   4. lift--.f64 N/A

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

   5. cos-diff N/A

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

   6. lift-cos.f64 N/A

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

   7. lift-cos.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. associate-+l+ N/A

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

   10. lower-+.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. lower-+.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 N/A

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

   17. lower-sin.f64 N/A

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

   18. lower-sin.f64 79.0%

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

3. Applied rewrites 79.0%

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

   1. lift-cos.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-rgt-identity N/A

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

   4. lift--.f64 N/A

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

   5. cos-diff N/A

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

   6. *-commutative N/A

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

   7. *-commutative N/A

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

   8. sum-to-mult N/A

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

   9. lower-unsound-*.f64 N/A

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

   10. lower-unsound-+.f64 N/A

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

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

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. lower-sin.f64 N/A

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

   15. lower-sin.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 N/A

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

   18. lower-cos.f64 N/A

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

   19. lower-cos.f64 N/A

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

   20. *-commutative N/A

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

5. Applied rewrites 99.1%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. add-flip N/A

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

   4. metadata-eval N/A

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

   5. lower--.f64 99.1%

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

   6. lift-/.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. times-frac N/A

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

   12. lower-*.f64 N/A

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

   13. lift-sin.f64 N/A

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

   14. lift-cos.f64 N/A

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

   15. quot-tan N/A

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

   16. lower-tan.f64 N/A

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

   17. lift-sin.f64 N/A

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

   18. lift-cos.f64 N/A

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

   19. quot-tan N/A

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

   20. lower-tan.f64 99.1%

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

7. Applied rewrites 99.1%

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

   1. lift-*.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. cos-neg-rev N/A

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

   4. lift-neg.f64 N/A

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

   5. sin-+PI/2-rev N/A

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

   6. lift-PI.f64 N/A

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

   7. mult-flip-rev N/A

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

   8. metadata-eval N/A

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

   9. lift-*.f64 N/A

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

   10. add-flip N/A

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

   11. sin-diff N/A

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

9. Applied rewrites 99.1%

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

Alternative 2: 99.1% accurate, 0.4× speedup

Math

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

FPCore

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

C

double code(double z1, double z0, double z2, double z3) {
	return sqrt(((0.5 * ((cos(z0) * cos(z1)) + ((sin(z0) * sin(z1)) + 1.0))) - ((-0.5 * (-1.0 + (((tan(z3) * tan(z2)) - -1.0) * (cos(z2) * cos(z3))))) * (cos(z1) * cos(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(z1, z0, z2, z3)
use fmin_fmax_functions
    real(8), intent (in) :: z1
    real(8), intent (in) :: z0
    real(8), intent (in) :: z2
    real(8), intent (in) :: z3
    code = sqrt(((0.5d0 * ((cos(z0) * cos(z1)) + ((sin(z0) * sin(z1)) + 1.0d0))) - (((-0.5d0) * ((-1.0d0) + (((tan(z3) * tan(z2)) - (-1.0d0)) * (cos(z2) * cos(z3))))) * (cos(z1) * cos(z0)))))
end function

Java

public static double code(double z1, double z0, double z2, double z3) {
	return Math.sqrt(((0.5 * ((Math.cos(z0) * Math.cos(z1)) + ((Math.sin(z0) * Math.sin(z1)) + 1.0))) - ((-0.5 * (-1.0 + (((Math.tan(z3) * Math.tan(z2)) - -1.0) * (Math.cos(z2) * Math.cos(z3))))) * (Math.cos(z1) * Math.cos(z0)))));
}

Python

def code(z1, z0, z2, z3):
	return math.sqrt(((0.5 * ((math.cos(z0) * math.cos(z1)) + ((math.sin(z0) * math.sin(z1)) + 1.0))) - ((-0.5 * (-1.0 + (((math.tan(z3) * math.tan(z2)) - -1.0) * (math.cos(z2) * math.cos(z3))))) * (math.cos(z1) * math.cos(z0)))))

Julia

function code(z1, z0, z2, z3)
	return sqrt(Float64(Float64(0.5 * Float64(Float64(cos(z0) * cos(z1)) + Float64(Float64(sin(z0) * sin(z1)) + 1.0))) - Float64(Float64(-0.5 * Float64(-1.0 + Float64(Float64(Float64(tan(z3) * tan(z2)) - -1.0) * Float64(cos(z2) * cos(z3))))) * Float64(cos(z1) * cos(z0)))))
end

MATLAB

function tmp = code(z1, z0, z2, z3)
	tmp = sqrt(((0.5 * ((cos(z0) * cos(z1)) + ((sin(z0) * sin(z1)) + 1.0))) - ((-0.5 * (-1.0 + (((tan(z3) * tan(z2)) - -1.0) * (cos(z2) * cos(z3))))) * (cos(z1) * cos(z0)))));
end

Wolfram

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

Derivation

1. Initial program 63.7%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-cos.f64 N/A

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

   4. lift--.f64 N/A

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

   5. cos-diff N/A

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

   6. lift-cos.f64 N/A

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

   7. lift-cos.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. associate-+l+ N/A

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

   10. lower-+.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. lower-+.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 N/A

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

   17. lower-sin.f64 N/A

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

   18. lower-sin.f64 79.0%

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

3. Applied rewrites 79.0%

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

   1. lift-cos.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-rgt-identity N/A

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

   4. lift--.f64 N/A

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

   5. cos-diff N/A

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

   6. *-commutative N/A

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

   7. *-commutative N/A

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

   8. sum-to-mult N/A

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

   9. lower-unsound-*.f64 N/A

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

   10. lower-unsound-+.f64 N/A

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

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

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. lower-sin.f64 N/A

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

   15. lower-sin.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 N/A

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

   18. lower-cos.f64 N/A

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

   19. lower-cos.f64 N/A

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

   20. *-commutative N/A

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

5. Applied rewrites 99.1%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. add-flip N/A

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

   4. metadata-eval N/A

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

   5. lower--.f64 99.1%

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

   6. lift-/.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. times-frac N/A

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

   12. lower-*.f64 N/A

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

   13. lift-sin.f64 N/A

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

   14. lift-cos.f64 N/A

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

   15. quot-tan N/A

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

   16. lower-tan.f64 N/A

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

   17. lift-sin.f64 N/A

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

   18. lift-cos.f64 N/A

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

   19. quot-tan N/A

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

   20. lower-tan.f64 99.1%

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

7. Applied rewrites 99.1%

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

Alternative 3: 99.1% accurate, 0.4× speedup

Math

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

FPCore

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

C

double code(double z1, double z0, double z2, double z3) {
	return sqrt(((0.5 * ((cos(z0) * cos(z1)) + ((sin(z0) * sin(z1)) + 1.0))) - ((-0.5 * ((-1.0 + (cos(z2) * cos(z3))) + (sin(z2) * sin(z3)))) * (cos(z1) * cos(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(z1, z0, z2, z3)
use fmin_fmax_functions
    real(8), intent (in) :: z1
    real(8), intent (in) :: z0
    real(8), intent (in) :: z2
    real(8), intent (in) :: z3
    code = sqrt(((0.5d0 * ((cos(z0) * cos(z1)) + ((sin(z0) * sin(z1)) + 1.0d0))) - (((-0.5d0) * (((-1.0d0) + (cos(z2) * cos(z3))) + (sin(z2) * sin(z3)))) * (cos(z1) * cos(z0)))))
end function

Java

public static double code(double z1, double z0, double z2, double z3) {
	return Math.sqrt(((0.5 * ((Math.cos(z0) * Math.cos(z1)) + ((Math.sin(z0) * Math.sin(z1)) + 1.0))) - ((-0.5 * ((-1.0 + (Math.cos(z2) * Math.cos(z3))) + (Math.sin(z2) * Math.sin(z3)))) * (Math.cos(z1) * Math.cos(z0)))));
}

Python

def code(z1, z0, z2, z3):
	return math.sqrt(((0.5 * ((math.cos(z0) * math.cos(z1)) + ((math.sin(z0) * math.sin(z1)) + 1.0))) - ((-0.5 * ((-1.0 + (math.cos(z2) * math.cos(z3))) + (math.sin(z2) * math.sin(z3)))) * (math.cos(z1) * math.cos(z0)))))

Julia

function code(z1, z0, z2, z3)
	return sqrt(Float64(Float64(0.5 * Float64(Float64(cos(z0) * cos(z1)) + Float64(Float64(sin(z0) * sin(z1)) + 1.0))) - Float64(Float64(-0.5 * Float64(Float64(-1.0 + Float64(cos(z2) * cos(z3))) + Float64(sin(z2) * sin(z3)))) * Float64(cos(z1) * cos(z0)))))
end

MATLAB

function tmp = code(z1, z0, z2, z3)
	tmp = sqrt(((0.5 * ((cos(z0) * cos(z1)) + ((sin(z0) * sin(z1)) + 1.0))) - ((-0.5 * ((-1.0 + (cos(z2) * cos(z3))) + (sin(z2) * sin(z3)))) * (cos(z1) * cos(z0)))));
end

Wolfram

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

Derivation

1. Initial program 63.7%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-cos.f64 N/A

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

   4. lift--.f64 N/A

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

   5. cos-diff N/A

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

   6. lift-cos.f64 N/A

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

   7. lift-cos.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. associate-+l+ N/A

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

   10. lower-+.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. lower-+.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 N/A

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

   17. lower-sin.f64 N/A

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

   18. lower-sin.f64 79.0%

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

3. Applied rewrites 79.0%

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

   1. lift-+.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. *-rgt-identity N/A

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

   5. lift--.f64 N/A

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

   6. cos-diff N/A

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

   7. *-commutative N/A

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

   8. *-commutative N/A

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

   9. associate-+r+ N/A

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

   10. lower-+.f64 N/A

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

   11. lower-+.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. lower-cos.f64 N/A

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

   15. lower-cos.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 N/A

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

   18. lower-sin.f64 N/A

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

   19. lower-sin.f64 99.1%

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

5. Applied rewrites 99.1%

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

Alternative 4: 88.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\mathsf{max}\left(z1, z0\right)\right)\\ t_1 := \cos \left(\mathsf{min}\left(z1, z0\right)\right)\\ t_2 := t\_0 \cdot t\_1\\ t_3 := \sqrt{\frac{1}{2} + \left(\mathsf{304\_z0z1z2z3z4}\left(\frac{1}{2}, t\_1, t\_0, \sin \left(\mathsf{min}\left(z1, z0\right)\right), \sin \left(-\mathsf{max}\left(z1, z0\right)\right)\right) - \left(t\_2 \cdot \frac{-1}{2}\right) \cdot \left(\cos \left(z3 - z2\right) - 1\right)\right)}\\ \mathbf{if}\;\mathsf{max}\left(z1, z0\right) \leq \frac{-6165521680034609}{604462909807314587353088}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;\mathsf{max}\left(z1, z0\right) \leq \frac{7791904696734915}{590295810358705651712}:\\ \;\;\;\;\sqrt{\left(\cos \left(\mathsf{max}\left(z1, z0\right) - \mathsf{min}\left(z1, z0\right)\right) - -1\right) \cdot \frac{1}{2} - \left(\left(\left(\tan z2 \cdot \tan z3 - -1\right) \cdot \left(\cos z2 \cdot \cos z3\right) - 1\right) \cdot \frac{-1}{2}\right) \cdot t\_2}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \]

FPCore

(FPCore (z1 z0 z2 z3)
  :precision binary64
  (let* ((t_0 (cos (fmax z1 z0)))
       (t_1 (cos (fmin z1 z0)))
       (t_2 (* t_0 t_1))
       (t_3
        (sqrt
         (+
          1/2
          (-
           (304-z0z1z2z3z4
            1/2
            t_1
            t_0
            (sin (fmin z1 z0))
            (sin (- (fmax z1 z0))))
           (* (* t_2 -1/2) (- (cos (- z3 z2)) 1)))))))
  (if (<= (fmax z1 z0) -6165521680034609/604462909807314587353088)
    t_3
    (if (<= (fmax z1 z0) 7791904696734915/590295810358705651712)
      (sqrt
       (-
        (* (- (cos (- (fmax z1 z0) (fmin z1 z0))) -1) 1/2)
        (*
         (*
          (- (* (- (* (tan z2) (tan z3)) -1) (* (cos z2) (cos z3))) 1)
          -1/2)
         t_2)))
      t_3))))

Derivation

1. Split input into 2 regimes

2. if z0 < -1.02e-8 or 1.3200000000000001e-5 < z0

   1. Initial program 63.7%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. distribute-rgt-in N/A

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

      5. metadata-eval N/A

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

      6. associate--l+ N/A

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

      7. lower-+.f64 N/A

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

      8. lower--.f64 N/A

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

   3. Applied rewrites 63.8%

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

   5. Applied rewrites 79.0%

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

   if -1.02e-8 < z0 < 1.3200000000000001e-5

   1. Initial program 63.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-cos.f64 N/A

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

      4. lift--.f64 N/A

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

      5. cos-diff N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-+l+ N/A

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

      10. lower-+.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-+.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-sin.f64 N/A

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

      18. lower-sin.f64 79.0%

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

   3. Applied rewrites 79.0%

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

      1. lift-cos.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-rgt-identity N/A

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

      4. lift--.f64 N/A

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

      5. cos-diff N/A

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

      6. *-commutative N/A

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

      7. *-commutative N/A

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

      8. sum-to-mult N/A

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

      9. lower-unsound-*.f64 N/A

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

      10. lower-unsound-+.f64 N/A

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

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

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lower-sin.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-cos.f64 N/A

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

      19. lower-cos.f64 N/A

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

      20. *-commutative N/A

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

   5. Applied rewrites 99.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. metadata-eval N/A

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

      5. lower--.f64 99.1%

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lift-sin.f64 N/A

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

      14. lift-cos.f64 N/A

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

      15. quot-tan N/A

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

      16. lower-tan.f64 N/A

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

      17. lift-sin.f64 N/A

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

      18. lift-cos.f64 N/A

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

      19. quot-tan N/A

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

      20. lower-tan.f64 99.1%

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

   7. Applied rewrites 99.1%

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

   9. Applied rewrites 78.8%

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

Alternative 5: 88.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\mathsf{max}\left(z1, z0\right)\right)\\ t_1 := \cos \left(\mathsf{min}\left(z1, z0\right)\right)\\ t_2 := \sqrt{\frac{1}{2} + \left(\mathsf{304\_z0z1z2z3z4}\left(\frac{1}{2}, t\_1, t\_0, \sin \left(\mathsf{min}\left(z1, z0\right)\right), \sin \left(-\mathsf{max}\left(z1, z0\right)\right)\right) - \left(\left(t\_0 \cdot t\_1\right) \cdot \frac{-1}{2}\right) \cdot \left(\cos \left(z3 - z2\right) - 1\right)\right)}\\ \mathbf{if}\;\mathsf{max}\left(z1, z0\right) \leq \frac{-6165521680034609}{604462909807314587353088}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\mathsf{max}\left(z1, z0\right) \leq \frac{7791904696734915}{590295810358705651712}:\\ \;\;\;\;\sqrt{\frac{1}{2} \cdot \left(1 + \cos \left(\mathsf{min}\left(z1, z0\right) - \mathsf{max}\left(z1, z0\right)\right)\right) - \left(\frac{-1}{2} \cdot \left(-1 + \left(\cos z3 \cdot \cos z2 + \sin z3 \cdot \sin z2\right)\right)\right) \cdot \left(t\_1 \cdot t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (z1 z0 z2 z3)
  :precision binary64
  (let* ((t_0 (cos (fmax z1 z0)))
       (t_1 (cos (fmin z1 z0)))
       (t_2
        (sqrt
         (+
          1/2
          (-
           (304-z0z1z2z3z4
            1/2
            t_1
            t_0
            (sin (fmin z1 z0))
            (sin (- (fmax z1 z0))))
           (* (* (* t_0 t_1) -1/2) (- (cos (- z3 z2)) 1)))))))
  (if (<= (fmax z1 z0) -6165521680034609/604462909807314587353088)
    t_2
    (if (<= (fmax z1 z0) 7791904696734915/590295810358705651712)
      (sqrt
       (-
        (* 1/2 (+ 1 (cos (- (fmin z1 z0) (fmax z1 z0)))))
        (*
         (*
          -1/2
          (+ -1 (+ (* (cos z3) (cos z2)) (* (sin z3) (sin z2)))))
         (* t_1 t_0))))
      t_2))))

Derivation

1. Split input into 2 regimes

2. if z0 < -1.02e-8 or 1.3200000000000001e-5 < z0

   1. Initial program 63.7%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. distribute-rgt-in N/A

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

      5. metadata-eval N/A

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

      6. associate--l+ N/A

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

      7. lower-+.f64 N/A

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

      8. lower--.f64 N/A

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

   3. Applied rewrites 63.8%

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

   5. Applied rewrites 79.0%

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

   if -1.02e-8 < z0 < 1.3200000000000001e-5

   1. Initial program 63.7%

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

      1. lift-cos.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-rgt-identity N/A

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

      4. lift--.f64 N/A

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

      5. sub-negate-rev N/A

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

      6. cos-neg N/A

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

      7. cos-diff N/A

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

      8. cos-neg-rev N/A

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

      9. lower-+.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. cos-neg-rev N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-sin.f64 N/A

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

      16. lower-sin.f64 78.8%

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

   3. Applied rewrites 78.8%

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

Alternative 6: 88.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\mathsf{max}\left(z1, z0\right)\right)\\ t_1 := \cos \left(\mathsf{min}\left(z1, z0\right)\right)\\ t_2 := \sin \left(\mathsf{max}\left(z1, z0\right)\right)\\ t_3 := \sin \left(\mathsf{min}\left(z1, z0\right)\right)\\ t_4 := \cos \left(z3 - z2\right) - 1\\ t_5 := t\_0 \cdot t\_1\\ \mathbf{if}\;\mathsf{max}\left(z1, z0\right) \leq \frac{-6165521680034609}{604462909807314587353088}:\\ \;\;\;\;\sqrt{\frac{1}{2} \cdot \left(t\_5 + \left(t\_2 \cdot t\_3 + 1\right)\right) - t\_4 \cdot \left(\left(\frac{-1}{2} \cdot t\_0\right) \cdot t\_1\right)}\\ \mathbf{elif}\;\mathsf{max}\left(z1, z0\right) \leq \frac{7791904696734915}{590295810358705651712}:\\ \;\;\;\;\sqrt{\frac{1}{2} \cdot \left(1 + \cos \left(\mathsf{min}\left(z1, z0\right) - \mathsf{max}\left(z1, z0\right)\right)\right) - \left(\frac{-1}{2} \cdot \left(-1 + \left(\cos z3 \cdot \cos z2 + \sin z3 \cdot \sin z2\right)\right)\right) \cdot \left(t\_1 \cdot t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{2} \cdot \left(1 + \left(t\_5 + t\_3 \cdot t\_2\right)\right) - \left(t\_1 \cdot \left(t\_4 \cdot \frac{-1}{2}\right)\right) \cdot t\_0}\\ \end{array} \]

FPCore

(FPCore (z1 z0 z2 z3)
  :precision binary64
  (let* ((t_0 (cos (fmax z1 z0)))
       (t_1 (cos (fmin z1 z0)))
       (t_2 (sin (fmax z1 z0)))
       (t_3 (sin (fmin z1 z0)))
       (t_4 (- (cos (- z3 z2)) 1))
       (t_5 (* t_0 t_1)))
  (if (<= (fmax z1 z0) -6165521680034609/604462909807314587353088)
    (sqrt
     (-
      (* 1/2 (+ t_5 (+ (* t_2 t_3) 1)))
      (* t_4 (* (* -1/2 t_0) t_1))))
    (if (<= (fmax z1 z0) 7791904696734915/590295810358705651712)
      (sqrt
       (-
        (* 1/2 (+ 1 (cos (- (fmin z1 z0) (fmax z1 z0)))))
        (*
         (*
          -1/2
          (+ -1 (+ (* (cos z3) (cos z2)) (* (sin z3) (sin z2)))))
         (* t_1 t_0))))
      (sqrt
       (-
        (* 1/2 (+ 1 (+ t_5 (* t_3 t_2))))
        (* (* t_1 (* t_4 -1/2)) t_0)))))))

C

double code(double z1, double z0, double z2, double z3) {
	double t_0 = cos(fmax(z1, z0));
	double t_1 = cos(fmin(z1, z0));
	double t_2 = sin(fmax(z1, z0));
	double t_3 = sin(fmin(z1, z0));
	double t_4 = cos((z3 - z2)) - 1.0;
	double t_5 = t_0 * t_1;
	double tmp;
	if (fmax(z1, z0) <= -1.02e-8) {
		tmp = sqrt(((0.5 * (t_5 + ((t_2 * t_3) + 1.0))) - (t_4 * ((-0.5 * t_0) * t_1))));
	} else if (fmax(z1, z0) <= 1.32e-5) {
		tmp = sqrt(((0.5 * (1.0 + cos((fmin(z1, z0) - fmax(z1, z0))))) - ((-0.5 * (-1.0 + ((cos(z3) * cos(z2)) + (sin(z3) * sin(z2))))) * (t_1 * t_0))));
	} else {
		tmp = sqrt(((0.5 * (1.0 + (t_5 + (t_3 * t_2)))) - ((t_1 * (t_4 * -0.5)) * t_0)));
	}
	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(z1, z0, z2, z3)
use fmin_fmax_functions
    real(8), intent (in) :: z1
    real(8), intent (in) :: z0
    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) :: t_4
    real(8) :: t_5
    real(8) :: tmp
    t_0 = cos(fmax(z1, z0))
    t_1 = cos(fmin(z1, z0))
    t_2 = sin(fmax(z1, z0))
    t_3 = sin(fmin(z1, z0))
    t_4 = cos((z3 - z2)) - 1.0d0
    t_5 = t_0 * t_1
    if (fmax(z1, z0) <= (-1.02d-8)) then
        tmp = sqrt(((0.5d0 * (t_5 + ((t_2 * t_3) + 1.0d0))) - (t_4 * (((-0.5d0) * t_0) * t_1))))
    else if (fmax(z1, z0) <= 1.32d-5) then
        tmp = sqrt(((0.5d0 * (1.0d0 + cos((fmin(z1, z0) - fmax(z1, z0))))) - (((-0.5d0) * ((-1.0d0) + ((cos(z3) * cos(z2)) + (sin(z3) * sin(z2))))) * (t_1 * t_0))))
    else
        tmp = sqrt(((0.5d0 * (1.0d0 + (t_5 + (t_3 * t_2)))) - ((t_1 * (t_4 * (-0.5d0))) * t_0)))
    end if
    code = tmp
end function

Java

public static double code(double z1, double z0, double z2, double z3) {
	double t_0 = Math.cos(fmax(z1, z0));
	double t_1 = Math.cos(fmin(z1, z0));
	double t_2 = Math.sin(fmax(z1, z0));
	double t_3 = Math.sin(fmin(z1, z0));
	double t_4 = Math.cos((z3 - z2)) - 1.0;
	double t_5 = t_0 * t_1;
	double tmp;
	if (fmax(z1, z0) <= -1.02e-8) {
		tmp = Math.sqrt(((0.5 * (t_5 + ((t_2 * t_3) + 1.0))) - (t_4 * ((-0.5 * t_0) * t_1))));
	} else if (fmax(z1, z0) <= 1.32e-5) {
		tmp = Math.sqrt(((0.5 * (1.0 + Math.cos((fmin(z1, z0) - fmax(z1, z0))))) - ((-0.5 * (-1.0 + ((Math.cos(z3) * Math.cos(z2)) + (Math.sin(z3) * Math.sin(z2))))) * (t_1 * t_0))));
	} else {
		tmp = Math.sqrt(((0.5 * (1.0 + (t_5 + (t_3 * t_2)))) - ((t_1 * (t_4 * -0.5)) * t_0)));
	}
	return tmp;
}

Python

def code(z1, z0, z2, z3):
	t_0 = math.cos(fmax(z1, z0))
	t_1 = math.cos(fmin(z1, z0))
	t_2 = math.sin(fmax(z1, z0))
	t_3 = math.sin(fmin(z1, z0))
	t_4 = math.cos((z3 - z2)) - 1.0
	t_5 = t_0 * t_1
	tmp = 0
	if fmax(z1, z0) <= -1.02e-8:
		tmp = math.sqrt(((0.5 * (t_5 + ((t_2 * t_3) + 1.0))) - (t_4 * ((-0.5 * t_0) * t_1))))
	elif fmax(z1, z0) <= 1.32e-5:
		tmp = math.sqrt(((0.5 * (1.0 + math.cos((fmin(z1, z0) - fmax(z1, z0))))) - ((-0.5 * (-1.0 + ((math.cos(z3) * math.cos(z2)) + (math.sin(z3) * math.sin(z2))))) * (t_1 * t_0))))
	else:
		tmp = math.sqrt(((0.5 * (1.0 + (t_5 + (t_3 * t_2)))) - ((t_1 * (t_4 * -0.5)) * t_0)))
	return tmp

Julia

function code(z1, z0, z2, z3)
	t_0 = cos(fmax(z1, z0))
	t_1 = cos(fmin(z1, z0))
	t_2 = sin(fmax(z1, z0))
	t_3 = sin(fmin(z1, z0))
	t_4 = Float64(cos(Float64(z3 - z2)) - 1.0)
	t_5 = Float64(t_0 * t_1)
	tmp = 0.0
	if (fmax(z1, z0) <= -1.02e-8)
		tmp = sqrt(Float64(Float64(0.5 * Float64(t_5 + Float64(Float64(t_2 * t_3) + 1.0))) - Float64(t_4 * Float64(Float64(-0.5 * t_0) * t_1))));
	elseif (fmax(z1, z0) <= 1.32e-5)
		tmp = sqrt(Float64(Float64(0.5 * Float64(1.0 + cos(Float64(fmin(z1, z0) - fmax(z1, z0))))) - Float64(Float64(-0.5 * Float64(-1.0 + Float64(Float64(cos(z3) * cos(z2)) + Float64(sin(z3) * sin(z2))))) * Float64(t_1 * t_0))));
	else
		tmp = sqrt(Float64(Float64(0.5 * Float64(1.0 + Float64(t_5 + Float64(t_3 * t_2)))) - Float64(Float64(t_1 * Float64(t_4 * -0.5)) * t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z1, z0, z2, z3)
	t_0 = cos(max(z1, z0));
	t_1 = cos(min(z1, z0));
	t_2 = sin(max(z1, z0));
	t_3 = sin(min(z1, z0));
	t_4 = cos((z3 - z2)) - 1.0;
	t_5 = t_0 * t_1;
	tmp = 0.0;
	if (max(z1, z0) <= -1.02e-8)
		tmp = sqrt(((0.5 * (t_5 + ((t_2 * t_3) + 1.0))) - (t_4 * ((-0.5 * t_0) * t_1))));
	elseif (max(z1, z0) <= 1.32e-5)
		tmp = sqrt(((0.5 * (1.0 + cos((min(z1, z0) - max(z1, z0))))) - ((-0.5 * (-1.0 + ((cos(z3) * cos(z2)) + (sin(z3) * sin(z2))))) * (t_1 * t_0))));
	else
		tmp = sqrt(((0.5 * (1.0 + (t_5 + (t_3 * t_2)))) - ((t_1 * (t_4 * -0.5)) * t_0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[z1_, z0_, z2_, z3_] := Block[{t$95$0 = N[Cos[N[Max[z1, z0], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[Min[z1, z0], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[Max[z1, z0], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sin[N[Min[z1, z0], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(N[Cos[N[(z3 - z2), $MachinePrecision]], $MachinePrecision] - 1), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$0 * t$95$1), $MachinePrecision]}, If[LessEqual[N[Max[z1, z0], $MachinePrecision], -6165521680034609/604462909807314587353088], N[Sqrt[N[(N[(1/2 * N[(t$95$5 + N[(N[(t$95$2 * t$95$3), $MachinePrecision] + 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(t$95$4 * N[(N[(-1/2 * t$95$0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Max[z1, z0], $MachinePrecision], 7791904696734915/590295810358705651712], N[Sqrt[N[(N[(1/2 * N[(1 + N[Cos[N[(N[Min[z1, z0], $MachinePrecision] - N[Max[z1, z0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(-1/2 * N[(-1 + N[(N[(N[Cos[z3], $MachinePrecision] * N[Cos[z2], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[z3], $MachinePrecision] * N[Sin[z2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(1/2 * N[(1 + N[(t$95$5 + N[(t$95$3 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(t$95$1 * N[(t$95$4 * -1/2), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if z0 < -1.02e-8

   1. Initial program 63.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-cos.f64 N/A

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

      4. lift--.f64 N/A

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

      5. cos-diff N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-+l+ N/A

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

      10. lower-+.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-+.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-sin.f64 N/A

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

      18. lower-sin.f64 79.0%

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

   3. Applied rewrites 79.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lift-*.f64 N/A

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

      10. lower-*.f64 79.0%

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

   5. Applied rewrites 79.0%

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

   if -1.02e-8 < z0 < 1.3200000000000001e-5

   1. Initial program 63.7%

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

      1. lift-cos.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-rgt-identity N/A

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

      4. lift--.f64 N/A

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

      5. sub-negate-rev N/A

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

      6. cos-neg N/A

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

      7. cos-diff N/A

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

      8. cos-neg-rev N/A

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

      9. lower-+.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. cos-neg-rev N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-sin.f64 N/A

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

      16. lower-sin.f64 78.8%

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

   3. Applied rewrites 78.8%

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

   if 1.3200000000000001e-5 < z0

   1. Initial program 63.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

   3. Applied rewrites 63.7%

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

      1. lift-cos.f64 N/A

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

      2. lift--.f64 N/A

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

      3. cos-diff N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-+.f64 79.0%

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 79.0%

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 79.0%

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

   5. Applied rewrites 79.0%

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

Alternative 7: 88.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\mathsf{max}\left(z1, z0\right)\right)\\ t_1 := \cos \left(\mathsf{min}\left(z1, z0\right)\right)\\ t_2 := \sin \left(\mathsf{max}\left(z1, z0\right)\right)\\ t_3 := \sin \left(\mathsf{min}\left(z1, z0\right)\right)\\ t_4 := \left(t\_1 \cdot \left(\left(\cos \left(z3 - z2\right) - 1\right) \cdot \frac{-1}{2}\right)\right) \cdot t\_0\\ t_5 := t\_0 \cdot t\_1\\ \mathbf{if}\;\mathsf{max}\left(z1, z0\right) \leq \frac{-6165521680034609}{604462909807314587353088}:\\ \;\;\;\;\sqrt{\frac{1}{2} \cdot \left(t\_5 + \left(t\_2 \cdot t\_3 + 1\right)\right) - t\_4}\\ \mathbf{elif}\;\mathsf{max}\left(z1, z0\right) \leq \frac{7791904696734915}{590295810358705651712}:\\ \;\;\;\;\sqrt{\frac{1}{2} \cdot \left(1 + \cos \left(\mathsf{min}\left(z1, z0\right) - \mathsf{max}\left(z1, z0\right)\right)\right) - \left(\frac{-1}{2} \cdot \left(-1 + \left(\cos z3 \cdot \cos z2 + \sin z3 \cdot \sin z2\right)\right)\right) \cdot \left(t\_1 \cdot t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{2} \cdot \left(1 + \left(t\_5 + t\_3 \cdot t\_2\right)\right) - t\_4}\\ \end{array} \]

FPCore

(FPCore (z1 z0 z2 z3)
  :precision binary64
  (let* ((t_0 (cos (fmax z1 z0)))
       (t_1 (cos (fmin z1 z0)))
       (t_2 (sin (fmax z1 z0)))
       (t_3 (sin (fmin z1 z0)))
       (t_4 (* (* t_1 (* (- (cos (- z3 z2)) 1) -1/2)) t_0))
       (t_5 (* t_0 t_1)))
  (if (<= (fmax z1 z0) -6165521680034609/604462909807314587353088)
    (sqrt (- (* 1/2 (+ t_5 (+ (* t_2 t_3) 1))) t_4))
    (if (<= (fmax z1 z0) 7791904696734915/590295810358705651712)
      (sqrt
       (-
        (* 1/2 (+ 1 (cos (- (fmin z1 z0) (fmax z1 z0)))))
        (*
         (*
          -1/2
          (+ -1 (+ (* (cos z3) (cos z2)) (* (sin z3) (sin z2)))))
         (* t_1 t_0))))
      (sqrt (- (* 1/2 (+ 1 (+ t_5 (* t_3 t_2)))) t_4))))))

C

double code(double z1, double z0, double z2, double z3) {
	double t_0 = cos(fmax(z1, z0));
	double t_1 = cos(fmin(z1, z0));
	double t_2 = sin(fmax(z1, z0));
	double t_3 = sin(fmin(z1, z0));
	double t_4 = (t_1 * ((cos((z3 - z2)) - 1.0) * -0.5)) * t_0;
	double t_5 = t_0 * t_1;
	double tmp;
	if (fmax(z1, z0) <= -1.02e-8) {
		tmp = sqrt(((0.5 * (t_5 + ((t_2 * t_3) + 1.0))) - t_4));
	} else if (fmax(z1, z0) <= 1.32e-5) {
		tmp = sqrt(((0.5 * (1.0 + cos((fmin(z1, z0) - fmax(z1, z0))))) - ((-0.5 * (-1.0 + ((cos(z3) * cos(z2)) + (sin(z3) * sin(z2))))) * (t_1 * t_0))));
	} else {
		tmp = sqrt(((0.5 * (1.0 + (t_5 + (t_3 * t_2)))) - t_4));
	}
	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(z1, z0, z2, z3)
use fmin_fmax_functions
    real(8), intent (in) :: z1
    real(8), intent (in) :: z0
    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) :: t_4
    real(8) :: t_5
    real(8) :: tmp
    t_0 = cos(fmax(z1, z0))
    t_1 = cos(fmin(z1, z0))
    t_2 = sin(fmax(z1, z0))
    t_3 = sin(fmin(z1, z0))
    t_4 = (t_1 * ((cos((z3 - z2)) - 1.0d0) * (-0.5d0))) * t_0
    t_5 = t_0 * t_1
    if (fmax(z1, z0) <= (-1.02d-8)) then
        tmp = sqrt(((0.5d0 * (t_5 + ((t_2 * t_3) + 1.0d0))) - t_4))
    else if (fmax(z1, z0) <= 1.32d-5) then
        tmp = sqrt(((0.5d0 * (1.0d0 + cos((fmin(z1, z0) - fmax(z1, z0))))) - (((-0.5d0) * ((-1.0d0) + ((cos(z3) * cos(z2)) + (sin(z3) * sin(z2))))) * (t_1 * t_0))))
    else
        tmp = sqrt(((0.5d0 * (1.0d0 + (t_5 + (t_3 * t_2)))) - t_4))
    end if
    code = tmp
end function

Java

public static double code(double z1, double z0, double z2, double z3) {
	double t_0 = Math.cos(fmax(z1, z0));
	double t_1 = Math.cos(fmin(z1, z0));
	double t_2 = Math.sin(fmax(z1, z0));
	double t_3 = Math.sin(fmin(z1, z0));
	double t_4 = (t_1 * ((Math.cos((z3 - z2)) - 1.0) * -0.5)) * t_0;
	double t_5 = t_0 * t_1;
	double tmp;
	if (fmax(z1, z0) <= -1.02e-8) {
		tmp = Math.sqrt(((0.5 * (t_5 + ((t_2 * t_3) + 1.0))) - t_4));
	} else if (fmax(z1, z0) <= 1.32e-5) {
		tmp = Math.sqrt(((0.5 * (1.0 + Math.cos((fmin(z1, z0) - fmax(z1, z0))))) - ((-0.5 * (-1.0 + ((Math.cos(z3) * Math.cos(z2)) + (Math.sin(z3) * Math.sin(z2))))) * (t_1 * t_0))));
	} else {
		tmp = Math.sqrt(((0.5 * (1.0 + (t_5 + (t_3 * t_2)))) - t_4));
	}
	return tmp;
}

Python

def code(z1, z0, z2, z3):
	t_0 = math.cos(fmax(z1, z0))
	t_1 = math.cos(fmin(z1, z0))
	t_2 = math.sin(fmax(z1, z0))
	t_3 = math.sin(fmin(z1, z0))
	t_4 = (t_1 * ((math.cos((z3 - z2)) - 1.0) * -0.5)) * t_0
	t_5 = t_0 * t_1
	tmp = 0
	if fmax(z1, z0) <= -1.02e-8:
		tmp = math.sqrt(((0.5 * (t_5 + ((t_2 * t_3) + 1.0))) - t_4))
	elif fmax(z1, z0) <= 1.32e-5:
		tmp = math.sqrt(((0.5 * (1.0 + math.cos((fmin(z1, z0) - fmax(z1, z0))))) - ((-0.5 * (-1.0 + ((math.cos(z3) * math.cos(z2)) + (math.sin(z3) * math.sin(z2))))) * (t_1 * t_0))))
	else:
		tmp = math.sqrt(((0.5 * (1.0 + (t_5 + (t_3 * t_2)))) - t_4))
	return tmp

Julia

function code(z1, z0, z2, z3)
	t_0 = cos(fmax(z1, z0))
	t_1 = cos(fmin(z1, z0))
	t_2 = sin(fmax(z1, z0))
	t_3 = sin(fmin(z1, z0))
	t_4 = Float64(Float64(t_1 * Float64(Float64(cos(Float64(z3 - z2)) - 1.0) * -0.5)) * t_0)
	t_5 = Float64(t_0 * t_1)
	tmp = 0.0
	if (fmax(z1, z0) <= -1.02e-8)
		tmp = sqrt(Float64(Float64(0.5 * Float64(t_5 + Float64(Float64(t_2 * t_3) + 1.0))) - t_4));
	elseif (fmax(z1, z0) <= 1.32e-5)
		tmp = sqrt(Float64(Float64(0.5 * Float64(1.0 + cos(Float64(fmin(z1, z0) - fmax(z1, z0))))) - Float64(Float64(-0.5 * Float64(-1.0 + Float64(Float64(cos(z3) * cos(z2)) + Float64(sin(z3) * sin(z2))))) * Float64(t_1 * t_0))));
	else
		tmp = sqrt(Float64(Float64(0.5 * Float64(1.0 + Float64(t_5 + Float64(t_3 * t_2)))) - t_4));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z1, z0, z2, z3)
	t_0 = cos(max(z1, z0));
	t_1 = cos(min(z1, z0));
	t_2 = sin(max(z1, z0));
	t_3 = sin(min(z1, z0));
	t_4 = (t_1 * ((cos((z3 - z2)) - 1.0) * -0.5)) * t_0;
	t_5 = t_0 * t_1;
	tmp = 0.0;
	if (max(z1, z0) <= -1.02e-8)
		tmp = sqrt(((0.5 * (t_5 + ((t_2 * t_3) + 1.0))) - t_4));
	elseif (max(z1, z0) <= 1.32e-5)
		tmp = sqrt(((0.5 * (1.0 + cos((min(z1, z0) - max(z1, z0))))) - ((-0.5 * (-1.0 + ((cos(z3) * cos(z2)) + (sin(z3) * sin(z2))))) * (t_1 * t_0))));
	else
		tmp = sqrt(((0.5 * (1.0 + (t_5 + (t_3 * t_2)))) - t_4));
	end
	tmp_2 = tmp;
end

Wolfram

code[z1_, z0_, z2_, z3_] := Block[{t$95$0 = N[Cos[N[Max[z1, z0], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[Min[z1, z0], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[Max[z1, z0], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sin[N[Min[z1, z0], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(N[(t$95$1 * N[(N[(N[Cos[N[(z3 - z2), $MachinePrecision]], $MachinePrecision] - 1), $MachinePrecision] * -1/2), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$0 * t$95$1), $MachinePrecision]}, If[LessEqual[N[Max[z1, z0], $MachinePrecision], -6165521680034609/604462909807314587353088], N[Sqrt[N[(N[(1/2 * N[(t$95$5 + N[(N[(t$95$2 * t$95$3), $MachinePrecision] + 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$4), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Max[z1, z0], $MachinePrecision], 7791904696734915/590295810358705651712], N[Sqrt[N[(N[(1/2 * N[(1 + N[Cos[N[(N[Min[z1, z0], $MachinePrecision] - N[Max[z1, z0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(-1/2 * N[(-1 + N[(N[(N[Cos[z3], $MachinePrecision] * N[Cos[z2], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[z3], $MachinePrecision] * N[Sin[z2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(1/2 * N[(1 + N[(t$95$5 + N[(t$95$3 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$4), $MachinePrecision]], $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if z0 < -1.02e-8

   1. Initial program 63.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-cos.f64 N/A

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

      4. lift--.f64 N/A

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

      5. cos-diff N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-+l+ N/A

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

      10. lower-+.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-+.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-sin.f64 N/A

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

      18. lower-sin.f64 79.0%

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

   3. Applied rewrites 79.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 79.0%

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

   if -1.02e-8 < z0 < 1.3200000000000001e-5

   1. Initial program 63.7%

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

      1. lift-cos.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-rgt-identity N/A

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

      4. lift--.f64 N/A

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

      5. sub-negate-rev N/A

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

      6. cos-neg N/A

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

      7. cos-diff N/A

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

      8. cos-neg-rev N/A

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

      9. lower-+.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. cos-neg-rev N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-sin.f64 N/A

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

      16. lower-sin.f64 78.8%

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

   3. Applied rewrites 78.8%

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

   if 1.3200000000000001e-5 < z0

   1. Initial program 63.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

   3. Applied rewrites 63.7%

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

      1. lift-cos.f64 N/A

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

      2. lift--.f64 N/A

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

      3. cos-diff N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-+.f64 79.0%

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 79.0%

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 79.0%

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

   5. Applied rewrites 79.0%

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

Alternative 8: 88.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\mathsf{max}\left(z1, z0\right)\right)\\ t_1 := \cos \left(\mathsf{min}\left(z1, z0\right)\right)\\ t_2 := \sqrt{\frac{1}{2} \cdot \left(1 + \left(t\_0 \cdot t\_1 + \sin \left(\mathsf{min}\left(z1, z0\right)\right) \cdot \sin \left(\mathsf{max}\left(z1, z0\right)\right)\right)\right) - \left(t\_1 \cdot \left(\left(\cos \left(z3 - z2\right) - 1\right) \cdot \frac{-1}{2}\right)\right) \cdot t\_0}\\ \mathbf{if}\;\mathsf{max}\left(z1, z0\right) \leq \frac{-6165521680034609}{604462909807314587353088}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\mathsf{max}\left(z1, z0\right) \leq \frac{7791904696734915}{590295810358705651712}:\\ \;\;\;\;\sqrt{\frac{1}{2} \cdot \left(1 + \cos \left(\mathsf{min}\left(z1, z0\right) - \mathsf{max}\left(z1, z0\right)\right)\right) - \left(\frac{-1}{2} \cdot \left(-1 + \left(\cos z3 \cdot \cos z2 + \sin z3 \cdot \sin z2\right)\right)\right) \cdot \left(t\_1 \cdot t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (z1 z0 z2 z3)
  :precision binary64
  (let* ((t_0 (cos (fmax z1 z0)))
       (t_1 (cos (fmin z1 z0)))
       (t_2
        (sqrt
         (-
          (*
           1/2
           (+
            1
            (+
             (* t_0 t_1)
             (* (sin (fmin z1 z0)) (sin (fmax z1 z0))))))
          (* (* t_1 (* (- (cos (- z3 z2)) 1) -1/2)) t_0)))))
  (if (<= (fmax z1 z0) -6165521680034609/604462909807314587353088)
    t_2
    (if (<= (fmax z1 z0) 7791904696734915/590295810358705651712)
      (sqrt
       (-
        (* 1/2 (+ 1 (cos (- (fmin z1 z0) (fmax z1 z0)))))
        (*
         (*
          -1/2
          (+ -1 (+ (* (cos z3) (cos z2)) (* (sin z3) (sin z2)))))
         (* t_1 t_0))))
      t_2))))

C

double code(double z1, double z0, double z2, double z3) {
	double t_0 = cos(fmax(z1, z0));
	double t_1 = cos(fmin(z1, z0));
	double t_2 = sqrt(((0.5 * (1.0 + ((t_0 * t_1) + (sin(fmin(z1, z0)) * sin(fmax(z1, z0)))))) - ((t_1 * ((cos((z3 - z2)) - 1.0) * -0.5)) * t_0)));
	double tmp;
	if (fmax(z1, z0) <= -1.02e-8) {
		tmp = t_2;
	} else if (fmax(z1, z0) <= 1.32e-5) {
		tmp = sqrt(((0.5 * (1.0 + cos((fmin(z1, z0) - fmax(z1, z0))))) - ((-0.5 * (-1.0 + ((cos(z3) * cos(z2)) + (sin(z3) * sin(z2))))) * (t_1 * t_0))));
	} else {
		tmp = t_2;
	}
	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(z1, z0, z2, z3)
use fmin_fmax_functions
    real(8), intent (in) :: z1
    real(8), intent (in) :: z0
    real(8), intent (in) :: z2
    real(8), intent (in) :: z3
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = cos(fmax(z1, z0))
    t_1 = cos(fmin(z1, z0))
    t_2 = sqrt(((0.5d0 * (1.0d0 + ((t_0 * t_1) + (sin(fmin(z1, z0)) * sin(fmax(z1, z0)))))) - ((t_1 * ((cos((z3 - z2)) - 1.0d0) * (-0.5d0))) * t_0)))
    if (fmax(z1, z0) <= (-1.02d-8)) then
        tmp = t_2
    else if (fmax(z1, z0) <= 1.32d-5) then
        tmp = sqrt(((0.5d0 * (1.0d0 + cos((fmin(z1, z0) - fmax(z1, z0))))) - (((-0.5d0) * ((-1.0d0) + ((cos(z3) * cos(z2)) + (sin(z3) * sin(z2))))) * (t_1 * t_0))))
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double z1, double z0, double z2, double z3) {
	double t_0 = Math.cos(fmax(z1, z0));
	double t_1 = Math.cos(fmin(z1, z0));
	double t_2 = Math.sqrt(((0.5 * (1.0 + ((t_0 * t_1) + (Math.sin(fmin(z1, z0)) * Math.sin(fmax(z1, z0)))))) - ((t_1 * ((Math.cos((z3 - z2)) - 1.0) * -0.5)) * t_0)));
	double tmp;
	if (fmax(z1, z0) <= -1.02e-8) {
		tmp = t_2;
	} else if (fmax(z1, z0) <= 1.32e-5) {
		tmp = Math.sqrt(((0.5 * (1.0 + Math.cos((fmin(z1, z0) - fmax(z1, z0))))) - ((-0.5 * (-1.0 + ((Math.cos(z3) * Math.cos(z2)) + (Math.sin(z3) * Math.sin(z2))))) * (t_1 * t_0))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(z1, z0, z2, z3):
	t_0 = math.cos(fmax(z1, z0))
	t_1 = math.cos(fmin(z1, z0))
	t_2 = math.sqrt(((0.5 * (1.0 + ((t_0 * t_1) + (math.sin(fmin(z1, z0)) * math.sin(fmax(z1, z0)))))) - ((t_1 * ((math.cos((z3 - z2)) - 1.0) * -0.5)) * t_0)))
	tmp = 0
	if fmax(z1, z0) <= -1.02e-8:
		tmp = t_2
	elif fmax(z1, z0) <= 1.32e-5:
		tmp = math.sqrt(((0.5 * (1.0 + math.cos((fmin(z1, z0) - fmax(z1, z0))))) - ((-0.5 * (-1.0 + ((math.cos(z3) * math.cos(z2)) + (math.sin(z3) * math.sin(z2))))) * (t_1 * t_0))))
	else:
		tmp = t_2
	return tmp

Julia

function code(z1, z0, z2, z3)
	t_0 = cos(fmax(z1, z0))
	t_1 = cos(fmin(z1, z0))
	t_2 = sqrt(Float64(Float64(0.5 * Float64(1.0 + Float64(Float64(t_0 * t_1) + Float64(sin(fmin(z1, z0)) * sin(fmax(z1, z0)))))) - Float64(Float64(t_1 * Float64(Float64(cos(Float64(z3 - z2)) - 1.0) * -0.5)) * t_0)))
	tmp = 0.0
	if (fmax(z1, z0) <= -1.02e-8)
		tmp = t_2;
	elseif (fmax(z1, z0) <= 1.32e-5)
		tmp = sqrt(Float64(Float64(0.5 * Float64(1.0 + cos(Float64(fmin(z1, z0) - fmax(z1, z0))))) - Float64(Float64(-0.5 * Float64(-1.0 + Float64(Float64(cos(z3) * cos(z2)) + Float64(sin(z3) * sin(z2))))) * Float64(t_1 * t_0))));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(z1, z0, z2, z3)
	t_0 = cos(max(z1, z0));
	t_1 = cos(min(z1, z0));
	t_2 = sqrt(((0.5 * (1.0 + ((t_0 * t_1) + (sin(min(z1, z0)) * sin(max(z1, z0)))))) - ((t_1 * ((cos((z3 - z2)) - 1.0) * -0.5)) * t_0)));
	tmp = 0.0;
	if (max(z1, z0) <= -1.02e-8)
		tmp = t_2;
	elseif (max(z1, z0) <= 1.32e-5)
		tmp = sqrt(((0.5 * (1.0 + cos((min(z1, z0) - max(z1, z0))))) - ((-0.5 * (-1.0 + ((cos(z3) * cos(z2)) + (sin(z3) * sin(z2))))) * (t_1 * t_0))));
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[z1_, z0_, z2_, z3_] := Block[{t$95$0 = N[Cos[N[Max[z1, z0], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[Min[z1, z0], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(N[(1/2 * N[(1 + N[(N[(t$95$0 * t$95$1), $MachinePrecision] + N[(N[Sin[N[Min[z1, z0], $MachinePrecision]], $MachinePrecision] * N[Sin[N[Max[z1, z0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(t$95$1 * N[(N[(N[Cos[N[(z3 - z2), $MachinePrecision]], $MachinePrecision] - 1), $MachinePrecision] * -1/2), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Max[z1, z0], $MachinePrecision], -6165521680034609/604462909807314587353088], t$95$2, If[LessEqual[N[Max[z1, z0], $MachinePrecision], 7791904696734915/590295810358705651712], N[Sqrt[N[(N[(1/2 * N[(1 + N[Cos[N[(N[Min[z1, z0], $MachinePrecision] - N[Max[z1, z0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(-1/2 * N[(-1 + N[(N[(N[Cos[z3], $MachinePrecision] * N[Cos[z2], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[z3], $MachinePrecision] * N[Sin[z2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if z0 < -1.02e-8 or 1.3200000000000001e-5 < z0

   1. Initial program 63.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

   3. Applied rewrites 63.7%

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

      1. lift-cos.f64 N/A

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

      2. lift--.f64 N/A

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

      3. cos-diff N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-+.f64 79.0%

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 79.0%

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 79.0%

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

   5. Applied rewrites 79.0%

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

   if -1.02e-8 < z0 < 1.3200000000000001e-5

   1. Initial program 63.7%

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

      1. lift-cos.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-rgt-identity N/A

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

      4. lift--.f64 N/A

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

      5. sub-negate-rev N/A

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

      6. cos-neg N/A

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

      7. cos-diff N/A

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

      8. cos-neg-rev N/A

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

      9. lower-+.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. cos-neg-rev N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-sin.f64 N/A

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

      16. lower-sin.f64 78.8%

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

   3. Applied rewrites 78.8%

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

Alternative 9: 78.8% accurate, 0.6× speedup

Math

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

FPCore

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

C

double code(double z1, double z0, double z2, double z3) {
	return sqrt(((0.5 * (1.0 + cos((z1 - z0)))) - ((-0.5 * (-1.0 + ((cos(z3) * cos(z2)) + (sin(z3) * sin(z2))))) * (cos(z1) * cos(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(z1, z0, z2, z3)
use fmin_fmax_functions
    real(8), intent (in) :: z1
    real(8), intent (in) :: z0
    real(8), intent (in) :: z2
    real(8), intent (in) :: z3
    code = sqrt(((0.5d0 * (1.0d0 + cos((z1 - z0)))) - (((-0.5d0) * ((-1.0d0) + ((cos(z3) * cos(z2)) + (sin(z3) * sin(z2))))) * (cos(z1) * cos(z0)))))
end function

Java

public static double code(double z1, double z0, double z2, double z3) {
	return Math.sqrt(((0.5 * (1.0 + Math.cos((z1 - z0)))) - ((-0.5 * (-1.0 + ((Math.cos(z3) * Math.cos(z2)) + (Math.sin(z3) * Math.sin(z2))))) * (Math.cos(z1) * Math.cos(z0)))));
}

Python

def code(z1, z0, z2, z3):
	return math.sqrt(((0.5 * (1.0 + math.cos((z1 - z0)))) - ((-0.5 * (-1.0 + ((math.cos(z3) * math.cos(z2)) + (math.sin(z3) * math.sin(z2))))) * (math.cos(z1) * math.cos(z0)))))

Julia

function code(z1, z0, z2, z3)
	return sqrt(Float64(Float64(0.5 * Float64(1.0 + cos(Float64(z1 - z0)))) - Float64(Float64(-0.5 * Float64(-1.0 + Float64(Float64(cos(z3) * cos(z2)) + Float64(sin(z3) * sin(z2))))) * Float64(cos(z1) * cos(z0)))))
end

MATLAB

function tmp = code(z1, z0, z2, z3)
	tmp = sqrt(((0.5 * (1.0 + cos((z1 - z0)))) - ((-0.5 * (-1.0 + ((cos(z3) * cos(z2)) + (sin(z3) * sin(z2))))) * (cos(z1) * cos(z0)))));
end

Wolfram

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

Derivation

1. Initial program 63.7%

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

   1. lift-cos.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-rgt-identity N/A

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

   4. lift--.f64 N/A

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

   5. sub-negate-rev N/A

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

   6. cos-neg N/A

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

   7. cos-diff N/A

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

   8. cos-neg-rev N/A

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

   9. lower-+.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. cos-neg-rev N/A

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

   12. lower-cos.f64 N/A

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

   13. lower-cos.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-sin.f64 N/A

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

   16. lower-sin.f64 78.8%

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

3. Applied rewrites 78.8%

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

Alternative 10: 64.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double z1, double z0, double z2, double z3) {
	return sqrt(((0.5 * (1.0 + cos((z1 - z0)))) - ((-0.5 * (-1.0 + cos(((z2 - z3) * 1.0)))) * ((cos((z0 + z1)) + cos((z0 - z1))) * 0.5))));
}

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(z1, z0, z2, z3)
use fmin_fmax_functions
    real(8), intent (in) :: z1
    real(8), intent (in) :: z0
    real(8), intent (in) :: z2
    real(8), intent (in) :: z3
    code = sqrt(((0.5d0 * (1.0d0 + cos((z1 - z0)))) - (((-0.5d0) * ((-1.0d0) + cos(((z2 - z3) * 1.0d0)))) * ((cos((z0 + z1)) + cos((z0 - z1))) * 0.5d0))))
end function

Java

public static double code(double z1, double z0, double z2, double z3) {
	return Math.sqrt(((0.5 * (1.0 + Math.cos((z1 - z0)))) - ((-0.5 * (-1.0 + Math.cos(((z2 - z3) * 1.0)))) * ((Math.cos((z0 + z1)) + Math.cos((z0 - z1))) * 0.5))));
}

Python

def code(z1, z0, z2, z3):
	return math.sqrt(((0.5 * (1.0 + math.cos((z1 - z0)))) - ((-0.5 * (-1.0 + math.cos(((z2 - z3) * 1.0)))) * ((math.cos((z0 + z1)) + math.cos((z0 - z1))) * 0.5))))

Julia

function code(z1, z0, z2, z3)
	return sqrt(Float64(Float64(0.5 * Float64(1.0 + cos(Float64(z1 - z0)))) - Float64(Float64(-0.5 * Float64(-1.0 + cos(Float64(Float64(z2 - z3) * 1.0)))) * Float64(Float64(cos(Float64(z0 + z1)) + cos(Float64(z0 - z1))) * 0.5))))
end

MATLAB

function tmp = code(z1, z0, z2, z3)
	tmp = sqrt(((0.5 * (1.0 + cos((z1 - z0)))) - ((-0.5 * (-1.0 + cos(((z2 - z3) * 1.0)))) * ((cos((z0 + z1)) + cos((z0 - z1))) * 0.5))));
end

Wolfram

code[z1_, z0_, z2_, z3_] := N[Sqrt[N[(N[(1/2 * N[(1 + N[Cos[N[(z1 - z0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(-1/2 * N[(-1 + N[Cos[N[(N[(z2 - z3), $MachinePrecision] * 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[N[(z0 + z1), $MachinePrecision]], $MachinePrecision] + N[Cos[N[(z0 - z1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1/2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 63.7%

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

   1. lift-*.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. cos-mult N/A

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

   5. mult-flip N/A

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

   6. metadata-eval N/A

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

   7. lower-*.f64 N/A

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

   8. lift--.f64 N/A

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

   9. lift-cos.f64 N/A

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

   10. lower-+.f64 N/A

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

   11. lower-cos.f64 N/A

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

   12. +-commutative N/A

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

   13. lower-+.f64 64.3%

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

   14. lift-cos.f64 N/A

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

   15. cos-neg-rev N/A

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

   16. lower-cos.f64 N/A

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

   17. lift--.f64 N/A

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

   18. sub-negate-rev N/A

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

   19. lower--.f64 64.3%

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

3. Applied rewrites 64.3%

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

Alternative 11: 64.2% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (z1 z0 z2 z3)
  :precision binary64
  (sqrt
 (fabs
  (-
   (* (- (cos (- z0 z1)) -1) 1/2)
   (* (* (* (cos z0) (cos z1)) -1/2) (- (cos (- z3 z2)) 1))))))

C

double code(double z1, double z0, double z2, double z3) {
	return sqrt(fabs((((cos((z0 - z1)) - -1.0) * 0.5) - (((cos(z0) * cos(z1)) * -0.5) * (cos((z3 - z2)) - 1.0)))));
}

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(z1, z0, z2, z3)
use fmin_fmax_functions
    real(8), intent (in) :: z1
    real(8), intent (in) :: z0
    real(8), intent (in) :: z2
    real(8), intent (in) :: z3
    code = sqrt(abs((((cos((z0 - z1)) - (-1.0d0)) * 0.5d0) - (((cos(z0) * cos(z1)) * (-0.5d0)) * (cos((z3 - z2)) - 1.0d0)))))
end function

Java

public static double code(double z1, double z0, double z2, double z3) {
	return Math.sqrt(Math.abs((((Math.cos((z0 - z1)) - -1.0) * 0.5) - (((Math.cos(z0) * Math.cos(z1)) * -0.5) * (Math.cos((z3 - z2)) - 1.0)))));
}

Python

def code(z1, z0, z2, z3):
	return math.sqrt(math.fabs((((math.cos((z0 - z1)) - -1.0) * 0.5) - (((math.cos(z0) * math.cos(z1)) * -0.5) * (math.cos((z3 - z2)) - 1.0)))))

Julia

function code(z1, z0, z2, z3)
	return sqrt(abs(Float64(Float64(Float64(cos(Float64(z0 - z1)) - -1.0) * 0.5) - Float64(Float64(Float64(cos(z0) * cos(z1)) * -0.5) * Float64(cos(Float64(z3 - z2)) - 1.0)))))
end

MATLAB

function tmp = code(z1, z0, z2, z3)
	tmp = sqrt(abs((((cos((z0 - z1)) - -1.0) * 0.5) - (((cos(z0) * cos(z1)) * -0.5) * (cos((z3 - z2)) - 1.0)))));
end

Wolfram

code[z1_, z0_, z2_, z3_] := N[Sqrt[N[Abs[N[(N[(N[(N[Cos[N[(z0 - z1), $MachinePrecision]], $MachinePrecision] - -1), $MachinePrecision] * 1/2), $MachinePrecision] - N[(N[(N[(N[Cos[z0], $MachinePrecision] * N[Cos[z1], $MachinePrecision]), $MachinePrecision] * -1/2), $MachinePrecision] * N[(N[Cos[N[(z3 - z2), $MachinePrecision]], $MachinePrecision] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 63.7%

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

   1. rem-square-sqrt N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift-sqrt.f64 N/A

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

   4. sqr-abs-rev N/A

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

   5. mul-fabs N/A

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

3. Applied rewrites 64.2%

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

Alternative 12: 63.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double z1, double z0, double z2, double z3) {
	return sqrt(((cos((z0 - z1)) * 0.5) - ((((-0.5 * cos(z0)) * cos(z1)) * (cos((z3 - z2)) - 1.0)) - 0.5)));
}

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(z1, z0, z2, z3)
use fmin_fmax_functions
    real(8), intent (in) :: z1
    real(8), intent (in) :: z0
    real(8), intent (in) :: z2
    real(8), intent (in) :: z3
    code = sqrt(((cos((z0 - z1)) * 0.5d0) - (((((-0.5d0) * cos(z0)) * cos(z1)) * (cos((z3 - z2)) - 1.0d0)) - 0.5d0)))
end function

Java

public static double code(double z1, double z0, double z2, double z3) {
	return Math.sqrt(((Math.cos((z0 - z1)) * 0.5) - ((((-0.5 * Math.cos(z0)) * Math.cos(z1)) * (Math.cos((z3 - z2)) - 1.0)) - 0.5)));
}

Python

def code(z1, z0, z2, z3):
	return math.sqrt(((math.cos((z0 - z1)) * 0.5) - ((((-0.5 * math.cos(z0)) * math.cos(z1)) * (math.cos((z3 - z2)) - 1.0)) - 0.5)))

Julia

function code(z1, z0, z2, z3)
	return sqrt(Float64(Float64(cos(Float64(z0 - z1)) * 0.5) - Float64(Float64(Float64(Float64(-0.5 * cos(z0)) * cos(z1)) * Float64(cos(Float64(z3 - z2)) - 1.0)) - 0.5)))
end

MATLAB

function tmp = code(z1, z0, z2, z3)
	tmp = sqrt(((cos((z0 - z1)) * 0.5) - ((((-0.5 * cos(z0)) * cos(z1)) * (cos((z3 - z2)) - 1.0)) - 0.5)));
end

Wolfram

code[z1_, z0_, z2_, z3_] := N[Sqrt[N[(N[(N[Cos[N[(z0 - z1), $MachinePrecision]], $MachinePrecision] * 1/2), $MachinePrecision] - N[(N[(N[(N[(-1/2 * N[Cos[z0], $MachinePrecision]), $MachinePrecision] * N[Cos[z1], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[N[(z3 - z2), $MachinePrecision]], $MachinePrecision] - 1), $MachinePrecision]), $MachinePrecision] - 1/2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 63.7%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. distribute-rgt-in N/A

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

   5. metadata-eval N/A

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

   6. associate--l+ N/A

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

   7. lower-+.f64 N/A

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

   8. lower--.f64 N/A

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

3. Applied rewrites 63.8%

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

5. Applied rewrites 63.7%

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

Alternative 13: 63.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double z1, double z0, double z2, double z3) {
	return sqrt((0.5 + ((cos((z0 - z1)) * 0.5) - (((cos(z0) * cos(z1)) * -0.5) * (cos((z3 - z2)) - 1.0)))));
}

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(z1, z0, z2, z3)
use fmin_fmax_functions
    real(8), intent (in) :: z1
    real(8), intent (in) :: z0
    real(8), intent (in) :: z2
    real(8), intent (in) :: z3
    code = sqrt((0.5d0 + ((cos((z0 - z1)) * 0.5d0) - (((cos(z0) * cos(z1)) * (-0.5d0)) * (cos((z3 - z2)) - 1.0d0)))))
end function

Java

public static double code(double z1, double z0, double z2, double z3) {
	return Math.sqrt((0.5 + ((Math.cos((z0 - z1)) * 0.5) - (((Math.cos(z0) * Math.cos(z1)) * -0.5) * (Math.cos((z3 - z2)) - 1.0)))));
}

Python

def code(z1, z0, z2, z3):
	return math.sqrt((0.5 + ((math.cos((z0 - z1)) * 0.5) - (((math.cos(z0) * math.cos(z1)) * -0.5) * (math.cos((z3 - z2)) - 1.0)))))

Julia

function code(z1, z0, z2, z3)
	return sqrt(Float64(0.5 + Float64(Float64(cos(Float64(z0 - z1)) * 0.5) - Float64(Float64(Float64(cos(z0) * cos(z1)) * -0.5) * Float64(cos(Float64(z3 - z2)) - 1.0)))))
end

MATLAB

function tmp = code(z1, z0, z2, z3)
	tmp = sqrt((0.5 + ((cos((z0 - z1)) * 0.5) - (((cos(z0) * cos(z1)) * -0.5) * (cos((z3 - z2)) - 1.0)))));
end

Wolfram

code[z1_, z0_, z2_, z3_] := N[Sqrt[N[(1/2 + N[(N[(N[Cos[N[(z0 - z1), $MachinePrecision]], $MachinePrecision] * 1/2), $MachinePrecision] - N[(N[(N[(N[Cos[z0], $MachinePrecision] * N[Cos[z1], $MachinePrecision]), $MachinePrecision] * -1/2), $MachinePrecision] * N[(N[Cos[N[(z3 - z2), $MachinePrecision]], $MachinePrecision] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 63.7%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. distribute-rgt-in N/A

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

   5. metadata-eval N/A

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

   6. associate--l+ N/A

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

   7. lower-+.f64 N/A

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

   8. lower--.f64 N/A

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

3. Applied rewrites 63.8%

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

Reproduce

herbie shell --seed 2025277 -o generate:taylor -o generate:evaluate
(FPCore (z1 z0 z2 z3)
  :name "(sqrt (- (* 1/2 (+ 1 (cos (- z1 z0)))) (* (* -1/2 (+ -1 (cos (* (- z2 z3) 1)))) (* (cos z1) (cos z0)))))"
  :precision binary64
  (sqrt (- (* 1/2 (+ 1 (cos (- z1 z0)))) (* (* -1/2 (+ -1 (cos (* (- z2 z3) 1)))) (* (cos z1) (cos z0))))))