(sin (- (* -314159265359/50000000000 z0) -7853981852531433/5000000000000000))

Percentage Accurate: 57.2% → 57.3%

Time: 2.9s

Alternatives: 11

Speedup: 0.9×

Specification

Math

\[\sin \left(-6.28318530718 \cdot z0 - -1.5707963705062866\right) \]

FPCore

(FPCore (z0)
  :precision binary64
  (sin (- (* -6.28318530718 z0) -1.5707963705062866)))

C

double code(double z0) {
	return sin(((-6.28318530718 * z0) - -1.5707963705062866));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(z0)
use fmin_fmax_functions
    real(8), intent (in) :: z0
    code = sin((((-6.28318530718d0) * z0) - (-1.5707963705062866d0)))
end function

Java

public static double code(double z0) {
	return Math.sin(((-6.28318530718 * z0) - -1.5707963705062866));
}

Python

def code(z0):
	return math.sin(((-6.28318530718 * z0) - -1.5707963705062866))

Julia

function code(z0)
	return sin(Float64(Float64(-6.28318530718 * z0) - -1.5707963705062866))
end

MATLAB

function tmp = code(z0)
	tmp = sin(((-6.28318530718 * z0) - -1.5707963705062866));
end

Wolfram

code[z0_] := N[Sin[N[(N[(-6.28318530718 * z0), $MachinePrecision] - -1.5707963705062866), $MachinePrecision]], $MachinePrecision]

Initial Program: 57.2% accurate, 1.0× speedup

Math

\[\sin \left(-6.28318530718 \cdot z0 - -1.5707963705062866\right) \]

FPCore

(FPCore (z0)
  :precision binary64
  (sin (- (* -6.28318530718 z0) -1.5707963705062866)))

C

double code(double z0) {
	return sin(((-6.28318530718 * z0) - -1.5707963705062866));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(z0)
use fmin_fmax_functions
    real(8), intent (in) :: z0
    code = sin((((-6.28318530718d0) * z0) - (-1.5707963705062866d0)))
end function

Java

public static double code(double z0) {
	return Math.sin(((-6.28318530718 * z0) - -1.5707963705062866));
}

Python

def code(z0):
	return math.sin(((-6.28318530718 * z0) - -1.5707963705062866))

Julia

function code(z0)
	return sin(Float64(Float64(-6.28318530718 * z0) - -1.5707963705062866))
end

MATLAB

function tmp = code(z0)
	tmp = sin(((-6.28318530718 * z0) - -1.5707963705062866));
end

Wolfram

code[z0_] := N[Sin[N[(N[(-6.28318530718 * z0), $MachinePrecision] - -1.5707963705062866), $MachinePrecision]], $MachinePrecision]

Alternative 1: 57.3% accurate, 0.2× speedup

Math

\[\left(1 - \cos \left(-6.28318530718 \cdot z0\right) \cdot \left(\sin -1.5707963705062866 \cdot \frac{1}{\cos \left(-6.28318530718 \cdot z0 - 0.5 \cdot \pi\right) \cdot \cos 1.5707963705062866}\right)\right) \cdot \left(\cos 1.5707963705062866 \cdot \cos \left(\left(1 - \frac{z0 \cdot -6.28318530718}{\pi \cdot 0.5}\right) \cdot \left(\pi \cdot 0.5\right)\right)\right) \]

FPCore

(FPCore (z0)
  :precision binary64
  (*
 (-
  1.0
  (*
   (cos (* -6.28318530718 z0))
   (*
    (sin -1.5707963705062866)
    (/
     1.0
     (*
      (cos (- (* -6.28318530718 z0) (* 0.5 PI)))
      (cos 1.5707963705062866))))))
 (*
  (cos 1.5707963705062866)
  (cos (* (- 1.0 (/ (* z0 -6.28318530718) (* PI 0.5))) (* PI 0.5))))))

C

double code(double z0) {
	return (1.0 - (cos((-6.28318530718 * z0)) * (sin(-1.5707963705062866) * (1.0 / (cos(((-6.28318530718 * z0) - (0.5 * ((double) M_PI)))) * cos(1.5707963705062866)))))) * (cos(1.5707963705062866) * cos(((1.0 - ((z0 * -6.28318530718) / (((double) M_PI) * 0.5))) * (((double) M_PI) * 0.5))));
}

Java

public static double code(double z0) {
	return (1.0 - (Math.cos((-6.28318530718 * z0)) * (Math.sin(-1.5707963705062866) * (1.0 / (Math.cos(((-6.28318530718 * z0) - (0.5 * Math.PI))) * Math.cos(1.5707963705062866)))))) * (Math.cos(1.5707963705062866) * Math.cos(((1.0 - ((z0 * -6.28318530718) / (Math.PI * 0.5))) * (Math.PI * 0.5))));
}

Python

def code(z0):
	return (1.0 - (math.cos((-6.28318530718 * z0)) * (math.sin(-1.5707963705062866) * (1.0 / (math.cos(((-6.28318530718 * z0) - (0.5 * math.pi))) * math.cos(1.5707963705062866)))))) * (math.cos(1.5707963705062866) * math.cos(((1.0 - ((z0 * -6.28318530718) / (math.pi * 0.5))) * (math.pi * 0.5))))

Julia

function code(z0)
	return Float64(Float64(1.0 - Float64(cos(Float64(-6.28318530718 * z0)) * Float64(sin(-1.5707963705062866) * Float64(1.0 / Float64(cos(Float64(Float64(-6.28318530718 * z0) - Float64(0.5 * pi))) * cos(1.5707963705062866)))))) * Float64(cos(1.5707963705062866) * cos(Float64(Float64(1.0 - Float64(Float64(z0 * -6.28318530718) / Float64(pi * 0.5))) * Float64(pi * 0.5)))))
end

MATLAB

function tmp = code(z0)
	tmp = (1.0 - (cos((-6.28318530718 * z0)) * (sin(-1.5707963705062866) * (1.0 / (cos(((-6.28318530718 * z0) - (0.5 * pi))) * cos(1.5707963705062866)))))) * (cos(1.5707963705062866) * cos(((1.0 - ((z0 * -6.28318530718) / (pi * 0.5))) * (pi * 0.5))));
end

Wolfram

code[z0_] := N[(N[(1.0 - N[(N[Cos[N[(-6.28318530718 * z0), $MachinePrecision]], $MachinePrecision] * N[(N[Sin[-1.5707963705062866], $MachinePrecision] * N[(1.0 / N[(N[Cos[N[(N[(-6.28318530718 * z0), $MachinePrecision] - N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[1.5707963705062866], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[1.5707963705062866], $MachinePrecision] * N[Cos[N[(N[(1.0 - N[(N[(z0 * -6.28318530718), $MachinePrecision] / N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 57.2%

   \[\sin \left(-6.28318530718 \cdot z0 - -1.5707963705062866\right) \]
2. Step-by-step derivation

   1. lift-sin.f64 N/A

      \[\leadsto \color{blue}{\sin \left(\frac{-314159265359}{50000000000} \cdot z0 - \frac{-7853981852531433}{5000000000000000}\right)} \]

   2. lift--.f64 N/A

      \[\leadsto \sin \color{blue}{\left(\frac{-314159265359}{50000000000} \cdot z0 - \frac{-7853981852531433}{5000000000000000}\right)} \]

   3. sin-diff N/A

      \[\leadsto \color{blue}{\sin \left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot \cos \frac{-7853981852531433}{5000000000000000} - \cos \left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot \sin \frac{-7853981852531433}{5000000000000000}} \]

   4. sub-to-mult N/A

      \[\leadsto \color{blue}{\left(1 - \frac{\cos \left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot \sin \frac{-7853981852531433}{5000000000000000}}{\sin \left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot \cos \frac{-7853981852531433}{5000000000000000}}\right) \cdot \left(\sin \left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot \cos \frac{-7853981852531433}{5000000000000000}\right)} \]

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

      \[\leadsto \color{blue}{\left(1 - \frac{\cos \left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot \sin \frac{-7853981852531433}{5000000000000000}}{\sin \left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot \cos \frac{-7853981852531433}{5000000000000000}}\right) \cdot \left(\sin \left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot \cos \frac{-7853981852531433}{5000000000000000}\right)} \]

3. Applied rewrites 56.0%

   \[\leadsto \color{blue}{\left(1 - \frac{\sin -1.5707963705062866 \cdot \cos \left(6.28318530718 \cdot z0\right)}{\cos 1.5707963705062866 \cdot \sin \left(z0 \cdot -6.28318530718\right)}\right) \cdot \left(\cos 1.5707963705062866 \cdot \sin \left(z0 \cdot -6.28318530718\right)\right)} \]
4. Step-by-step derivation

   1. lift-sin.f64 N/A

      \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \color{blue}{\sin \left(z0 \cdot \frac{-314159265359}{50000000000}\right)}}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(z0 \cdot \frac{-314159265359}{50000000000}\right)\right) \]

   2. lift-*.f64 N/A

      \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \color{blue}{\left(z0 \cdot \frac{-314159265359}{50000000000}\right)}}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(z0 \cdot \frac{-314159265359}{50000000000}\right)\right) \]

   3. metadata-eval N/A

      \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(z0 \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{314159265359}{50000000000}\right)\right)}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(z0 \cdot \frac{-314159265359}{50000000000}\right)\right) \]

   4. distribute-rgt-neg-in N/A

      \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \color{blue}{\left(\mathsf{neg}\left(z0 \cdot \frac{314159265359}{50000000000}\right)\right)}}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(z0 \cdot \frac{-314159265359}{50000000000}\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(\mathsf{neg}\left(\color{blue}{\frac{314159265359}{50000000000} \cdot z0}\right)\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(z0 \cdot \frac{-314159265359}{50000000000}\right)\right) \]

   6. lift-*.f64 N/A

      \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(\mathsf{neg}\left(\color{blue}{\frac{314159265359}{50000000000} \cdot z0}\right)\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(z0 \cdot \frac{-314159265359}{50000000000}\right)\right) \]

   7. sin-neg-rev N/A

      \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \color{blue}{\left(\mathsf{neg}\left(\sin \left(\frac{314159265359}{50000000000} \cdot z0\right)\right)\right)}}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(z0 \cdot \frac{-314159265359}{50000000000}\right)\right) \]

   8. cos-+PI/2-rev N/A

      \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \color{blue}{\cos \left(\frac{314159265359}{50000000000} \cdot z0 + \frac{\mathsf{PI}\left(\right)}{2}\right)}}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(z0 \cdot \frac{-314159265359}{50000000000}\right)\right) \]

   9. lower-cos.f64 N/A

      \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \color{blue}{\cos \left(\frac{314159265359}{50000000000} \cdot z0 + \frac{\mathsf{PI}\left(\right)}{2}\right)}}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(z0 \cdot \frac{-314159265359}{50000000000}\right)\right) \]

   10. +-commutative N/A

       \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{2} + \frac{314159265359}{50000000000} \cdot z0\right)}}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(z0 \cdot \frac{-314159265359}{50000000000}\right)\right) \]

   11. lower-+.f64 N/A

       \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{2} + \frac{314159265359}{50000000000} \cdot z0\right)}}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(z0 \cdot \frac{-314159265359}{50000000000}\right)\right) \]

   12. lift-PI.f64 N/A

       \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\frac{\color{blue}{\pi}}{2} + \frac{314159265359}{50000000000} \cdot z0\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(z0 \cdot \frac{-314159265359}{50000000000}\right)\right) \]

   13. mult-flip N/A

       \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\color{blue}{\pi \cdot \frac{1}{2}} + \frac{314159265359}{50000000000} \cdot z0\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(z0 \cdot \frac{-314159265359}{50000000000}\right)\right) \]

   14. lower-*.f64 N/A

       \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\color{blue}{\pi \cdot \frac{1}{2}} + \frac{314159265359}{50000000000} \cdot z0\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(z0 \cdot \frac{-314159265359}{50000000000}\right)\right) \]

   15. metadata-eval 9.9%

       \[\leadsto \left(1 - \frac{\sin -1.5707963705062866 \cdot \cos \left(6.28318530718 \cdot z0\right)}{\cos 1.5707963705062866 \cdot \cos \left(\pi \cdot \color{blue}{0.5} + 6.28318530718 \cdot z0\right)}\right) \cdot \left(\cos 1.5707963705062866 \cdot \sin \left(z0 \cdot -6.28318530718\right)\right) \]

   16. lift-*.f64 N/A

       \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + \color{blue}{\frac{314159265359}{50000000000} \cdot z0}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(z0 \cdot \frac{-314159265359}{50000000000}\right)\right) \]

   17. *-commutative N/A

       \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + \color{blue}{z0 \cdot \frac{314159265359}{50000000000}}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(z0 \cdot \frac{-314159265359}{50000000000}\right)\right) \]

   18. lower-*.f64 9.9%

       \[\leadsto \left(1 - \frac{\sin -1.5707963705062866 \cdot \cos \left(6.28318530718 \cdot z0\right)}{\cos 1.5707963705062866 \cdot \cos \left(\pi \cdot 0.5 + \color{blue}{z0 \cdot 6.28318530718}\right)}\right) \cdot \left(\cos 1.5707963705062866 \cdot \sin \left(z0 \cdot -6.28318530718\right)\right) \]

5. Applied rewrites 9.9%

   \[\leadsto \left(1 - \frac{\sin -1.5707963705062866 \cdot \cos \left(6.28318530718 \cdot z0\right)}{\cos 1.5707963705062866 \cdot \color{blue}{\cos \left(\pi \cdot 0.5 + z0 \cdot 6.28318530718\right)}}\right) \cdot \left(\cos 1.5707963705062866 \cdot \sin \left(z0 \cdot -6.28318530718\right)\right) \]
6. Step-by-step derivation

   1. lift-sin.f64 N/A

      \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \color{blue}{\sin \left(z0 \cdot \frac{-314159265359}{50000000000}\right)}\right) \]

   2. lift-*.f64 N/A

      \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \color{blue}{\left(z0 \cdot \frac{-314159265359}{50000000000}\right)}\right) \]

   3. metadata-eval N/A

      \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(z0 \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{314159265359}{50000000000}\right)\right)}\right)\right) \]

   4. distribute-rgt-neg-in N/A

      \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \color{blue}{\left(\mathsf{neg}\left(z0 \cdot \frac{314159265359}{50000000000}\right)\right)}\right) \]

   5. *-commutative N/A

      \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(\mathsf{neg}\left(\color{blue}{\frac{314159265359}{50000000000} \cdot z0}\right)\right)\right) \]

   6. lift-*.f64 N/A

      \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(\mathsf{neg}\left(\color{blue}{\frac{314159265359}{50000000000} \cdot z0}\right)\right)\right) \]

   7. sin-neg-rev N/A

      \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \color{blue}{\left(\mathsf{neg}\left(\sin \left(\frac{314159265359}{50000000000} \cdot z0\right)\right)\right)}\right) \]

   8. cos-+PI/2-rev N/A

      \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \color{blue}{\cos \left(\frac{314159265359}{50000000000} \cdot z0 + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) \]

   9. lower-cos.f64 N/A

      \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \color{blue}{\cos \left(\frac{314159265359}{50000000000} \cdot z0 + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) \]

   10. +-commutative N/A

       \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{2} + \frac{314159265359}{50000000000} \cdot z0\right)}\right) \]

   11. lower-+.f64 N/A

       \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{2} + \frac{314159265359}{50000000000} \cdot z0\right)}\right) \]

   12. lift-PI.f64 N/A

       \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\frac{\color{blue}{\pi}}{2} + \frac{314159265359}{50000000000} \cdot z0\right)\right) \]

   13. mult-flip N/A

       \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\color{blue}{\pi \cdot \frac{1}{2}} + \frac{314159265359}{50000000000} \cdot z0\right)\right) \]

   14. lower-*.f64 N/A

       \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\color{blue}{\pi \cdot \frac{1}{2}} + \frac{314159265359}{50000000000} \cdot z0\right)\right) \]

   15. metadata-eval 57.3%

       \[\leadsto \left(1 - \frac{\sin -1.5707963705062866 \cdot \cos \left(6.28318530718 \cdot z0\right)}{\cos 1.5707963705062866 \cdot \cos \left(\pi \cdot 0.5 + z0 \cdot 6.28318530718\right)}\right) \cdot \left(\cos 1.5707963705062866 \cdot \cos \left(\pi \cdot \color{blue}{0.5} + 6.28318530718 \cdot z0\right)\right) \]

   16. lift-*.f64 N/A

       \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + \color{blue}{\frac{314159265359}{50000000000} \cdot z0}\right)\right) \]

   17. *-commutative N/A

       \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + \color{blue}{z0 \cdot \frac{314159265359}{50000000000}}\right)\right) \]

   18. lower-*.f64 57.3%

       \[\leadsto \left(1 - \frac{\sin -1.5707963705062866 \cdot \cos \left(6.28318530718 \cdot z0\right)}{\cos 1.5707963705062866 \cdot \cos \left(\pi \cdot 0.5 + z0 \cdot 6.28318530718\right)}\right) \cdot \left(\cos 1.5707963705062866 \cdot \cos \left(\pi \cdot 0.5 + \color{blue}{z0 \cdot 6.28318530718}\right)\right) \]

7. Applied rewrites 57.3%

   \[\leadsto \left(1 - \frac{\sin -1.5707963705062866 \cdot \cos \left(6.28318530718 \cdot z0\right)}{\cos 1.5707963705062866 \cdot \cos \left(\pi \cdot 0.5 + z0 \cdot 6.28318530718\right)}\right) \cdot \left(\cos 1.5707963705062866 \cdot \color{blue}{\cos \left(\pi \cdot 0.5 + z0 \cdot 6.28318530718\right)}\right) \]
8. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \left(1 - \color{blue}{\frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)\right) \]

   2. mult-flip N/A

      \[\leadsto \left(1 - \color{blue}{\left(\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)\right) \cdot \frac{1}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)\right) \]

   3. lift-*.f64 N/A

      \[\leadsto \left(1 - \color{blue}{\left(\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)\right)} \cdot \frac{1}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)\right) \]

   4. *-commutative N/A

      \[\leadsto \left(1 - \color{blue}{\left(\cos \left(\frac{314159265359}{50000000000} \cdot z0\right) \cdot \sin \frac{-7853981852531433}{5000000000000000}\right)} \cdot \frac{1}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)\right) \]

   5. associate-*l* N/A

      \[\leadsto \left(1 - \color{blue}{\cos \left(\frac{314159265359}{50000000000} \cdot z0\right) \cdot \left(\sin \frac{-7853981852531433}{5000000000000000} \cdot \frac{1}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)\right) \]

   6. lower-*.f64 N/A

      \[\leadsto \left(1 - \color{blue}{\cos \left(\frac{314159265359}{50000000000} \cdot z0\right) \cdot \left(\sin \frac{-7853981852531433}{5000000000000000} \cdot \frac{1}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)\right) \]

   7. lift-cos.f64 N/A

      \[\leadsto \left(1 - \color{blue}{\cos \left(\frac{314159265359}{50000000000} \cdot z0\right)} \cdot \left(\sin \frac{-7853981852531433}{5000000000000000} \cdot \frac{1}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right)\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)\right) \]

   8. lift-*.f64 N/A

      \[\leadsto \left(1 - \cos \color{blue}{\left(\frac{314159265359}{50000000000} \cdot z0\right)} \cdot \left(\sin \frac{-7853981852531433}{5000000000000000} \cdot \frac{1}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right)\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)\right) \]

   9. metadata-eval N/A

      \[\leadsto \left(1 - \cos \left(\color{blue}{\left(\mathsf{neg}\left(\frac{-314159265359}{50000000000}\right)\right)} \cdot z0\right) \cdot \left(\sin \frac{-7853981852531433}{5000000000000000} \cdot \frac{1}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right)\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)\right) \]

   10. distribute-lft-neg-in N/A

       \[\leadsto \left(1 - \cos \color{blue}{\left(\mathsf{neg}\left(\frac{-314159265359}{50000000000} \cdot z0\right)\right)} \cdot \left(\sin \frac{-7853981852531433}{5000000000000000} \cdot \frac{1}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right)\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)\right) \]

   11. lift-*.f64 N/A

       \[\leadsto \left(1 - \cos \left(\mathsf{neg}\left(\color{blue}{\frac{-314159265359}{50000000000} \cdot z0}\right)\right) \cdot \left(\sin \frac{-7853981852531433}{5000000000000000} \cdot \frac{1}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right)\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)\right) \]

   12. cos-neg-rev N/A

       \[\leadsto \left(1 - \color{blue}{\cos \left(\frac{-314159265359}{50000000000} \cdot z0\right)} \cdot \left(\sin \frac{-7853981852531433}{5000000000000000} \cdot \frac{1}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right)\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)\right) \]

   13. lift-cos.f64 N/A

       \[\leadsto \left(1 - \color{blue}{\cos \left(\frac{-314159265359}{50000000000} \cdot z0\right)} \cdot \left(\sin \frac{-7853981852531433}{5000000000000000} \cdot \frac{1}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right)\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)\right) \]

   14. lower-*.f64 N/A

       \[\leadsto \left(1 - \cos \left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot \color{blue}{\left(\sin \frac{-7853981852531433}{5000000000000000} \cdot \frac{1}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)\right) \]

   15. lower-/.f64 57.3%

       \[\leadsto \left(1 - \cos \left(-6.28318530718 \cdot z0\right) \cdot \left(\sin -1.5707963705062866 \cdot \color{blue}{\frac{1}{\cos 1.5707963705062866 \cdot \cos \left(\pi \cdot 0.5 + z0 \cdot 6.28318530718\right)}}\right)\right) \cdot \left(\cos 1.5707963705062866 \cdot \cos \left(\pi \cdot 0.5 + z0 \cdot 6.28318530718\right)\right) \]

   16. lift-*.f64 N/A

       \[\leadsto \left(1 - \cos \left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot \left(\sin \frac{-7853981852531433}{5000000000000000} \cdot \frac{1}{\color{blue}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}}\right)\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)\right) \]

   17. *-commutative N/A

       \[\leadsto \left(1 - \cos \left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot \left(\sin \frac{-7853981852531433}{5000000000000000} \cdot \frac{1}{\color{blue}{\cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right) \cdot \cos \frac{7853981852531433}{5000000000000000}}}\right)\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)\right) \]

9. Applied rewrites 57.3%

   \[\leadsto \left(1 - \color{blue}{\cos \left(-6.28318530718 \cdot z0\right) \cdot \left(\sin -1.5707963705062866 \cdot \frac{1}{\cos \left(-6.28318530718 \cdot z0 - 0.5 \cdot \pi\right) \cdot \cos 1.5707963705062866}\right)}\right) \cdot \left(\cos 1.5707963705062866 \cdot \cos \left(\pi \cdot 0.5 + z0 \cdot 6.28318530718\right)\right) \]
10. Step-by-step derivation

    1. lift-+.f64 N/A

       \[\leadsto \left(1 - \cos \left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot \left(\sin \frac{-7853981852531433}{5000000000000000} \cdot \frac{1}{\cos \left(\frac{-314159265359}{50000000000} \cdot z0 - \frac{1}{2} \cdot \pi\right) \cdot \cos \frac{7853981852531433}{5000000000000000}}\right)\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \color{blue}{\left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right) \]

    2. lift-*.f64 N/A

       \[\leadsto \left(1 - \cos \left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot \left(\sin \frac{-7853981852531433}{5000000000000000} \cdot \frac{1}{\cos \left(\frac{-314159265359}{50000000000} \cdot z0 - \frac{1}{2} \cdot \pi\right) \cdot \cos \frac{7853981852531433}{5000000000000000}}\right)\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + \color{blue}{z0 \cdot \frac{314159265359}{50000000000}}\right)\right) \]

    3. *-commutative N/A

       \[\leadsto \left(1 - \cos \left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot \left(\sin \frac{-7853981852531433}{5000000000000000} \cdot \frac{1}{\cos \left(\frac{-314159265359}{50000000000} \cdot z0 - \frac{1}{2} \cdot \pi\right) \cdot \cos \frac{7853981852531433}{5000000000000000}}\right)\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + \color{blue}{\frac{314159265359}{50000000000} \cdot z0}\right)\right) \]

    4. metadata-eval N/A

       \[\leadsto \left(1 - \cos \left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot \left(\sin \frac{-7853981852531433}{5000000000000000} \cdot \frac{1}{\cos \left(\frac{-314159265359}{50000000000} \cdot z0 - \frac{1}{2} \cdot \pi\right) \cdot \cos \frac{7853981852531433}{5000000000000000}}\right)\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{-314159265359}{50000000000}\right)\right)} \cdot z0\right)\right) \]

    5. fp-cancel-sub-sign-inv N/A

       \[\leadsto \left(1 - \cos \left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot \left(\sin \frac{-7853981852531433}{5000000000000000} \cdot \frac{1}{\cos \left(\frac{-314159265359}{50000000000} \cdot z0 - \frac{1}{2} \cdot \pi\right) \cdot \cos \frac{7853981852531433}{5000000000000000}}\right)\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \color{blue}{\left(\pi \cdot \frac{1}{2} - \frac{-314159265359}{50000000000} \cdot z0\right)}\right) \]

    6. lift-*.f64 N/A

       \[\leadsto \left(1 - \cos \left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot \left(\sin \frac{-7853981852531433}{5000000000000000} \cdot \frac{1}{\cos \left(\frac{-314159265359}{50000000000} \cdot z0 - \frac{1}{2} \cdot \pi\right) \cdot \cos \frac{7853981852531433}{5000000000000000}}\right)\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} - \color{blue}{\frac{-314159265359}{50000000000} \cdot z0}\right)\right) \]

    7. sub-to-mult N/A

       \[\leadsto \left(1 - \cos \left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot \left(\sin \frac{-7853981852531433}{5000000000000000} \cdot \frac{1}{\cos \left(\frac{-314159265359}{50000000000} \cdot z0 - \frac{1}{2} \cdot \pi\right) \cdot \cos \frac{7853981852531433}{5000000000000000}}\right)\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \color{blue}{\left(\left(1 - \frac{\frac{-314159265359}{50000000000} \cdot z0}{\pi \cdot \frac{1}{2}}\right) \cdot \left(\pi \cdot \frac{1}{2}\right)\right)}\right) \]

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

       \[\leadsto \left(1 - \cos \left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot \left(\sin \frac{-7853981852531433}{5000000000000000} \cdot \frac{1}{\cos \left(\frac{-314159265359}{50000000000} \cdot z0 - \frac{1}{2} \cdot \pi\right) \cdot \cos \frac{7853981852531433}{5000000000000000}}\right)\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \color{blue}{\left(\left(1 - \frac{\frac{-314159265359}{50000000000} \cdot z0}{\pi \cdot \frac{1}{2}}\right) \cdot \left(\pi \cdot \frac{1}{2}\right)\right)}\right) \]

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

       \[\leadsto \left(1 - \cos \left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot \left(\sin \frac{-7853981852531433}{5000000000000000} \cdot \frac{1}{\cos \left(\frac{-314159265359}{50000000000} \cdot z0 - \frac{1}{2} \cdot \pi\right) \cdot \cos \frac{7853981852531433}{5000000000000000}}\right)\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\color{blue}{\left(1 - \frac{\frac{-314159265359}{50000000000} \cdot z0}{\pi \cdot \frac{1}{2}}\right)} \cdot \left(\pi \cdot \frac{1}{2}\right)\right)\right) \]

    10. lower-unsound-/.f64 57.0%

        \[\leadsto \left(1 - \cos \left(-6.28318530718 \cdot z0\right) \cdot \left(\sin -1.5707963705062866 \cdot \frac{1}{\cos \left(-6.28318530718 \cdot z0 - 0.5 \cdot \pi\right) \cdot \cos 1.5707963705062866}\right)\right) \cdot \left(\cos 1.5707963705062866 \cdot \cos \left(\left(1 - \color{blue}{\frac{-6.28318530718 \cdot z0}{\pi \cdot 0.5}}\right) \cdot \left(\pi \cdot 0.5\right)\right)\right) \]

    11. lift-*.f64 N/A

        \[\leadsto \left(1 - \cos \left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot \left(\sin \frac{-7853981852531433}{5000000000000000} \cdot \frac{1}{\cos \left(\frac{-314159265359}{50000000000} \cdot z0 - \frac{1}{2} \cdot \pi\right) \cdot \cos \frac{7853981852531433}{5000000000000000}}\right)\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\left(1 - \frac{\color{blue}{\frac{-314159265359}{50000000000} \cdot z0}}{\pi \cdot \frac{1}{2}}\right) \cdot \left(\pi \cdot \frac{1}{2}\right)\right)\right) \]

    12. *-commutative N/A

        \[\leadsto \left(1 - \cos \left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot \left(\sin \frac{-7853981852531433}{5000000000000000} \cdot \frac{1}{\cos \left(\frac{-314159265359}{50000000000} \cdot z0 - \frac{1}{2} \cdot \pi\right) \cdot \cos \frac{7853981852531433}{5000000000000000}}\right)\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\left(1 - \frac{\color{blue}{z0 \cdot \frac{-314159265359}{50000000000}}}{\pi \cdot \frac{1}{2}}\right) \cdot \left(\pi \cdot \frac{1}{2}\right)\right)\right) \]

    13. lift-*.f64 57.0%

        \[\leadsto \left(1 - \cos \left(-6.28318530718 \cdot z0\right) \cdot \left(\sin -1.5707963705062866 \cdot \frac{1}{\cos \left(-6.28318530718 \cdot z0 - 0.5 \cdot \pi\right) \cdot \cos 1.5707963705062866}\right)\right) \cdot \left(\cos 1.5707963705062866 \cdot \cos \left(\left(1 - \frac{\color{blue}{z0 \cdot -6.28318530718}}{\pi \cdot 0.5}\right) \cdot \left(\pi \cdot 0.5\right)\right)\right) \]

11. Applied rewrites 57.0%

    \[\leadsto \left(1 - \cos \left(-6.28318530718 \cdot z0\right) \cdot \left(\sin -1.5707963705062866 \cdot \frac{1}{\cos \left(-6.28318530718 \cdot z0 - 0.5 \cdot \pi\right) \cdot \cos 1.5707963705062866}\right)\right) \cdot \left(\cos 1.5707963705062866 \cdot \cos \color{blue}{\left(\left(1 - \frac{z0 \cdot -6.28318530718}{\pi \cdot 0.5}\right) \cdot \left(\pi \cdot 0.5\right)\right)}\right) \]
12. Add Preprocessing

Alternative 2: 57.3% accurate, 0.2× speedup

Math

\[\left(1 - \cos \left(-6.28318530718 \cdot z0\right) \cdot \left(\sin -1.5707963705062866 \cdot \frac{1}{\cos \left(-6.28318530718 \cdot z0 - 0.5 \cdot \pi\right) \cdot \cos 1.5707963705062866}\right)\right) \cdot \left(\cos 1.5707963705062866 \cdot \cos \left(\pi \cdot 0.5 + z0 \cdot 6.28318530718\right)\right) \]

FPCore

(FPCore (z0)
  :precision binary64
  (*
 (-
  1.0
  (*
   (cos (* -6.28318530718 z0))
   (*
    (sin -1.5707963705062866)
    (/
     1.0
     (*
      (cos (- (* -6.28318530718 z0) (* 0.5 PI)))
      (cos 1.5707963705062866))))))
 (*
  (cos 1.5707963705062866)
  (cos (+ (* PI 0.5) (* z0 6.28318530718))))))

C

double code(double z0) {
	return (1.0 - (cos((-6.28318530718 * z0)) * (sin(-1.5707963705062866) * (1.0 / (cos(((-6.28318530718 * z0) - (0.5 * ((double) M_PI)))) * cos(1.5707963705062866)))))) * (cos(1.5707963705062866) * cos(((((double) M_PI) * 0.5) + (z0 * 6.28318530718))));
}

Java

public static double code(double z0) {
	return (1.0 - (Math.cos((-6.28318530718 * z0)) * (Math.sin(-1.5707963705062866) * (1.0 / (Math.cos(((-6.28318530718 * z0) - (0.5 * Math.PI))) * Math.cos(1.5707963705062866)))))) * (Math.cos(1.5707963705062866) * Math.cos(((Math.PI * 0.5) + (z0 * 6.28318530718))));
}

Python

def code(z0):
	return (1.0 - (math.cos((-6.28318530718 * z0)) * (math.sin(-1.5707963705062866) * (1.0 / (math.cos(((-6.28318530718 * z0) - (0.5 * math.pi))) * math.cos(1.5707963705062866)))))) * (math.cos(1.5707963705062866) * math.cos(((math.pi * 0.5) + (z0 * 6.28318530718))))

Julia

function code(z0)
	return Float64(Float64(1.0 - Float64(cos(Float64(-6.28318530718 * z0)) * Float64(sin(-1.5707963705062866) * Float64(1.0 / Float64(cos(Float64(Float64(-6.28318530718 * z0) - Float64(0.5 * pi))) * cos(1.5707963705062866)))))) * Float64(cos(1.5707963705062866) * cos(Float64(Float64(pi * 0.5) + Float64(z0 * 6.28318530718)))))
end

MATLAB

function tmp = code(z0)
	tmp = (1.0 - (cos((-6.28318530718 * z0)) * (sin(-1.5707963705062866) * (1.0 / (cos(((-6.28318530718 * z0) - (0.5 * pi))) * cos(1.5707963705062866)))))) * (cos(1.5707963705062866) * cos(((pi * 0.5) + (z0 * 6.28318530718))));
end

Wolfram

code[z0_] := N[(N[(1.0 - N[(N[Cos[N[(-6.28318530718 * z0), $MachinePrecision]], $MachinePrecision] * N[(N[Sin[-1.5707963705062866], $MachinePrecision] * N[(1.0 / N[(N[Cos[N[(N[(-6.28318530718 * z0), $MachinePrecision] - N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[1.5707963705062866], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[1.5707963705062866], $MachinePrecision] * N[Cos[N[(N[(Pi * 0.5), $MachinePrecision] + N[(z0 * 6.28318530718), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 57.2%

   \[\sin \left(-6.28318530718 \cdot z0 - -1.5707963705062866\right) \]
2. Step-by-step derivation

   1. lift-sin.f64 N/A

      \[\leadsto \color{blue}{\sin \left(\frac{-314159265359}{50000000000} \cdot z0 - \frac{-7853981852531433}{5000000000000000}\right)} \]

   2. lift--.f64 N/A

      \[\leadsto \sin \color{blue}{\left(\frac{-314159265359}{50000000000} \cdot z0 - \frac{-7853981852531433}{5000000000000000}\right)} \]

   3. sin-diff N/A

      \[\leadsto \color{blue}{\sin \left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot \cos \frac{-7853981852531433}{5000000000000000} - \cos \left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot \sin \frac{-7853981852531433}{5000000000000000}} \]

   4. sub-to-mult N/A

      \[\leadsto \color{blue}{\left(1 - \frac{\cos \left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot \sin \frac{-7853981852531433}{5000000000000000}}{\sin \left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot \cos \frac{-7853981852531433}{5000000000000000}}\right) \cdot \left(\sin \left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot \cos \frac{-7853981852531433}{5000000000000000}\right)} \]

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

      \[\leadsto \color{blue}{\left(1 - \frac{\cos \left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot \sin \frac{-7853981852531433}{5000000000000000}}{\sin \left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot \cos \frac{-7853981852531433}{5000000000000000}}\right) \cdot \left(\sin \left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot \cos \frac{-7853981852531433}{5000000000000000}\right)} \]

3. Applied rewrites 56.0%

   \[\leadsto \color{blue}{\left(1 - \frac{\sin -1.5707963705062866 \cdot \cos \left(6.28318530718 \cdot z0\right)}{\cos 1.5707963705062866 \cdot \sin \left(z0 \cdot -6.28318530718\right)}\right) \cdot \left(\cos 1.5707963705062866 \cdot \sin \left(z0 \cdot -6.28318530718\right)\right)} \]
4. Step-by-step derivation

   1. lift-sin.f64 N/A

      \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \color{blue}{\sin \left(z0 \cdot \frac{-314159265359}{50000000000}\right)}}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(z0 \cdot \frac{-314159265359}{50000000000}\right)\right) \]

   2. lift-*.f64 N/A

      \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \color{blue}{\left(z0 \cdot \frac{-314159265359}{50000000000}\right)}}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(z0 \cdot \frac{-314159265359}{50000000000}\right)\right) \]

   3. metadata-eval N/A

      \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(z0 \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{314159265359}{50000000000}\right)\right)}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(z0 \cdot \frac{-314159265359}{50000000000}\right)\right) \]

   4. distribute-rgt-neg-in N/A

      \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \color{blue}{\left(\mathsf{neg}\left(z0 \cdot \frac{314159265359}{50000000000}\right)\right)}}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(z0 \cdot \frac{-314159265359}{50000000000}\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(\mathsf{neg}\left(\color{blue}{\frac{314159265359}{50000000000} \cdot z0}\right)\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(z0 \cdot \frac{-314159265359}{50000000000}\right)\right) \]

   6. lift-*.f64 N/A

      \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(\mathsf{neg}\left(\color{blue}{\frac{314159265359}{50000000000} \cdot z0}\right)\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(z0 \cdot \frac{-314159265359}{50000000000}\right)\right) \]

   7. sin-neg-rev N/A

      \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \color{blue}{\left(\mathsf{neg}\left(\sin \left(\frac{314159265359}{50000000000} \cdot z0\right)\right)\right)}}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(z0 \cdot \frac{-314159265359}{50000000000}\right)\right) \]

   8. cos-+PI/2-rev N/A

      \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \color{blue}{\cos \left(\frac{314159265359}{50000000000} \cdot z0 + \frac{\mathsf{PI}\left(\right)}{2}\right)}}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(z0 \cdot \frac{-314159265359}{50000000000}\right)\right) \]

   9. lower-cos.f64 N/A

      \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \color{blue}{\cos \left(\frac{314159265359}{50000000000} \cdot z0 + \frac{\mathsf{PI}\left(\right)}{2}\right)}}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(z0 \cdot \frac{-314159265359}{50000000000}\right)\right) \]

   10. +-commutative N/A

       \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{2} + \frac{314159265359}{50000000000} \cdot z0\right)}}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(z0 \cdot \frac{-314159265359}{50000000000}\right)\right) \]

   11. lower-+.f64 N/A

       \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{2} + \frac{314159265359}{50000000000} \cdot z0\right)}}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(z0 \cdot \frac{-314159265359}{50000000000}\right)\right) \]

   12. lift-PI.f64 N/A

       \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\frac{\color{blue}{\pi}}{2} + \frac{314159265359}{50000000000} \cdot z0\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(z0 \cdot \frac{-314159265359}{50000000000}\right)\right) \]

   13. mult-flip N/A

       \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\color{blue}{\pi \cdot \frac{1}{2}} + \frac{314159265359}{50000000000} \cdot z0\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(z0 \cdot \frac{-314159265359}{50000000000}\right)\right) \]

   14. lower-*.f64 N/A

       \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\color{blue}{\pi \cdot \frac{1}{2}} + \frac{314159265359}{50000000000} \cdot z0\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(z0 \cdot \frac{-314159265359}{50000000000}\right)\right) \]

   15. metadata-eval 9.9%

       \[\leadsto \left(1 - \frac{\sin -1.5707963705062866 \cdot \cos \left(6.28318530718 \cdot z0\right)}{\cos 1.5707963705062866 \cdot \cos \left(\pi \cdot \color{blue}{0.5} + 6.28318530718 \cdot z0\right)}\right) \cdot \left(\cos 1.5707963705062866 \cdot \sin \left(z0 \cdot -6.28318530718\right)\right) \]

   16. lift-*.f64 N/A

       \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + \color{blue}{\frac{314159265359}{50000000000} \cdot z0}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(z0 \cdot \frac{-314159265359}{50000000000}\right)\right) \]

   17. *-commutative N/A

       \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + \color{blue}{z0 \cdot \frac{314159265359}{50000000000}}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(z0 \cdot \frac{-314159265359}{50000000000}\right)\right) \]

   18. lower-*.f64 9.9%

       \[\leadsto \left(1 - \frac{\sin -1.5707963705062866 \cdot \cos \left(6.28318530718 \cdot z0\right)}{\cos 1.5707963705062866 \cdot \cos \left(\pi \cdot 0.5 + \color{blue}{z0 \cdot 6.28318530718}\right)}\right) \cdot \left(\cos 1.5707963705062866 \cdot \sin \left(z0 \cdot -6.28318530718\right)\right) \]

5. Applied rewrites 9.9%

   \[\leadsto \left(1 - \frac{\sin -1.5707963705062866 \cdot \cos \left(6.28318530718 \cdot z0\right)}{\cos 1.5707963705062866 \cdot \color{blue}{\cos \left(\pi \cdot 0.5 + z0 \cdot 6.28318530718\right)}}\right) \cdot \left(\cos 1.5707963705062866 \cdot \sin \left(z0 \cdot -6.28318530718\right)\right) \]
6. Step-by-step derivation

   1. lift-sin.f64 N/A

      \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \color{blue}{\sin \left(z0 \cdot \frac{-314159265359}{50000000000}\right)}\right) \]

   2. lift-*.f64 N/A

      \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \color{blue}{\left(z0 \cdot \frac{-314159265359}{50000000000}\right)}\right) \]

   3. metadata-eval N/A

      \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(z0 \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{314159265359}{50000000000}\right)\right)}\right)\right) \]

   4. distribute-rgt-neg-in N/A

      \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \color{blue}{\left(\mathsf{neg}\left(z0 \cdot \frac{314159265359}{50000000000}\right)\right)}\right) \]

   5. *-commutative N/A

      \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(\mathsf{neg}\left(\color{blue}{\frac{314159265359}{50000000000} \cdot z0}\right)\right)\right) \]

   6. lift-*.f64 N/A

      \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(\mathsf{neg}\left(\color{blue}{\frac{314159265359}{50000000000} \cdot z0}\right)\right)\right) \]

   7. sin-neg-rev N/A

      \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \color{blue}{\left(\mathsf{neg}\left(\sin \left(\frac{314159265359}{50000000000} \cdot z0\right)\right)\right)}\right) \]

   8. cos-+PI/2-rev N/A

      \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \color{blue}{\cos \left(\frac{314159265359}{50000000000} \cdot z0 + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) \]

   9. lower-cos.f64 N/A

      \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \color{blue}{\cos \left(\frac{314159265359}{50000000000} \cdot z0 + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) \]

   10. +-commutative N/A

       \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{2} + \frac{314159265359}{50000000000} \cdot z0\right)}\right) \]

   11. lower-+.f64 N/A

       \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{2} + \frac{314159265359}{50000000000} \cdot z0\right)}\right) \]

   12. lift-PI.f64 N/A

       \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\frac{\color{blue}{\pi}}{2} + \frac{314159265359}{50000000000} \cdot z0\right)\right) \]

   13. mult-flip N/A

       \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\color{blue}{\pi \cdot \frac{1}{2}} + \frac{314159265359}{50000000000} \cdot z0\right)\right) \]

   14. lower-*.f64 N/A

       \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\color{blue}{\pi \cdot \frac{1}{2}} + \frac{314159265359}{50000000000} \cdot z0\right)\right) \]

   15. metadata-eval 57.3%

       \[\leadsto \left(1 - \frac{\sin -1.5707963705062866 \cdot \cos \left(6.28318530718 \cdot z0\right)}{\cos 1.5707963705062866 \cdot \cos \left(\pi \cdot 0.5 + z0 \cdot 6.28318530718\right)}\right) \cdot \left(\cos 1.5707963705062866 \cdot \cos \left(\pi \cdot \color{blue}{0.5} + 6.28318530718 \cdot z0\right)\right) \]

   16. lift-*.f64 N/A

       \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + \color{blue}{\frac{314159265359}{50000000000} \cdot z0}\right)\right) \]

   17. *-commutative N/A

       \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + \color{blue}{z0 \cdot \frac{314159265359}{50000000000}}\right)\right) \]

   18. lower-*.f64 57.3%

       \[\leadsto \left(1 - \frac{\sin -1.5707963705062866 \cdot \cos \left(6.28318530718 \cdot z0\right)}{\cos 1.5707963705062866 \cdot \cos \left(\pi \cdot 0.5 + z0 \cdot 6.28318530718\right)}\right) \cdot \left(\cos 1.5707963705062866 \cdot \cos \left(\pi \cdot 0.5 + \color{blue}{z0 \cdot 6.28318530718}\right)\right) \]

7. Applied rewrites 57.3%

   \[\leadsto \left(1 - \frac{\sin -1.5707963705062866 \cdot \cos \left(6.28318530718 \cdot z0\right)}{\cos 1.5707963705062866 \cdot \cos \left(\pi \cdot 0.5 + z0 \cdot 6.28318530718\right)}\right) \cdot \left(\cos 1.5707963705062866 \cdot \color{blue}{\cos \left(\pi \cdot 0.5 + z0 \cdot 6.28318530718\right)}\right) \]
8. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \left(1 - \color{blue}{\frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)\right) \]

   2. mult-flip N/A

      \[\leadsto \left(1 - \color{blue}{\left(\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)\right) \cdot \frac{1}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)\right) \]

   3. lift-*.f64 N/A

      \[\leadsto \left(1 - \color{blue}{\left(\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)\right)} \cdot \frac{1}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)\right) \]

   4. *-commutative N/A

      \[\leadsto \left(1 - \color{blue}{\left(\cos \left(\frac{314159265359}{50000000000} \cdot z0\right) \cdot \sin \frac{-7853981852531433}{5000000000000000}\right)} \cdot \frac{1}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)\right) \]

   5. associate-*l* N/A

      \[\leadsto \left(1 - \color{blue}{\cos \left(\frac{314159265359}{50000000000} \cdot z0\right) \cdot \left(\sin \frac{-7853981852531433}{5000000000000000} \cdot \frac{1}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)\right) \]

   6. lower-*.f64 N/A

      \[\leadsto \left(1 - \color{blue}{\cos \left(\frac{314159265359}{50000000000} \cdot z0\right) \cdot \left(\sin \frac{-7853981852531433}{5000000000000000} \cdot \frac{1}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)\right) \]

   7. lift-cos.f64 N/A

      \[\leadsto \left(1 - \color{blue}{\cos \left(\frac{314159265359}{50000000000} \cdot z0\right)} \cdot \left(\sin \frac{-7853981852531433}{5000000000000000} \cdot \frac{1}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right)\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)\right) \]

   8. lift-*.f64 N/A

      \[\leadsto \left(1 - \cos \color{blue}{\left(\frac{314159265359}{50000000000} \cdot z0\right)} \cdot \left(\sin \frac{-7853981852531433}{5000000000000000} \cdot \frac{1}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right)\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)\right) \]

   9. metadata-eval N/A

      \[\leadsto \left(1 - \cos \left(\color{blue}{\left(\mathsf{neg}\left(\frac{-314159265359}{50000000000}\right)\right)} \cdot z0\right) \cdot \left(\sin \frac{-7853981852531433}{5000000000000000} \cdot \frac{1}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right)\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)\right) \]

   10. distribute-lft-neg-in N/A

       \[\leadsto \left(1 - \cos \color{blue}{\left(\mathsf{neg}\left(\frac{-314159265359}{50000000000} \cdot z0\right)\right)} \cdot \left(\sin \frac{-7853981852531433}{5000000000000000} \cdot \frac{1}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right)\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)\right) \]

   11. lift-*.f64 N/A

       \[\leadsto \left(1 - \cos \left(\mathsf{neg}\left(\color{blue}{\frac{-314159265359}{50000000000} \cdot z0}\right)\right) \cdot \left(\sin \frac{-7853981852531433}{5000000000000000} \cdot \frac{1}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right)\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)\right) \]

   12. cos-neg-rev N/A

       \[\leadsto \left(1 - \color{blue}{\cos \left(\frac{-314159265359}{50000000000} \cdot z0\right)} \cdot \left(\sin \frac{-7853981852531433}{5000000000000000} \cdot \frac{1}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right)\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)\right) \]

   13. lift-cos.f64 N/A

       \[\leadsto \left(1 - \color{blue}{\cos \left(\frac{-314159265359}{50000000000} \cdot z0\right)} \cdot \left(\sin \frac{-7853981852531433}{5000000000000000} \cdot \frac{1}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right)\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)\right) \]

   14. lower-*.f64 N/A

       \[\leadsto \left(1 - \cos \left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot \color{blue}{\left(\sin \frac{-7853981852531433}{5000000000000000} \cdot \frac{1}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)\right) \]

   15. lower-/.f64 57.3%

       \[\leadsto \left(1 - \cos \left(-6.28318530718 \cdot z0\right) \cdot \left(\sin -1.5707963705062866 \cdot \color{blue}{\frac{1}{\cos 1.5707963705062866 \cdot \cos \left(\pi \cdot 0.5 + z0 \cdot 6.28318530718\right)}}\right)\right) \cdot \left(\cos 1.5707963705062866 \cdot \cos \left(\pi \cdot 0.5 + z0 \cdot 6.28318530718\right)\right) \]

   16. lift-*.f64 N/A

       \[\leadsto \left(1 - \cos \left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot \left(\sin \frac{-7853981852531433}{5000000000000000} \cdot \frac{1}{\color{blue}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}}\right)\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)\right) \]

   17. *-commutative N/A

       \[\leadsto \left(1 - \cos \left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot \left(\sin \frac{-7853981852531433}{5000000000000000} \cdot \frac{1}{\color{blue}{\cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right) \cdot \cos \frac{7853981852531433}{5000000000000000}}}\right)\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)\right) \]

9. Applied rewrites 57.3%

   \[\leadsto \left(1 - \color{blue}{\cos \left(-6.28318530718 \cdot z0\right) \cdot \left(\sin -1.5707963705062866 \cdot \frac{1}{\cos \left(-6.28318530718 \cdot z0 - 0.5 \cdot \pi\right) \cdot \cos 1.5707963705062866}\right)}\right) \cdot \left(\cos 1.5707963705062866 \cdot \cos \left(\pi \cdot 0.5 + z0 \cdot 6.28318530718\right)\right) \]
10. Add Preprocessing

Alternative 3: 57.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \left(6.28318530718 \cdot z0 - -0.5 \cdot \pi\right) - -0.5 \cdot \pi\\ t_1 := \pi \cdot 0.5 + 1.5707963705062866\\ \frac{\cos \left(t\_0 - t\_1\right) - \cos \left(t\_0 + t\_1\right)}{2} - \cos \left(-6.28318530718 \cdot z0\right) \cdot \sin -1.5707963705062866 \end{array} \]

FPCore

(FPCore (z0)
  :precision binary64
  (let* ((t_0 (- (- (* 6.28318530718 z0) (* -0.5 PI)) (* -0.5 PI)))
       (t_1 (+ (* PI 0.5) 1.5707963705062866)))
  (-
   (/ (- (cos (- t_0 t_1)) (cos (+ t_0 t_1))) 2.0)
   (* (cos (* -6.28318530718 z0)) (sin -1.5707963705062866)))))

C

double code(double z0) {
	double t_0 = ((6.28318530718 * z0) - (-0.5 * ((double) M_PI))) - (-0.5 * ((double) M_PI));
	double t_1 = (((double) M_PI) * 0.5) + 1.5707963705062866;
	return ((cos((t_0 - t_1)) - cos((t_0 + t_1))) / 2.0) - (cos((-6.28318530718 * z0)) * sin(-1.5707963705062866));
}

Java

public static double code(double z0) {
	double t_0 = ((6.28318530718 * z0) - (-0.5 * Math.PI)) - (-0.5 * Math.PI);
	double t_1 = (Math.PI * 0.5) + 1.5707963705062866;
	return ((Math.cos((t_0 - t_1)) - Math.cos((t_0 + t_1))) / 2.0) - (Math.cos((-6.28318530718 * z0)) * Math.sin(-1.5707963705062866));
}

Python

def code(z0):
	t_0 = ((6.28318530718 * z0) - (-0.5 * math.pi)) - (-0.5 * math.pi)
	t_1 = (math.pi * 0.5) + 1.5707963705062866
	return ((math.cos((t_0 - t_1)) - math.cos((t_0 + t_1))) / 2.0) - (math.cos((-6.28318530718 * z0)) * math.sin(-1.5707963705062866))

Julia

function code(z0)
	t_0 = Float64(Float64(Float64(6.28318530718 * z0) - Float64(-0.5 * pi)) - Float64(-0.5 * pi))
	t_1 = Float64(Float64(pi * 0.5) + 1.5707963705062866)
	return Float64(Float64(Float64(cos(Float64(t_0 - t_1)) - cos(Float64(t_0 + t_1))) / 2.0) - Float64(cos(Float64(-6.28318530718 * z0)) * sin(-1.5707963705062866)))
end

MATLAB

function tmp = code(z0)
	t_0 = ((6.28318530718 * z0) - (-0.5 * pi)) - (-0.5 * pi);
	t_1 = (pi * 0.5) + 1.5707963705062866;
	tmp = ((cos((t_0 - t_1)) - cos((t_0 + t_1))) / 2.0) - (cos((-6.28318530718 * z0)) * sin(-1.5707963705062866));
end

Wolfram

code[z0_] := Block[{t$95$0 = N[(N[(N[(6.28318530718 * z0), $MachinePrecision] - N[(-0.5 * Pi), $MachinePrecision]), $MachinePrecision] - N[(-0.5 * Pi), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(Pi * 0.5), $MachinePrecision] + 1.5707963705062866), $MachinePrecision]}, N[(N[(N[(N[Cos[N[(t$95$0 - t$95$1), $MachinePrecision]], $MachinePrecision] - N[Cos[N[(t$95$0 + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] - N[(N[Cos[N[(-6.28318530718 * z0), $MachinePrecision]], $MachinePrecision] * N[Sin[-1.5707963705062866], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 57.2%

   \[\sin \left(-6.28318530718 \cdot z0 - -1.5707963705062866\right) \]
2. Step-by-step derivation

   1. lift-sin.f64 N/A

      \[\leadsto \color{blue}{\sin \left(\frac{-314159265359}{50000000000} \cdot z0 - \frac{-7853981852531433}{5000000000000000}\right)} \]

   2. lift--.f64 N/A

      \[\leadsto \sin \color{blue}{\left(\frac{-314159265359}{50000000000} \cdot z0 - \frac{-7853981852531433}{5000000000000000}\right)} \]

   3. sin-diff N/A

      \[\leadsto \color{blue}{\sin \left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot \cos \frac{-7853981852531433}{5000000000000000} - \cos \left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot \sin \frac{-7853981852531433}{5000000000000000}} \]

   4. sub-to-mult N/A

      \[\leadsto \color{blue}{\left(1 - \frac{\cos \left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot \sin \frac{-7853981852531433}{5000000000000000}}{\sin \left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot \cos \frac{-7853981852531433}{5000000000000000}}\right) \cdot \left(\sin \left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot \cos \frac{-7853981852531433}{5000000000000000}\right)} \]

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

      \[\leadsto \color{blue}{\left(1 - \frac{\cos \left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot \sin \frac{-7853981852531433}{5000000000000000}}{\sin \left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot \cos \frac{-7853981852531433}{5000000000000000}}\right) \cdot \left(\sin \left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot \cos \frac{-7853981852531433}{5000000000000000}\right)} \]

3. Applied rewrites 56.0%

   \[\leadsto \color{blue}{\left(1 - \frac{\sin -1.5707963705062866 \cdot \cos \left(6.28318530718 \cdot z0\right)}{\cos 1.5707963705062866 \cdot \sin \left(z0 \cdot -6.28318530718\right)}\right) \cdot \left(\cos 1.5707963705062866 \cdot \sin \left(z0 \cdot -6.28318530718\right)\right)} \]
4. Step-by-step derivation

   1. lift-sin.f64 N/A

      \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \color{blue}{\sin \left(z0 \cdot \frac{-314159265359}{50000000000}\right)}}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(z0 \cdot \frac{-314159265359}{50000000000}\right)\right) \]

   2. lift-*.f64 N/A

      \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \color{blue}{\left(z0 \cdot \frac{-314159265359}{50000000000}\right)}}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(z0 \cdot \frac{-314159265359}{50000000000}\right)\right) \]

   3. metadata-eval N/A

      \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(z0 \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{314159265359}{50000000000}\right)\right)}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(z0 \cdot \frac{-314159265359}{50000000000}\right)\right) \]

   4. distribute-rgt-neg-in N/A

      \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \color{blue}{\left(\mathsf{neg}\left(z0 \cdot \frac{314159265359}{50000000000}\right)\right)}}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(z0 \cdot \frac{-314159265359}{50000000000}\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(\mathsf{neg}\left(\color{blue}{\frac{314159265359}{50000000000} \cdot z0}\right)\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(z0 \cdot \frac{-314159265359}{50000000000}\right)\right) \]

   6. lift-*.f64 N/A

      \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(\mathsf{neg}\left(\color{blue}{\frac{314159265359}{50000000000} \cdot z0}\right)\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(z0 \cdot \frac{-314159265359}{50000000000}\right)\right) \]

   7. sin-neg-rev N/A

      \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \color{blue}{\left(\mathsf{neg}\left(\sin \left(\frac{314159265359}{50000000000} \cdot z0\right)\right)\right)}}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(z0 \cdot \frac{-314159265359}{50000000000}\right)\right) \]

   8. cos-+PI/2-rev N/A

      \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \color{blue}{\cos \left(\frac{314159265359}{50000000000} \cdot z0 + \frac{\mathsf{PI}\left(\right)}{2}\right)}}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(z0 \cdot \frac{-314159265359}{50000000000}\right)\right) \]

   9. lower-cos.f64 N/A

      \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \color{blue}{\cos \left(\frac{314159265359}{50000000000} \cdot z0 + \frac{\mathsf{PI}\left(\right)}{2}\right)}}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(z0 \cdot \frac{-314159265359}{50000000000}\right)\right) \]

   10. +-commutative N/A

       \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{2} + \frac{314159265359}{50000000000} \cdot z0\right)}}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(z0 \cdot \frac{-314159265359}{50000000000}\right)\right) \]

   11. lower-+.f64 N/A

       \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{2} + \frac{314159265359}{50000000000} \cdot z0\right)}}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(z0 \cdot \frac{-314159265359}{50000000000}\right)\right) \]

   12. lift-PI.f64 N/A

       \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\frac{\color{blue}{\pi}}{2} + \frac{314159265359}{50000000000} \cdot z0\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(z0 \cdot \frac{-314159265359}{50000000000}\right)\right) \]

   13. mult-flip N/A

       \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\color{blue}{\pi \cdot \frac{1}{2}} + \frac{314159265359}{50000000000} \cdot z0\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(z0 \cdot \frac{-314159265359}{50000000000}\right)\right) \]

   14. lower-*.f64 N/A

       \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\color{blue}{\pi \cdot \frac{1}{2}} + \frac{314159265359}{50000000000} \cdot z0\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(z0 \cdot \frac{-314159265359}{50000000000}\right)\right) \]

   15. metadata-eval 9.9%

       \[\leadsto \left(1 - \frac{\sin -1.5707963705062866 \cdot \cos \left(6.28318530718 \cdot z0\right)}{\cos 1.5707963705062866 \cdot \cos \left(\pi \cdot \color{blue}{0.5} + 6.28318530718 \cdot z0\right)}\right) \cdot \left(\cos 1.5707963705062866 \cdot \sin \left(z0 \cdot -6.28318530718\right)\right) \]

   16. lift-*.f64 N/A

       \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + \color{blue}{\frac{314159265359}{50000000000} \cdot z0}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(z0 \cdot \frac{-314159265359}{50000000000}\right)\right) \]

   17. *-commutative N/A

       \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + \color{blue}{z0 \cdot \frac{314159265359}{50000000000}}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(z0 \cdot \frac{-314159265359}{50000000000}\right)\right) \]

   18. lower-*.f64 9.9%

       \[\leadsto \left(1 - \frac{\sin -1.5707963705062866 \cdot \cos \left(6.28318530718 \cdot z0\right)}{\cos 1.5707963705062866 \cdot \cos \left(\pi \cdot 0.5 + \color{blue}{z0 \cdot 6.28318530718}\right)}\right) \cdot \left(\cos 1.5707963705062866 \cdot \sin \left(z0 \cdot -6.28318530718\right)\right) \]

5. Applied rewrites 9.9%

   \[\leadsto \left(1 - \frac{\sin -1.5707963705062866 \cdot \cos \left(6.28318530718 \cdot z0\right)}{\cos 1.5707963705062866 \cdot \color{blue}{\cos \left(\pi \cdot 0.5 + z0 \cdot 6.28318530718\right)}}\right) \cdot \left(\cos 1.5707963705062866 \cdot \sin \left(z0 \cdot -6.28318530718\right)\right) \]
6. Step-by-step derivation

   1. lift-sin.f64 N/A

      \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \color{blue}{\sin \left(z0 \cdot \frac{-314159265359}{50000000000}\right)}\right) \]

   2. lift-*.f64 N/A

      \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \color{blue}{\left(z0 \cdot \frac{-314159265359}{50000000000}\right)}\right) \]

   3. metadata-eval N/A

      \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(z0 \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{314159265359}{50000000000}\right)\right)}\right)\right) \]

   4. distribute-rgt-neg-in N/A

      \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \color{blue}{\left(\mathsf{neg}\left(z0 \cdot \frac{314159265359}{50000000000}\right)\right)}\right) \]

   5. *-commutative N/A

      \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(\mathsf{neg}\left(\color{blue}{\frac{314159265359}{50000000000} \cdot z0}\right)\right)\right) \]

   6. lift-*.f64 N/A

      \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(\mathsf{neg}\left(\color{blue}{\frac{314159265359}{50000000000} \cdot z0}\right)\right)\right) \]

   7. sin-neg-rev N/A

      \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \color{blue}{\left(\mathsf{neg}\left(\sin \left(\frac{314159265359}{50000000000} \cdot z0\right)\right)\right)}\right) \]

   8. cos-+PI/2-rev N/A

      \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \color{blue}{\cos \left(\frac{314159265359}{50000000000} \cdot z0 + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) \]

   9. lower-cos.f64 N/A

      \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \color{blue}{\cos \left(\frac{314159265359}{50000000000} \cdot z0 + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) \]

   10. +-commutative N/A

       \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{2} + \frac{314159265359}{50000000000} \cdot z0\right)}\right) \]

   11. lower-+.f64 N/A

       \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{2} + \frac{314159265359}{50000000000} \cdot z0\right)}\right) \]

   12. lift-PI.f64 N/A

       \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\frac{\color{blue}{\pi}}{2} + \frac{314159265359}{50000000000} \cdot z0\right)\right) \]

   13. mult-flip N/A

       \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\color{blue}{\pi \cdot \frac{1}{2}} + \frac{314159265359}{50000000000} \cdot z0\right)\right) \]

   14. lower-*.f64 N/A

       \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\color{blue}{\pi \cdot \frac{1}{2}} + \frac{314159265359}{50000000000} \cdot z0\right)\right) \]

   15. metadata-eval 57.3%

       \[\leadsto \left(1 - \frac{\sin -1.5707963705062866 \cdot \cos \left(6.28318530718 \cdot z0\right)}{\cos 1.5707963705062866 \cdot \cos \left(\pi \cdot 0.5 + z0 \cdot 6.28318530718\right)}\right) \cdot \left(\cos 1.5707963705062866 \cdot \cos \left(\pi \cdot \color{blue}{0.5} + 6.28318530718 \cdot z0\right)\right) \]

   16. lift-*.f64 N/A

       \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + \color{blue}{\frac{314159265359}{50000000000} \cdot z0}\right)\right) \]

   17. *-commutative N/A

       \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + \color{blue}{z0 \cdot \frac{314159265359}{50000000000}}\right)\right) \]

   18. lower-*.f64 57.3%

       \[\leadsto \left(1 - \frac{\sin -1.5707963705062866 \cdot \cos \left(6.28318530718 \cdot z0\right)}{\cos 1.5707963705062866 \cdot \cos \left(\pi \cdot 0.5 + z0 \cdot 6.28318530718\right)}\right) \cdot \left(\cos 1.5707963705062866 \cdot \cos \left(\pi \cdot 0.5 + \color{blue}{z0 \cdot 6.28318530718}\right)\right) \]

7. Applied rewrites 57.3%

   \[\leadsto \left(1 - \frac{\sin -1.5707963705062866 \cdot \cos \left(6.28318530718 \cdot z0\right)}{\cos 1.5707963705062866 \cdot \cos \left(\pi \cdot 0.5 + z0 \cdot 6.28318530718\right)}\right) \cdot \left(\cos 1.5707963705062866 \cdot \color{blue}{\cos \left(\pi \cdot 0.5 + z0 \cdot 6.28318530718\right)}\right) \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)\right)} \]

   2. lift--.f64 N/A

      \[\leadsto \color{blue}{\left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right)} \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)\right) \]

   3. lift-/.f64 N/A

      \[\leadsto \left(1 - \color{blue}{\frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)\right) \]

   4. sub-to-mult-rev N/A

      \[\leadsto \color{blue}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right) - \sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)} \]

   5. lower--.f64 57.3%

      \[\leadsto \color{blue}{\cos 1.5707963705062866 \cdot \cos \left(\pi \cdot 0.5 + z0 \cdot 6.28318530718\right) - \sin -1.5707963705062866 \cdot \cos \left(6.28318530718 \cdot z0\right)} \]

9. Applied rewrites 57.3%

   \[\leadsto \color{blue}{\cos \left(-6.28318530718 \cdot z0 - 0.5 \cdot \pi\right) \cdot \cos 1.5707963705062866 - \cos \left(-6.28318530718 \cdot z0\right) \cdot \sin -1.5707963705062866} \]

11. Applied rewrites 57.3%

    \[\leadsto \color{blue}{\frac{\cos \left(\left(\left(6.28318530718 \cdot z0 - -0.5 \cdot \pi\right) - -0.5 \cdot \pi\right) - \left(\pi \cdot 0.5 + 1.5707963705062866\right)\right) - \cos \left(\left(\left(6.28318530718 \cdot z0 - -0.5 \cdot \pi\right) - -0.5 \cdot \pi\right) + \left(\pi \cdot 0.5 + 1.5707963705062866\right)\right)}{2}} - \cos \left(-6.28318530718 \cdot z0\right) \cdot \sin -1.5707963705062866 \]
12. Add Preprocessing

Alternative 4: 57.3% accurate, 0.2× speedup

Math

\[\left(1 - \cos \left(-6.28318530718 \cdot z0\right) \cdot \left(\sin -1.5707963705062866 \cdot \frac{1}{\cos \left(-6.28318530718 \cdot z0 - 0.5 \cdot \pi\right) \cdot -4.3711389980768665 \cdot 10^{-8}}\right)\right) \cdot \left(-4.3711389980768665 \cdot 10^{-8} \cdot \cos \left(\pi \cdot 0.5 + z0 \cdot 6.28318530718\right)\right) \]

FPCore

(FPCore (z0)
  :precision binary64
  (*
 (-
  1.0
  (*
   (cos (* -6.28318530718 z0))
   (*
    (sin -1.5707963705062866)
    (/
     1.0
     (*
      (cos (- (* -6.28318530718 z0) (* 0.5 PI)))
      -4.3711389980768665e-8)))))
 (* -4.3711389980768665e-8 (cos (+ (* PI 0.5) (* z0 6.28318530718))))))

C

double code(double z0) {
	return (1.0 - (cos((-6.28318530718 * z0)) * (sin(-1.5707963705062866) * (1.0 / (cos(((-6.28318530718 * z0) - (0.5 * ((double) M_PI)))) * -4.3711389980768665e-8))))) * (-4.3711389980768665e-8 * cos(((((double) M_PI) * 0.5) + (z0 * 6.28318530718))));
}

Java

public static double code(double z0) {
	return (1.0 - (Math.cos((-6.28318530718 * z0)) * (Math.sin(-1.5707963705062866) * (1.0 / (Math.cos(((-6.28318530718 * z0) - (0.5 * Math.PI))) * -4.3711389980768665e-8))))) * (-4.3711389980768665e-8 * Math.cos(((Math.PI * 0.5) + (z0 * 6.28318530718))));
}

Python

def code(z0):
	return (1.0 - (math.cos((-6.28318530718 * z0)) * (math.sin(-1.5707963705062866) * (1.0 / (math.cos(((-6.28318530718 * z0) - (0.5 * math.pi))) * -4.3711389980768665e-8))))) * (-4.3711389980768665e-8 * math.cos(((math.pi * 0.5) + (z0 * 6.28318530718))))

Julia

function code(z0)
	return Float64(Float64(1.0 - Float64(cos(Float64(-6.28318530718 * z0)) * Float64(sin(-1.5707963705062866) * Float64(1.0 / Float64(cos(Float64(Float64(-6.28318530718 * z0) - Float64(0.5 * pi))) * -4.3711389980768665e-8))))) * Float64(-4.3711389980768665e-8 * cos(Float64(Float64(pi * 0.5) + Float64(z0 * 6.28318530718)))))
end

MATLAB

function tmp = code(z0)
	tmp = (1.0 - (cos((-6.28318530718 * z0)) * (sin(-1.5707963705062866) * (1.0 / (cos(((-6.28318530718 * z0) - (0.5 * pi))) * -4.3711389980768665e-8))))) * (-4.3711389980768665e-8 * cos(((pi * 0.5) + (z0 * 6.28318530718))));
end

Wolfram

code[z0_] := N[(N[(1.0 - N[(N[Cos[N[(-6.28318530718 * z0), $MachinePrecision]], $MachinePrecision] * N[(N[Sin[-1.5707963705062866], $MachinePrecision] * N[(1.0 / N[(N[Cos[N[(N[(-6.28318530718 * z0), $MachinePrecision] - N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * -4.3711389980768665e-8), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-4.3711389980768665e-8 * N[Cos[N[(N[(Pi * 0.5), $MachinePrecision] + N[(z0 * 6.28318530718), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 57.2%

   \[\sin \left(-6.28318530718 \cdot z0 - -1.5707963705062866\right) \]
2. Step-by-step derivation

   1. lift-sin.f64 N/A

      \[\leadsto \color{blue}{\sin \left(\frac{-314159265359}{50000000000} \cdot z0 - \frac{-7853981852531433}{5000000000000000}\right)} \]

   2. lift--.f64 N/A

      \[\leadsto \sin \color{blue}{\left(\frac{-314159265359}{50000000000} \cdot z0 - \frac{-7853981852531433}{5000000000000000}\right)} \]

   3. sin-diff N/A

      \[\leadsto \color{blue}{\sin \left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot \cos \frac{-7853981852531433}{5000000000000000} - \cos \left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot \sin \frac{-7853981852531433}{5000000000000000}} \]

   4. sub-to-mult N/A

      \[\leadsto \color{blue}{\left(1 - \frac{\cos \left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot \sin \frac{-7853981852531433}{5000000000000000}}{\sin \left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot \cos \frac{-7853981852531433}{5000000000000000}}\right) \cdot \left(\sin \left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot \cos \frac{-7853981852531433}{5000000000000000}\right)} \]

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

      \[\leadsto \color{blue}{\left(1 - \frac{\cos \left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot \sin \frac{-7853981852531433}{5000000000000000}}{\sin \left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot \cos \frac{-7853981852531433}{5000000000000000}}\right) \cdot \left(\sin \left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot \cos \frac{-7853981852531433}{5000000000000000}\right)} \]

3. Applied rewrites 56.0%

   \[\leadsto \color{blue}{\left(1 - \frac{\sin -1.5707963705062866 \cdot \cos \left(6.28318530718 \cdot z0\right)}{\cos 1.5707963705062866 \cdot \sin \left(z0 \cdot -6.28318530718\right)}\right) \cdot \left(\cos 1.5707963705062866 \cdot \sin \left(z0 \cdot -6.28318530718\right)\right)} \]
4. Step-by-step derivation

   1. lift-sin.f64 N/A

      \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \color{blue}{\sin \left(z0 \cdot \frac{-314159265359}{50000000000}\right)}}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(z0 \cdot \frac{-314159265359}{50000000000}\right)\right) \]

   2. lift-*.f64 N/A

      \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \color{blue}{\left(z0 \cdot \frac{-314159265359}{50000000000}\right)}}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(z0 \cdot \frac{-314159265359}{50000000000}\right)\right) \]

   3. metadata-eval N/A

      \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(z0 \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{314159265359}{50000000000}\right)\right)}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(z0 \cdot \frac{-314159265359}{50000000000}\right)\right) \]

   4. distribute-rgt-neg-in N/A

      \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \color{blue}{\left(\mathsf{neg}\left(z0 \cdot \frac{314159265359}{50000000000}\right)\right)}}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(z0 \cdot \frac{-314159265359}{50000000000}\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(\mathsf{neg}\left(\color{blue}{\frac{314159265359}{50000000000} \cdot z0}\right)\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(z0 \cdot \frac{-314159265359}{50000000000}\right)\right) \]

   6. lift-*.f64 N/A

      \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(\mathsf{neg}\left(\color{blue}{\frac{314159265359}{50000000000} \cdot z0}\right)\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(z0 \cdot \frac{-314159265359}{50000000000}\right)\right) \]

   7. sin-neg-rev N/A

      \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \color{blue}{\left(\mathsf{neg}\left(\sin \left(\frac{314159265359}{50000000000} \cdot z0\right)\right)\right)}}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(z0 \cdot \frac{-314159265359}{50000000000}\right)\right) \]

   8. cos-+PI/2-rev N/A

      \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \color{blue}{\cos \left(\frac{314159265359}{50000000000} \cdot z0 + \frac{\mathsf{PI}\left(\right)}{2}\right)}}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(z0 \cdot \frac{-314159265359}{50000000000}\right)\right) \]

   9. lower-cos.f64 N/A

      \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \color{blue}{\cos \left(\frac{314159265359}{50000000000} \cdot z0 + \frac{\mathsf{PI}\left(\right)}{2}\right)}}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(z0 \cdot \frac{-314159265359}{50000000000}\right)\right) \]

   10. +-commutative N/A

       \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{2} + \frac{314159265359}{50000000000} \cdot z0\right)}}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(z0 \cdot \frac{-314159265359}{50000000000}\right)\right) \]

   11. lower-+.f64 N/A

       \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{2} + \frac{314159265359}{50000000000} \cdot z0\right)}}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(z0 \cdot \frac{-314159265359}{50000000000}\right)\right) \]

   12. lift-PI.f64 N/A

       \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\frac{\color{blue}{\pi}}{2} + \frac{314159265359}{50000000000} \cdot z0\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(z0 \cdot \frac{-314159265359}{50000000000}\right)\right) \]

   13. mult-flip N/A

       \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\color{blue}{\pi \cdot \frac{1}{2}} + \frac{314159265359}{50000000000} \cdot z0\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(z0 \cdot \frac{-314159265359}{50000000000}\right)\right) \]

   14. lower-*.f64 N/A

       \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\color{blue}{\pi \cdot \frac{1}{2}} + \frac{314159265359}{50000000000} \cdot z0\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(z0 \cdot \frac{-314159265359}{50000000000}\right)\right) \]

   15. metadata-eval 9.9%

       \[\leadsto \left(1 - \frac{\sin -1.5707963705062866 \cdot \cos \left(6.28318530718 \cdot z0\right)}{\cos 1.5707963705062866 \cdot \cos \left(\pi \cdot \color{blue}{0.5} + 6.28318530718 \cdot z0\right)}\right) \cdot \left(\cos 1.5707963705062866 \cdot \sin \left(z0 \cdot -6.28318530718\right)\right) \]

   16. lift-*.f64 N/A

       \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + \color{blue}{\frac{314159265359}{50000000000} \cdot z0}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(z0 \cdot \frac{-314159265359}{50000000000}\right)\right) \]

   17. *-commutative N/A

       \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + \color{blue}{z0 \cdot \frac{314159265359}{50000000000}}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(z0 \cdot \frac{-314159265359}{50000000000}\right)\right) \]

   18. lower-*.f64 9.9%

       \[\leadsto \left(1 - \frac{\sin -1.5707963705062866 \cdot \cos \left(6.28318530718 \cdot z0\right)}{\cos 1.5707963705062866 \cdot \cos \left(\pi \cdot 0.5 + \color{blue}{z0 \cdot 6.28318530718}\right)}\right) \cdot \left(\cos 1.5707963705062866 \cdot \sin \left(z0 \cdot -6.28318530718\right)\right) \]

5. Applied rewrites 9.9%

   \[\leadsto \left(1 - \frac{\sin -1.5707963705062866 \cdot \cos \left(6.28318530718 \cdot z0\right)}{\cos 1.5707963705062866 \cdot \color{blue}{\cos \left(\pi \cdot 0.5 + z0 \cdot 6.28318530718\right)}}\right) \cdot \left(\cos 1.5707963705062866 \cdot \sin \left(z0 \cdot -6.28318530718\right)\right) \]
6. Step-by-step derivation

   1. lift-sin.f64 N/A

      \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \color{blue}{\sin \left(z0 \cdot \frac{-314159265359}{50000000000}\right)}\right) \]

   2. lift-*.f64 N/A

      \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \color{blue}{\left(z0 \cdot \frac{-314159265359}{50000000000}\right)}\right) \]

   3. metadata-eval N/A

      \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(z0 \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{314159265359}{50000000000}\right)\right)}\right)\right) \]

   4. distribute-rgt-neg-in N/A

      \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \color{blue}{\left(\mathsf{neg}\left(z0 \cdot \frac{314159265359}{50000000000}\right)\right)}\right) \]

   5. *-commutative N/A

      \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(\mathsf{neg}\left(\color{blue}{\frac{314159265359}{50000000000} \cdot z0}\right)\right)\right) \]

   6. lift-*.f64 N/A

      \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(\mathsf{neg}\left(\color{blue}{\frac{314159265359}{50000000000} \cdot z0}\right)\right)\right) \]

   7. sin-neg-rev N/A

      \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \color{blue}{\left(\mathsf{neg}\left(\sin \left(\frac{314159265359}{50000000000} \cdot z0\right)\right)\right)}\right) \]

   8. cos-+PI/2-rev N/A

      \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \color{blue}{\cos \left(\frac{314159265359}{50000000000} \cdot z0 + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) \]

   9. lower-cos.f64 N/A

      \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \color{blue}{\cos \left(\frac{314159265359}{50000000000} \cdot z0 + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) \]

   10. +-commutative N/A

       \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{2} + \frac{314159265359}{50000000000} \cdot z0\right)}\right) \]

   11. lower-+.f64 N/A

       \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{2} + \frac{314159265359}{50000000000} \cdot z0\right)}\right) \]

   12. lift-PI.f64 N/A

       \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\frac{\color{blue}{\pi}}{2} + \frac{314159265359}{50000000000} \cdot z0\right)\right) \]

   13. mult-flip N/A

       \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\color{blue}{\pi \cdot \frac{1}{2}} + \frac{314159265359}{50000000000} \cdot z0\right)\right) \]

   14. lower-*.f64 N/A

       \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\color{blue}{\pi \cdot \frac{1}{2}} + \frac{314159265359}{50000000000} \cdot z0\right)\right) \]

   15. metadata-eval 57.3%

       \[\leadsto \left(1 - \frac{\sin -1.5707963705062866 \cdot \cos \left(6.28318530718 \cdot z0\right)}{\cos 1.5707963705062866 \cdot \cos \left(\pi \cdot 0.5 + z0 \cdot 6.28318530718\right)}\right) \cdot \left(\cos 1.5707963705062866 \cdot \cos \left(\pi \cdot \color{blue}{0.5} + 6.28318530718 \cdot z0\right)\right) \]

   16. lift-*.f64 N/A

       \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + \color{blue}{\frac{314159265359}{50000000000} \cdot z0}\right)\right) \]

   17. *-commutative N/A

       \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + \color{blue}{z0 \cdot \frac{314159265359}{50000000000}}\right)\right) \]

   18. lower-*.f64 57.3%

       \[\leadsto \left(1 - \frac{\sin -1.5707963705062866 \cdot \cos \left(6.28318530718 \cdot z0\right)}{\cos 1.5707963705062866 \cdot \cos \left(\pi \cdot 0.5 + z0 \cdot 6.28318530718\right)}\right) \cdot \left(\cos 1.5707963705062866 \cdot \cos \left(\pi \cdot 0.5 + \color{blue}{z0 \cdot 6.28318530718}\right)\right) \]

7. Applied rewrites 57.3%

   \[\leadsto \left(1 - \frac{\sin -1.5707963705062866 \cdot \cos \left(6.28318530718 \cdot z0\right)}{\cos 1.5707963705062866 \cdot \cos \left(\pi \cdot 0.5 + z0 \cdot 6.28318530718\right)}\right) \cdot \left(\cos 1.5707963705062866 \cdot \color{blue}{\cos \left(\pi \cdot 0.5 + z0 \cdot 6.28318530718\right)}\right) \]
8. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \left(1 - \color{blue}{\frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)\right) \]

   2. mult-flip N/A

      \[\leadsto \left(1 - \color{blue}{\left(\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)\right) \cdot \frac{1}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)\right) \]

   3. lift-*.f64 N/A

      \[\leadsto \left(1 - \color{blue}{\left(\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)\right)} \cdot \frac{1}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)\right) \]

   4. *-commutative N/A

      \[\leadsto \left(1 - \color{blue}{\left(\cos \left(\frac{314159265359}{50000000000} \cdot z0\right) \cdot \sin \frac{-7853981852531433}{5000000000000000}\right)} \cdot \frac{1}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)\right) \]

   5. associate-*l* N/A

      \[\leadsto \left(1 - \color{blue}{\cos \left(\frac{314159265359}{50000000000} \cdot z0\right) \cdot \left(\sin \frac{-7853981852531433}{5000000000000000} \cdot \frac{1}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)\right) \]

   6. lower-*.f64 N/A

      \[\leadsto \left(1 - \color{blue}{\cos \left(\frac{314159265359}{50000000000} \cdot z0\right) \cdot \left(\sin \frac{-7853981852531433}{5000000000000000} \cdot \frac{1}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)\right) \]

   7. lift-cos.f64 N/A

      \[\leadsto \left(1 - \color{blue}{\cos \left(\frac{314159265359}{50000000000} \cdot z0\right)} \cdot \left(\sin \frac{-7853981852531433}{5000000000000000} \cdot \frac{1}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right)\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)\right) \]

   8. lift-*.f64 N/A

      \[\leadsto \left(1 - \cos \color{blue}{\left(\frac{314159265359}{50000000000} \cdot z0\right)} \cdot \left(\sin \frac{-7853981852531433}{5000000000000000} \cdot \frac{1}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right)\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)\right) \]

   9. metadata-eval N/A

      \[\leadsto \left(1 - \cos \left(\color{blue}{\left(\mathsf{neg}\left(\frac{-314159265359}{50000000000}\right)\right)} \cdot z0\right) \cdot \left(\sin \frac{-7853981852531433}{5000000000000000} \cdot \frac{1}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right)\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)\right) \]

   10. distribute-lft-neg-in N/A

       \[\leadsto \left(1 - \cos \color{blue}{\left(\mathsf{neg}\left(\frac{-314159265359}{50000000000} \cdot z0\right)\right)} \cdot \left(\sin \frac{-7853981852531433}{5000000000000000} \cdot \frac{1}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right)\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)\right) \]

   11. lift-*.f64 N/A

       \[\leadsto \left(1 - \cos \left(\mathsf{neg}\left(\color{blue}{\frac{-314159265359}{50000000000} \cdot z0}\right)\right) \cdot \left(\sin \frac{-7853981852531433}{5000000000000000} \cdot \frac{1}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right)\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)\right) \]

   12. cos-neg-rev N/A

       \[\leadsto \left(1 - \color{blue}{\cos \left(\frac{-314159265359}{50000000000} \cdot z0\right)} \cdot \left(\sin \frac{-7853981852531433}{5000000000000000} \cdot \frac{1}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right)\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)\right) \]

   13. lift-cos.f64 N/A

       \[\leadsto \left(1 - \color{blue}{\cos \left(\frac{-314159265359}{50000000000} \cdot z0\right)} \cdot \left(\sin \frac{-7853981852531433}{5000000000000000} \cdot \frac{1}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right)\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)\right) \]

   14. lower-*.f64 N/A

       \[\leadsto \left(1 - \cos \left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot \color{blue}{\left(\sin \frac{-7853981852531433}{5000000000000000} \cdot \frac{1}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)\right) \]

   15. lower-/.f64 57.3%

       \[\leadsto \left(1 - \cos \left(-6.28318530718 \cdot z0\right) \cdot \left(\sin -1.5707963705062866 \cdot \color{blue}{\frac{1}{\cos 1.5707963705062866 \cdot \cos \left(\pi \cdot 0.5 + z0 \cdot 6.28318530718\right)}}\right)\right) \cdot \left(\cos 1.5707963705062866 \cdot \cos \left(\pi \cdot 0.5 + z0 \cdot 6.28318530718\right)\right) \]

   16. lift-*.f64 N/A

       \[\leadsto \left(1 - \cos \left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot \left(\sin \frac{-7853981852531433}{5000000000000000} \cdot \frac{1}{\color{blue}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}}\right)\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)\right) \]

   17. *-commutative N/A

       \[\leadsto \left(1 - \cos \left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot \left(\sin \frac{-7853981852531433}{5000000000000000} \cdot \frac{1}{\color{blue}{\cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right) \cdot \cos \frac{7853981852531433}{5000000000000000}}}\right)\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)\right) \]

9. Applied rewrites 57.3%

   \[\leadsto \left(1 - \color{blue}{\cos \left(-6.28318530718 \cdot z0\right) \cdot \left(\sin -1.5707963705062866 \cdot \frac{1}{\cos \left(-6.28318530718 \cdot z0 - 0.5 \cdot \pi\right) \cdot \cos 1.5707963705062866}\right)}\right) \cdot \left(\cos 1.5707963705062866 \cdot \cos \left(\pi \cdot 0.5 + z0 \cdot 6.28318530718\right)\right) \]

10. Evaluated real constant 40.5%

    \[\leadsto \left(1 - \cos \left(-6.28318530718 \cdot z0\right) \cdot \left(\sin -1.5707963705062866 \cdot \frac{1}{\cos \left(-6.28318530718 \cdot z0 - 0.5 \cdot \pi\right) \cdot \color{blue}{-4.3711389980768665 \cdot 10^{-8}}}\right)\right) \cdot \left(\cos 1.5707963705062866 \cdot \cos \left(\pi \cdot 0.5 + z0 \cdot 6.28318530718\right)\right) \]

11. Evaluated real constant 57.3%

    \[\leadsto \left(1 - \cos \left(-6.28318530718 \cdot z0\right) \cdot \left(\sin -1.5707963705062866 \cdot \frac{1}{\cos \left(-6.28318530718 \cdot z0 - 0.5 \cdot \pi\right) \cdot -4.3711389980768665 \cdot 10^{-8}}\right)\right) \cdot \left(\color{blue}{-4.3711389980768665 \cdot 10^{-8}} \cdot \cos \left(\pi \cdot 0.5 + z0 \cdot 6.28318530718\right)\right) \]
12. Add Preprocessing

Alternative 5: 57.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := -4.3711389980768665 \cdot 10^{-8} \cdot \cos \left(\pi \cdot 0.5 + z0 \cdot 6.28318530718\right)\\ \left(1 - \frac{\sin -1.5707963705062866 \cdot \cos \left(6.28318530718 \cdot z0\right)}{t\_0}\right) \cdot t\_0 \end{array} \]

FPCore

(FPCore (z0)
  :precision binary64
  (let* ((t_0
        (*
         -4.3711389980768665e-8
         (cos (+ (* PI 0.5) (* z0 6.28318530718))))))
  (*
   (-
    1.0
    (/ (* (sin -1.5707963705062866) (cos (* 6.28318530718 z0))) t_0))
   t_0)))

C

double code(double z0) {
	double t_0 = -4.3711389980768665e-8 * cos(((((double) M_PI) * 0.5) + (z0 * 6.28318530718)));
	return (1.0 - ((sin(-1.5707963705062866) * cos((6.28318530718 * z0))) / t_0)) * t_0;
}

Java

public static double code(double z0) {
	double t_0 = -4.3711389980768665e-8 * Math.cos(((Math.PI * 0.5) + (z0 * 6.28318530718)));
	return (1.0 - ((Math.sin(-1.5707963705062866) * Math.cos((6.28318530718 * z0))) / t_0)) * t_0;
}

Python

def code(z0):
	t_0 = -4.3711389980768665e-8 * math.cos(((math.pi * 0.5) + (z0 * 6.28318530718)))
	return (1.0 - ((math.sin(-1.5707963705062866) * math.cos((6.28318530718 * z0))) / t_0)) * t_0

Julia

function code(z0)
	t_0 = Float64(-4.3711389980768665e-8 * cos(Float64(Float64(pi * 0.5) + Float64(z0 * 6.28318530718))))
	return Float64(Float64(1.0 - Float64(Float64(sin(-1.5707963705062866) * cos(Float64(6.28318530718 * z0))) / t_0)) * t_0)
end

MATLAB

function tmp = code(z0)
	t_0 = -4.3711389980768665e-8 * cos(((pi * 0.5) + (z0 * 6.28318530718)));
	tmp = (1.0 - ((sin(-1.5707963705062866) * cos((6.28318530718 * z0))) / t_0)) * t_0;
end

Wolfram

code[z0_] := Block[{t$95$0 = N[(-4.3711389980768665e-8 * N[Cos[N[(N[(Pi * 0.5), $MachinePrecision] + N[(z0 * 6.28318530718), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(N[(1.0 - N[(N[(N[Sin[-1.5707963705062866], $MachinePrecision] * N[Cos[N[(6.28318530718 * z0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]]

Derivation

1. Initial program 57.2%

   \[\sin \left(-6.28318530718 \cdot z0 - -1.5707963705062866\right) \]
2. Step-by-step derivation

   1. lift-sin.f64 N/A

      \[\leadsto \color{blue}{\sin \left(\frac{-314159265359}{50000000000} \cdot z0 - \frac{-7853981852531433}{5000000000000000}\right)} \]

   2. lift--.f64 N/A

      \[\leadsto \sin \color{blue}{\left(\frac{-314159265359}{50000000000} \cdot z0 - \frac{-7853981852531433}{5000000000000000}\right)} \]

   3. sin-diff N/A

      \[\leadsto \color{blue}{\sin \left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot \cos \frac{-7853981852531433}{5000000000000000} - \cos \left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot \sin \frac{-7853981852531433}{5000000000000000}} \]

   4. sub-to-mult N/A

      \[\leadsto \color{blue}{\left(1 - \frac{\cos \left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot \sin \frac{-7853981852531433}{5000000000000000}}{\sin \left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot \cos \frac{-7853981852531433}{5000000000000000}}\right) \cdot \left(\sin \left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot \cos \frac{-7853981852531433}{5000000000000000}\right)} \]

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

      \[\leadsto \color{blue}{\left(1 - \frac{\cos \left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot \sin \frac{-7853981852531433}{5000000000000000}}{\sin \left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot \cos \frac{-7853981852531433}{5000000000000000}}\right) \cdot \left(\sin \left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot \cos \frac{-7853981852531433}{5000000000000000}\right)} \]

3. Applied rewrites 56.0%

   \[\leadsto \color{blue}{\left(1 - \frac{\sin -1.5707963705062866 \cdot \cos \left(6.28318530718 \cdot z0\right)}{\cos 1.5707963705062866 \cdot \sin \left(z0 \cdot -6.28318530718\right)}\right) \cdot \left(\cos 1.5707963705062866 \cdot \sin \left(z0 \cdot -6.28318530718\right)\right)} \]
4. Step-by-step derivation

   1. lift-sin.f64 N/A

      \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \color{blue}{\sin \left(z0 \cdot \frac{-314159265359}{50000000000}\right)}}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(z0 \cdot \frac{-314159265359}{50000000000}\right)\right) \]

   2. lift-*.f64 N/A

      \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \color{blue}{\left(z0 \cdot \frac{-314159265359}{50000000000}\right)}}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(z0 \cdot \frac{-314159265359}{50000000000}\right)\right) \]

   3. metadata-eval N/A

      \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(z0 \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{314159265359}{50000000000}\right)\right)}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(z0 \cdot \frac{-314159265359}{50000000000}\right)\right) \]

   4. distribute-rgt-neg-in N/A

      \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \color{blue}{\left(\mathsf{neg}\left(z0 \cdot \frac{314159265359}{50000000000}\right)\right)}}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(z0 \cdot \frac{-314159265359}{50000000000}\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(\mathsf{neg}\left(\color{blue}{\frac{314159265359}{50000000000} \cdot z0}\right)\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(z0 \cdot \frac{-314159265359}{50000000000}\right)\right) \]

   6. lift-*.f64 N/A

      \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(\mathsf{neg}\left(\color{blue}{\frac{314159265359}{50000000000} \cdot z0}\right)\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(z0 \cdot \frac{-314159265359}{50000000000}\right)\right) \]

   7. sin-neg-rev N/A

      \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \color{blue}{\left(\mathsf{neg}\left(\sin \left(\frac{314159265359}{50000000000} \cdot z0\right)\right)\right)}}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(z0 \cdot \frac{-314159265359}{50000000000}\right)\right) \]

   8. cos-+PI/2-rev N/A

      \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \color{blue}{\cos \left(\frac{314159265359}{50000000000} \cdot z0 + \frac{\mathsf{PI}\left(\right)}{2}\right)}}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(z0 \cdot \frac{-314159265359}{50000000000}\right)\right) \]

   9. lower-cos.f64 N/A

      \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \color{blue}{\cos \left(\frac{314159265359}{50000000000} \cdot z0 + \frac{\mathsf{PI}\left(\right)}{2}\right)}}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(z0 \cdot \frac{-314159265359}{50000000000}\right)\right) \]

   10. +-commutative N/A

       \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{2} + \frac{314159265359}{50000000000} \cdot z0\right)}}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(z0 \cdot \frac{-314159265359}{50000000000}\right)\right) \]

   11. lower-+.f64 N/A

       \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{2} + \frac{314159265359}{50000000000} \cdot z0\right)}}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(z0 \cdot \frac{-314159265359}{50000000000}\right)\right) \]

   12. lift-PI.f64 N/A

       \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\frac{\color{blue}{\pi}}{2} + \frac{314159265359}{50000000000} \cdot z0\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(z0 \cdot \frac{-314159265359}{50000000000}\right)\right) \]

   13. mult-flip N/A

       \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\color{blue}{\pi \cdot \frac{1}{2}} + \frac{314159265359}{50000000000} \cdot z0\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(z0 \cdot \frac{-314159265359}{50000000000}\right)\right) \]

   14. lower-*.f64 N/A

       \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\color{blue}{\pi \cdot \frac{1}{2}} + \frac{314159265359}{50000000000} \cdot z0\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(z0 \cdot \frac{-314159265359}{50000000000}\right)\right) \]

   15. metadata-eval 9.9%

       \[\leadsto \left(1 - \frac{\sin -1.5707963705062866 \cdot \cos \left(6.28318530718 \cdot z0\right)}{\cos 1.5707963705062866 \cdot \cos \left(\pi \cdot \color{blue}{0.5} + 6.28318530718 \cdot z0\right)}\right) \cdot \left(\cos 1.5707963705062866 \cdot \sin \left(z0 \cdot -6.28318530718\right)\right) \]

   16. lift-*.f64 N/A

       \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + \color{blue}{\frac{314159265359}{50000000000} \cdot z0}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(z0 \cdot \frac{-314159265359}{50000000000}\right)\right) \]

   17. *-commutative N/A

       \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + \color{blue}{z0 \cdot \frac{314159265359}{50000000000}}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(z0 \cdot \frac{-314159265359}{50000000000}\right)\right) \]

   18. lower-*.f64 9.9%

       \[\leadsto \left(1 - \frac{\sin -1.5707963705062866 \cdot \cos \left(6.28318530718 \cdot z0\right)}{\cos 1.5707963705062866 \cdot \cos \left(\pi \cdot 0.5 + \color{blue}{z0 \cdot 6.28318530718}\right)}\right) \cdot \left(\cos 1.5707963705062866 \cdot \sin \left(z0 \cdot -6.28318530718\right)\right) \]

5. Applied rewrites 9.9%

   \[\leadsto \left(1 - \frac{\sin -1.5707963705062866 \cdot \cos \left(6.28318530718 \cdot z0\right)}{\cos 1.5707963705062866 \cdot \color{blue}{\cos \left(\pi \cdot 0.5 + z0 \cdot 6.28318530718\right)}}\right) \cdot \left(\cos 1.5707963705062866 \cdot \sin \left(z0 \cdot -6.28318530718\right)\right) \]
6. Step-by-step derivation

   1. lift-sin.f64 N/A

      \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \color{blue}{\sin \left(z0 \cdot \frac{-314159265359}{50000000000}\right)}\right) \]

   2. lift-*.f64 N/A

      \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \color{blue}{\left(z0 \cdot \frac{-314159265359}{50000000000}\right)}\right) \]

   3. metadata-eval N/A

      \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(z0 \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{314159265359}{50000000000}\right)\right)}\right)\right) \]

   4. distribute-rgt-neg-in N/A

      \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \color{blue}{\left(\mathsf{neg}\left(z0 \cdot \frac{314159265359}{50000000000}\right)\right)}\right) \]

   5. *-commutative N/A

      \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(\mathsf{neg}\left(\color{blue}{\frac{314159265359}{50000000000} \cdot z0}\right)\right)\right) \]

   6. lift-*.f64 N/A

      \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(\mathsf{neg}\left(\color{blue}{\frac{314159265359}{50000000000} \cdot z0}\right)\right)\right) \]

   7. sin-neg-rev N/A

      \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \color{blue}{\left(\mathsf{neg}\left(\sin \left(\frac{314159265359}{50000000000} \cdot z0\right)\right)\right)}\right) \]

   8. cos-+PI/2-rev N/A

      \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \color{blue}{\cos \left(\frac{314159265359}{50000000000} \cdot z0 + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) \]

   9. lower-cos.f64 N/A

      \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \color{blue}{\cos \left(\frac{314159265359}{50000000000} \cdot z0 + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) \]

   10. +-commutative N/A

       \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{2} + \frac{314159265359}{50000000000} \cdot z0\right)}\right) \]

   11. lower-+.f64 N/A

       \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{2} + \frac{314159265359}{50000000000} \cdot z0\right)}\right) \]

   12. lift-PI.f64 N/A

       \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\frac{\color{blue}{\pi}}{2} + \frac{314159265359}{50000000000} \cdot z0\right)\right) \]

   13. mult-flip N/A

       \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\color{blue}{\pi \cdot \frac{1}{2}} + \frac{314159265359}{50000000000} \cdot z0\right)\right) \]

   14. lower-*.f64 N/A

       \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\color{blue}{\pi \cdot \frac{1}{2}} + \frac{314159265359}{50000000000} \cdot z0\right)\right) \]

   15. metadata-eval 57.3%

       \[\leadsto \left(1 - \frac{\sin -1.5707963705062866 \cdot \cos \left(6.28318530718 \cdot z0\right)}{\cos 1.5707963705062866 \cdot \cos \left(\pi \cdot 0.5 + z0 \cdot 6.28318530718\right)}\right) \cdot \left(\cos 1.5707963705062866 \cdot \cos \left(\pi \cdot \color{blue}{0.5} + 6.28318530718 \cdot z0\right)\right) \]

   16. lift-*.f64 N/A

       \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + \color{blue}{\frac{314159265359}{50000000000} \cdot z0}\right)\right) \]

   17. *-commutative N/A

       \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + \color{blue}{z0 \cdot \frac{314159265359}{50000000000}}\right)\right) \]

   18. lower-*.f64 57.3%

       \[\leadsto \left(1 - \frac{\sin -1.5707963705062866 \cdot \cos \left(6.28318530718 \cdot z0\right)}{\cos 1.5707963705062866 \cdot \cos \left(\pi \cdot 0.5 + z0 \cdot 6.28318530718\right)}\right) \cdot \left(\cos 1.5707963705062866 \cdot \cos \left(\pi \cdot 0.5 + \color{blue}{z0 \cdot 6.28318530718}\right)\right) \]

7. Applied rewrites 57.3%

   \[\leadsto \left(1 - \frac{\sin -1.5707963705062866 \cdot \cos \left(6.28318530718 \cdot z0\right)}{\cos 1.5707963705062866 \cdot \cos \left(\pi \cdot 0.5 + z0 \cdot 6.28318530718\right)}\right) \cdot \left(\cos 1.5707963705062866 \cdot \color{blue}{\cos \left(\pi \cdot 0.5 + z0 \cdot 6.28318530718\right)}\right) \]

8. Evaluated real constant 40.5%

   \[\leadsto \left(1 - \frac{\sin -1.5707963705062866 \cdot \cos \left(6.28318530718 \cdot z0\right)}{\color{blue}{-4.3711389980768665 \cdot 10^{-8}} \cdot \cos \left(\pi \cdot 0.5 + z0 \cdot 6.28318530718\right)}\right) \cdot \left(\cos 1.5707963705062866 \cdot \cos \left(\pi \cdot 0.5 + z0 \cdot 6.28318530718\right)\right) \]

9. Evaluated real constant 57.3%

   \[\leadsto \left(1 - \frac{\sin -1.5707963705062866 \cdot \cos \left(6.28318530718 \cdot z0\right)}{-4.3711389980768665 \cdot 10^{-8} \cdot \cos \left(\pi \cdot 0.5 + z0 \cdot 6.28318530718\right)}\right) \cdot \left(\color{blue}{-4.3711389980768665 \cdot 10^{-8}} \cdot \cos \left(\pi \cdot 0.5 + z0 \cdot 6.28318530718\right)\right) \]
10. Add Preprocessing

Alternative 6: 57.3% accurate, 0.3× speedup

Math

\[\cos \left(-6.28318530718 \cdot z0 - 0.5 \cdot \pi\right) \cdot \cos 1.5707963705062866 - \cos \left(-6.28318530718 \cdot z0\right) \cdot -0.999999999999999 \]

FPCore

(FPCore (z0)
  :precision binary64
  (-
 (*
  (cos (- (* -6.28318530718 z0) (* 0.5 PI)))
  (cos 1.5707963705062866))
 (* (cos (* -6.28318530718 z0)) -0.999999999999999)))

C

double code(double z0) {
	return (cos(((-6.28318530718 * z0) - (0.5 * ((double) M_PI)))) * cos(1.5707963705062866)) - (cos((-6.28318530718 * z0)) * -0.999999999999999);
}

Java

public static double code(double z0) {
	return (Math.cos(((-6.28318530718 * z0) - (0.5 * Math.PI))) * Math.cos(1.5707963705062866)) - (Math.cos((-6.28318530718 * z0)) * -0.999999999999999);
}

Python

def code(z0):
	return (math.cos(((-6.28318530718 * z0) - (0.5 * math.pi))) * math.cos(1.5707963705062866)) - (math.cos((-6.28318530718 * z0)) * -0.999999999999999)

Julia

function code(z0)
	return Float64(Float64(cos(Float64(Float64(-6.28318530718 * z0) - Float64(0.5 * pi))) * cos(1.5707963705062866)) - Float64(cos(Float64(-6.28318530718 * z0)) * -0.999999999999999))
end

MATLAB

function tmp = code(z0)
	tmp = (cos(((-6.28318530718 * z0) - (0.5 * pi))) * cos(1.5707963705062866)) - (cos((-6.28318530718 * z0)) * -0.999999999999999);
end

Wolfram

code[z0_] := N[(N[(N[Cos[N[(N[(-6.28318530718 * z0), $MachinePrecision] - N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[1.5707963705062866], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[N[(-6.28318530718 * z0), $MachinePrecision]], $MachinePrecision] * -0.999999999999999), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 57.2%

   \[\sin \left(-6.28318530718 \cdot z0 - -1.5707963705062866\right) \]
2. Step-by-step derivation

   1. lift-sin.f64 N/A

      \[\leadsto \color{blue}{\sin \left(\frac{-314159265359}{50000000000} \cdot z0 - \frac{-7853981852531433}{5000000000000000}\right)} \]

   2. lift--.f64 N/A

      \[\leadsto \sin \color{blue}{\left(\frac{-314159265359}{50000000000} \cdot z0 - \frac{-7853981852531433}{5000000000000000}\right)} \]

   3. sin-diff N/A

      \[\leadsto \color{blue}{\sin \left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot \cos \frac{-7853981852531433}{5000000000000000} - \cos \left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot \sin \frac{-7853981852531433}{5000000000000000}} \]

   4. sub-to-mult N/A

      \[\leadsto \color{blue}{\left(1 - \frac{\cos \left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot \sin \frac{-7853981852531433}{5000000000000000}}{\sin \left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot \cos \frac{-7853981852531433}{5000000000000000}}\right) \cdot \left(\sin \left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot \cos \frac{-7853981852531433}{5000000000000000}\right)} \]

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

      \[\leadsto \color{blue}{\left(1 - \frac{\cos \left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot \sin \frac{-7853981852531433}{5000000000000000}}{\sin \left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot \cos \frac{-7853981852531433}{5000000000000000}}\right) \cdot \left(\sin \left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot \cos \frac{-7853981852531433}{5000000000000000}\right)} \]

3. Applied rewrites 56.0%

   \[\leadsto \color{blue}{\left(1 - \frac{\sin -1.5707963705062866 \cdot \cos \left(6.28318530718 \cdot z0\right)}{\cos 1.5707963705062866 \cdot \sin \left(z0 \cdot -6.28318530718\right)}\right) \cdot \left(\cos 1.5707963705062866 \cdot \sin \left(z0 \cdot -6.28318530718\right)\right)} \]
4. Step-by-step derivation

   1. lift-sin.f64 N/A

      \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \color{blue}{\sin \left(z0 \cdot \frac{-314159265359}{50000000000}\right)}}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(z0 \cdot \frac{-314159265359}{50000000000}\right)\right) \]

   2. lift-*.f64 N/A

      \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \color{blue}{\left(z0 \cdot \frac{-314159265359}{50000000000}\right)}}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(z0 \cdot \frac{-314159265359}{50000000000}\right)\right) \]

   3. metadata-eval N/A

      \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(z0 \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{314159265359}{50000000000}\right)\right)}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(z0 \cdot \frac{-314159265359}{50000000000}\right)\right) \]

   4. distribute-rgt-neg-in N/A

      \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \color{blue}{\left(\mathsf{neg}\left(z0 \cdot \frac{314159265359}{50000000000}\right)\right)}}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(z0 \cdot \frac{-314159265359}{50000000000}\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(\mathsf{neg}\left(\color{blue}{\frac{314159265359}{50000000000} \cdot z0}\right)\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(z0 \cdot \frac{-314159265359}{50000000000}\right)\right) \]

   6. lift-*.f64 N/A

      \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(\mathsf{neg}\left(\color{blue}{\frac{314159265359}{50000000000} \cdot z0}\right)\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(z0 \cdot \frac{-314159265359}{50000000000}\right)\right) \]

   7. sin-neg-rev N/A

      \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \color{blue}{\left(\mathsf{neg}\left(\sin \left(\frac{314159265359}{50000000000} \cdot z0\right)\right)\right)}}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(z0 \cdot \frac{-314159265359}{50000000000}\right)\right) \]

   8. cos-+PI/2-rev N/A

      \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \color{blue}{\cos \left(\frac{314159265359}{50000000000} \cdot z0 + \frac{\mathsf{PI}\left(\right)}{2}\right)}}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(z0 \cdot \frac{-314159265359}{50000000000}\right)\right) \]

   9. lower-cos.f64 N/A

      \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \color{blue}{\cos \left(\frac{314159265359}{50000000000} \cdot z0 + \frac{\mathsf{PI}\left(\right)}{2}\right)}}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(z0 \cdot \frac{-314159265359}{50000000000}\right)\right) \]

   10. +-commutative N/A

       \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{2} + \frac{314159265359}{50000000000} \cdot z0\right)}}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(z0 \cdot \frac{-314159265359}{50000000000}\right)\right) \]

   11. lower-+.f64 N/A

       \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{2} + \frac{314159265359}{50000000000} \cdot z0\right)}}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(z0 \cdot \frac{-314159265359}{50000000000}\right)\right) \]

   12. lift-PI.f64 N/A

       \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\frac{\color{blue}{\pi}}{2} + \frac{314159265359}{50000000000} \cdot z0\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(z0 \cdot \frac{-314159265359}{50000000000}\right)\right) \]

   13. mult-flip N/A

       \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\color{blue}{\pi \cdot \frac{1}{2}} + \frac{314159265359}{50000000000} \cdot z0\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(z0 \cdot \frac{-314159265359}{50000000000}\right)\right) \]

   14. lower-*.f64 N/A

       \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\color{blue}{\pi \cdot \frac{1}{2}} + \frac{314159265359}{50000000000} \cdot z0\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(z0 \cdot \frac{-314159265359}{50000000000}\right)\right) \]

   15. metadata-eval 9.9%

       \[\leadsto \left(1 - \frac{\sin -1.5707963705062866 \cdot \cos \left(6.28318530718 \cdot z0\right)}{\cos 1.5707963705062866 \cdot \cos \left(\pi \cdot \color{blue}{0.5} + 6.28318530718 \cdot z0\right)}\right) \cdot \left(\cos 1.5707963705062866 \cdot \sin \left(z0 \cdot -6.28318530718\right)\right) \]

   16. lift-*.f64 N/A

       \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + \color{blue}{\frac{314159265359}{50000000000} \cdot z0}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(z0 \cdot \frac{-314159265359}{50000000000}\right)\right) \]

   17. *-commutative N/A

       \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + \color{blue}{z0 \cdot \frac{314159265359}{50000000000}}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(z0 \cdot \frac{-314159265359}{50000000000}\right)\right) \]

   18. lower-*.f64 9.9%

       \[\leadsto \left(1 - \frac{\sin -1.5707963705062866 \cdot \cos \left(6.28318530718 \cdot z0\right)}{\cos 1.5707963705062866 \cdot \cos \left(\pi \cdot 0.5 + \color{blue}{z0 \cdot 6.28318530718}\right)}\right) \cdot \left(\cos 1.5707963705062866 \cdot \sin \left(z0 \cdot -6.28318530718\right)\right) \]

5. Applied rewrites 9.9%

   \[\leadsto \left(1 - \frac{\sin -1.5707963705062866 \cdot \cos \left(6.28318530718 \cdot z0\right)}{\cos 1.5707963705062866 \cdot \color{blue}{\cos \left(\pi \cdot 0.5 + z0 \cdot 6.28318530718\right)}}\right) \cdot \left(\cos 1.5707963705062866 \cdot \sin \left(z0 \cdot -6.28318530718\right)\right) \]
6. Step-by-step derivation

   1. lift-sin.f64 N/A

      \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \color{blue}{\sin \left(z0 \cdot \frac{-314159265359}{50000000000}\right)}\right) \]

   2. lift-*.f64 N/A

      \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \color{blue}{\left(z0 \cdot \frac{-314159265359}{50000000000}\right)}\right) \]

   3. metadata-eval N/A

      \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(z0 \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{314159265359}{50000000000}\right)\right)}\right)\right) \]

   4. distribute-rgt-neg-in N/A

      \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \color{blue}{\left(\mathsf{neg}\left(z0 \cdot \frac{314159265359}{50000000000}\right)\right)}\right) \]

   5. *-commutative N/A

      \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(\mathsf{neg}\left(\color{blue}{\frac{314159265359}{50000000000} \cdot z0}\right)\right)\right) \]

   6. lift-*.f64 N/A

      \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \sin \left(\mathsf{neg}\left(\color{blue}{\frac{314159265359}{50000000000} \cdot z0}\right)\right)\right) \]

   7. sin-neg-rev N/A

      \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \color{blue}{\left(\mathsf{neg}\left(\sin \left(\frac{314159265359}{50000000000} \cdot z0\right)\right)\right)}\right) \]

   8. cos-+PI/2-rev N/A

      \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \color{blue}{\cos \left(\frac{314159265359}{50000000000} \cdot z0 + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) \]

   9. lower-cos.f64 N/A

      \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \color{blue}{\cos \left(\frac{314159265359}{50000000000} \cdot z0 + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) \]

   10. +-commutative N/A

       \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{2} + \frac{314159265359}{50000000000} \cdot z0\right)}\right) \]

   11. lower-+.f64 N/A

       \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{2} + \frac{314159265359}{50000000000} \cdot z0\right)}\right) \]

   12. lift-PI.f64 N/A

       \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\frac{\color{blue}{\pi}}{2} + \frac{314159265359}{50000000000} \cdot z0\right)\right) \]

   13. mult-flip N/A

       \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\color{blue}{\pi \cdot \frac{1}{2}} + \frac{314159265359}{50000000000} \cdot z0\right)\right) \]

   14. lower-*.f64 N/A

       \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\color{blue}{\pi \cdot \frac{1}{2}} + \frac{314159265359}{50000000000} \cdot z0\right)\right) \]

   15. metadata-eval 57.3%

       \[\leadsto \left(1 - \frac{\sin -1.5707963705062866 \cdot \cos \left(6.28318530718 \cdot z0\right)}{\cos 1.5707963705062866 \cdot \cos \left(\pi \cdot 0.5 + z0 \cdot 6.28318530718\right)}\right) \cdot \left(\cos 1.5707963705062866 \cdot \cos \left(\pi \cdot \color{blue}{0.5} + 6.28318530718 \cdot z0\right)\right) \]

   16. lift-*.f64 N/A

       \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + \color{blue}{\frac{314159265359}{50000000000} \cdot z0}\right)\right) \]

   17. *-commutative N/A

       \[\leadsto \left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + \color{blue}{z0 \cdot \frac{314159265359}{50000000000}}\right)\right) \]

   18. lower-*.f64 57.3%

       \[\leadsto \left(1 - \frac{\sin -1.5707963705062866 \cdot \cos \left(6.28318530718 \cdot z0\right)}{\cos 1.5707963705062866 \cdot \cos \left(\pi \cdot 0.5 + z0 \cdot 6.28318530718\right)}\right) \cdot \left(\cos 1.5707963705062866 \cdot \cos \left(\pi \cdot 0.5 + \color{blue}{z0 \cdot 6.28318530718}\right)\right) \]

7. Applied rewrites 57.3%

   \[\leadsto \left(1 - \frac{\sin -1.5707963705062866 \cdot \cos \left(6.28318530718 \cdot z0\right)}{\cos 1.5707963705062866 \cdot \cos \left(\pi \cdot 0.5 + z0 \cdot 6.28318530718\right)}\right) \cdot \left(\cos 1.5707963705062866 \cdot \color{blue}{\cos \left(\pi \cdot 0.5 + z0 \cdot 6.28318530718\right)}\right) \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)\right)} \]

   2. lift--.f64 N/A

      \[\leadsto \color{blue}{\left(1 - \frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}\right)} \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)\right) \]

   3. lift-/.f64 N/A

      \[\leadsto \left(1 - \color{blue}{\frac{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)}}\right) \cdot \left(\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right)\right) \]

   4. sub-to-mult-rev N/A

      \[\leadsto \color{blue}{\cos \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\pi \cdot \frac{1}{2} + z0 \cdot \frac{314159265359}{50000000000}\right) - \sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{314159265359}{50000000000} \cdot z0\right)} \]

   5. lower--.f64 57.3%

      \[\leadsto \color{blue}{\cos 1.5707963705062866 \cdot \cos \left(\pi \cdot 0.5 + z0 \cdot 6.28318530718\right) - \sin -1.5707963705062866 \cdot \cos \left(6.28318530718 \cdot z0\right)} \]

9. Applied rewrites 57.3%

   \[\leadsto \color{blue}{\cos \left(-6.28318530718 \cdot z0 - 0.5 \cdot \pi\right) \cdot \cos 1.5707963705062866 - \cos \left(-6.28318530718 \cdot z0\right) \cdot \sin -1.5707963705062866} \]

10. Evaluated real constant 57.3%

    \[\leadsto \cos \left(-6.28318530718 \cdot z0 - 0.5 \cdot \pi\right) \cdot \cos 1.5707963705062866 - \cos \left(-6.28318530718 \cdot z0\right) \cdot \color{blue}{-0.999999999999999} \]
11. Add Preprocessing

Alternative 7: 57.3% accurate, 0.3× speedup

Math

\[0.999999999999999 \cdot \cos \left(6.28318530718 \cdot z0\right) - \sin \left(6.28318530718 \cdot z0\right) \cdot \cos 1.5707963705062866 \]

FPCore

(FPCore (z0)
  :precision binary64
  (-
 (* 0.999999999999999 (cos (* 6.28318530718 z0)))
 (* (sin (* 6.28318530718 z0)) (cos 1.5707963705062866))))

C

double code(double z0) {
	return (0.999999999999999 * cos((6.28318530718 * z0))) - (sin((6.28318530718 * z0)) * cos(1.5707963705062866));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(z0)
use fmin_fmax_functions
    real(8), intent (in) :: z0
    code = (0.999999999999999d0 * cos((6.28318530718d0 * z0))) - (sin((6.28318530718d0 * z0)) * cos(1.5707963705062866d0))
end function

Java

public static double code(double z0) {
	return (0.999999999999999 * Math.cos((6.28318530718 * z0))) - (Math.sin((6.28318530718 * z0)) * Math.cos(1.5707963705062866));
}

Python

def code(z0):
	return (0.999999999999999 * math.cos((6.28318530718 * z0))) - (math.sin((6.28318530718 * z0)) * math.cos(1.5707963705062866))

Julia

function code(z0)
	return Float64(Float64(0.999999999999999 * cos(Float64(6.28318530718 * z0))) - Float64(sin(Float64(6.28318530718 * z0)) * cos(1.5707963705062866)))
end

MATLAB

function tmp = code(z0)
	tmp = (0.999999999999999 * cos((6.28318530718 * z0))) - (sin((6.28318530718 * z0)) * cos(1.5707963705062866));
end

Wolfram

code[z0_] := N[(N[(0.999999999999999 * N[Cos[N[(6.28318530718 * z0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[N[(6.28318530718 * z0), $MachinePrecision]], $MachinePrecision] * N[Cos[1.5707963705062866], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 57.2%

   \[\sin \left(-6.28318530718 \cdot z0 - -1.5707963705062866\right) \]
2. Step-by-step derivation

   1. lift-sin.f64 N/A

      \[\leadsto \color{blue}{\sin \left(\frac{-314159265359}{50000000000} \cdot z0 - \frac{-7853981852531433}{5000000000000000}\right)} \]

   2. lift--.f64 N/A

      \[\leadsto \sin \color{blue}{\left(\frac{-314159265359}{50000000000} \cdot z0 - \frac{-7853981852531433}{5000000000000000}\right)} \]

   3. sin-diff N/A

      \[\leadsto \color{blue}{\sin \left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot \cos \frac{-7853981852531433}{5000000000000000} - \cos \left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot \sin \frac{-7853981852531433}{5000000000000000}} \]

   4. fp-cancel-sub-sign-inv N/A

      \[\leadsto \color{blue}{\sin \left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot \cos \frac{-7853981852531433}{5000000000000000} + \left(\mathsf{neg}\left(\cos \left(\frac{-314159265359}{50000000000} \cdot z0\right)\right)\right) \cdot \sin \frac{-7853981852531433}{5000000000000000}} \]

   5. +-commutative N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\cos \left(\frac{-314159265359}{50000000000} \cdot z0\right)\right)\right) \cdot \sin \frac{-7853981852531433}{5000000000000000} + \sin \left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot \cos \frac{-7853981852531433}{5000000000000000}} \]

   6. add-flip N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\cos \left(\frac{-314159265359}{50000000000} \cdot z0\right)\right)\right) \cdot \sin \frac{-7853981852531433}{5000000000000000} - \left(\mathsf{neg}\left(\sin \left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot \cos \frac{-7853981852531433}{5000000000000000}\right)\right)} \]

   7. lower--.f64 N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\cos \left(\frac{-314159265359}{50000000000} \cdot z0\right)\right)\right) \cdot \sin \frac{-7853981852531433}{5000000000000000} - \left(\mathsf{neg}\left(\sin \left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot \cos \frac{-7853981852531433}{5000000000000000}\right)\right)} \]

   8. distribute-lft-neg-out N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\cos \left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot \sin \frac{-7853981852531433}{5000000000000000}\right)\right)} - \left(\mathsf{neg}\left(\sin \left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot \cos \frac{-7853981852531433}{5000000000000000}\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{-314159265359}{50000000000} \cdot z0\right)}\right)\right) - \left(\mathsf{neg}\left(\sin \left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot \cos \frac{-7853981852531433}{5000000000000000}\right)\right) \]

   10. distribute-lft-neg-in N/A

       \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\sin \frac{-7853981852531433}{5000000000000000}\right)\right) \cdot \cos \left(\frac{-314159265359}{50000000000} \cdot z0\right)} - \left(\mathsf{neg}\left(\sin \left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot \cos \frac{-7853981852531433}{5000000000000000}\right)\right) \]

   11. sin-neg N/A

       \[\leadsto \color{blue}{\sin \left(\mathsf{neg}\left(\frac{-7853981852531433}{5000000000000000}\right)\right)} \cdot \cos \left(\frac{-314159265359}{50000000000} \cdot z0\right) - \left(\mathsf{neg}\left(\sin \left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot \cos \frac{-7853981852531433}{5000000000000000}\right)\right) \]

   12. lower-*.f64 N/A

       \[\leadsto \color{blue}{\sin \left(\mathsf{neg}\left(\frac{-7853981852531433}{5000000000000000}\right)\right) \cdot \cos \left(\frac{-314159265359}{50000000000} \cdot z0\right)} - \left(\mathsf{neg}\left(\sin \left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot \cos \frac{-7853981852531433}{5000000000000000}\right)\right) \]

   13. lower-sin.f64 N/A

       \[\leadsto \color{blue}{\sin \left(\mathsf{neg}\left(\frac{-7853981852531433}{5000000000000000}\right)\right)} \cdot \cos \left(\frac{-314159265359}{50000000000} \cdot z0\right) - \left(\mathsf{neg}\left(\sin \left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot \cos \frac{-7853981852531433}{5000000000000000}\right)\right) \]

   14. metadata-eval N/A

       \[\leadsto \sin \color{blue}{\frac{7853981852531433}{5000000000000000}} \cdot \cos \left(\frac{-314159265359}{50000000000} \cdot z0\right) - \left(\mathsf{neg}\left(\sin \left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot \cos \frac{-7853981852531433}{5000000000000000}\right)\right) \]

   15. cos-neg-rev N/A

       \[\leadsto \sin \frac{7853981852531433}{5000000000000000} \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\frac{-314159265359}{50000000000} \cdot z0\right)\right)} - \left(\mathsf{neg}\left(\sin \left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot \cos \frac{-7853981852531433}{5000000000000000}\right)\right) \]

   16. lower-cos.f64 N/A

       \[\leadsto \sin \frac{7853981852531433}{5000000000000000} \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\frac{-314159265359}{50000000000} \cdot z0\right)\right)} - \left(\mathsf{neg}\left(\sin \left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot \cos \frac{-7853981852531433}{5000000000000000}\right)\right) \]

   17. lift-*.f64 N/A

       \[\leadsto \sin \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\mathsf{neg}\left(\color{blue}{\frac{-314159265359}{50000000000} \cdot z0}\right)\right) - \left(\mathsf{neg}\left(\sin \left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot \cos \frac{-7853981852531433}{5000000000000000}\right)\right) \]

   18. distribute-lft-neg-out N/A

       \[\leadsto \sin \frac{7853981852531433}{5000000000000000} \cdot \cos \color{blue}{\left(\left(\mathsf{neg}\left(\frac{-314159265359}{50000000000}\right)\right) \cdot z0\right)} - \left(\mathsf{neg}\left(\sin \left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot \cos \frac{-7853981852531433}{5000000000000000}\right)\right) \]

   19. lower-*.f64 N/A

       \[\leadsto \sin \frac{7853981852531433}{5000000000000000} \cdot \cos \color{blue}{\left(\left(\mathsf{neg}\left(\frac{-314159265359}{50000000000}\right)\right) \cdot z0\right)} - \left(\mathsf{neg}\left(\sin \left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot \cos \frac{-7853981852531433}{5000000000000000}\right)\right) \]

   20. metadata-eval N/A

       \[\leadsto \sin \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\color{blue}{\frac{314159265359}{50000000000}} \cdot z0\right) - \left(\mathsf{neg}\left(\sin \left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot \cos \frac{-7853981852531433}{5000000000000000}\right)\right) \]

3. Applied rewrites 57.3%

   \[\leadsto \color{blue}{\sin 1.5707963705062866 \cdot \cos \left(6.28318530718 \cdot z0\right) - \sin \left(6.28318530718 \cdot z0\right) \cdot \cos 1.5707963705062866} \]

4. Evaluated real constant 57.3%

   \[\leadsto \color{blue}{0.999999999999999} \cdot \cos \left(6.28318530718 \cdot z0\right) - \sin \left(6.28318530718 \cdot z0\right) \cdot \cos 1.5707963705062866 \]
5. Add Preprocessing

Alternative 8: 57.3% accurate, 0.9× speedup

Math

\[\cos \left(\left(-1.5707963705062866 - -6.28318530718 \cdot z0\right) - \left(\pi \cdot 0.5 + \pi\right)\right) \]

FPCore

(FPCore (z0)
  :precision binary64
  (cos
 (- (- -1.5707963705062866 (* -6.28318530718 z0)) (+ (* PI 0.5) PI))))

C

double code(double z0) {
	return cos(((-1.5707963705062866 - (-6.28318530718 * z0)) - ((((double) M_PI) * 0.5) + ((double) M_PI))));
}

Java

public static double code(double z0) {
	return Math.cos(((-1.5707963705062866 - (-6.28318530718 * z0)) - ((Math.PI * 0.5) + Math.PI)));
}

Python

def code(z0):
	return math.cos(((-1.5707963705062866 - (-6.28318530718 * z0)) - ((math.pi * 0.5) + math.pi)))

Julia

function code(z0)
	return cos(Float64(Float64(-1.5707963705062866 - Float64(-6.28318530718 * z0)) - Float64(Float64(pi * 0.5) + pi)))
end

MATLAB

function tmp = code(z0)
	tmp = cos(((-1.5707963705062866 - (-6.28318530718 * z0)) - ((pi * 0.5) + pi)));
end

Wolfram

code[z0_] := N[Cos[N[(N[(-1.5707963705062866 - N[(-6.28318530718 * z0), $MachinePrecision]), $MachinePrecision] - N[(N[(Pi * 0.5), $MachinePrecision] + Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 57.2%

   \[\sin \left(-6.28318530718 \cdot z0 - -1.5707963705062866\right) \]
2. Step-by-step derivation

   1. lift-sin.f64 N/A

      \[\leadsto \color{blue}{\sin \left(\frac{-314159265359}{50000000000} \cdot z0 - \frac{-7853981852531433}{5000000000000000}\right)} \]

   2. lift--.f64 N/A

      \[\leadsto \sin \color{blue}{\left(\frac{-314159265359}{50000000000} \cdot z0 - \frac{-7853981852531433}{5000000000000000}\right)} \]

   3. sin-diff N/A

      \[\leadsto \color{blue}{\sin \left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot \cos \frac{-7853981852531433}{5000000000000000} - \cos \left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot \sin \frac{-7853981852531433}{5000000000000000}} \]

   4. fp-cancel-sub-sign-inv N/A

      \[\leadsto \color{blue}{\sin \left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot \cos \frac{-7853981852531433}{5000000000000000} + \left(\mathsf{neg}\left(\cos \left(\frac{-314159265359}{50000000000} \cdot z0\right)\right)\right) \cdot \sin \frac{-7853981852531433}{5000000000000000}} \]

   5. +-commutative N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\cos \left(\frac{-314159265359}{50000000000} \cdot z0\right)\right)\right) \cdot \sin \frac{-7853981852531433}{5000000000000000} + \sin \left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot \cos \frac{-7853981852531433}{5000000000000000}} \]

   6. add-flip N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\cos \left(\frac{-314159265359}{50000000000} \cdot z0\right)\right)\right) \cdot \sin \frac{-7853981852531433}{5000000000000000} - \left(\mathsf{neg}\left(\sin \left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot \cos \frac{-7853981852531433}{5000000000000000}\right)\right)} \]

   7. lower--.f64 N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\cos \left(\frac{-314159265359}{50000000000} \cdot z0\right)\right)\right) \cdot \sin \frac{-7853981852531433}{5000000000000000} - \left(\mathsf{neg}\left(\sin \left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot \cos \frac{-7853981852531433}{5000000000000000}\right)\right)} \]

   8. distribute-lft-neg-out N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\cos \left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot \sin \frac{-7853981852531433}{5000000000000000}\right)\right)} - \left(\mathsf{neg}\left(\sin \left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot \cos \frac{-7853981852531433}{5000000000000000}\right)\right) \]

   9. *-commutative N/A

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\sin \frac{-7853981852531433}{5000000000000000} \cdot \cos \left(\frac{-314159265359}{50000000000} \cdot z0\right)}\right)\right) - \left(\mathsf{neg}\left(\sin \left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot \cos \frac{-7853981852531433}{5000000000000000}\right)\right) \]

   10. distribute-lft-neg-in N/A

       \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\sin \frac{-7853981852531433}{5000000000000000}\right)\right) \cdot \cos \left(\frac{-314159265359}{50000000000} \cdot z0\right)} - \left(\mathsf{neg}\left(\sin \left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot \cos \frac{-7853981852531433}{5000000000000000}\right)\right) \]

   11. sin-neg N/A

       \[\leadsto \color{blue}{\sin \left(\mathsf{neg}\left(\frac{-7853981852531433}{5000000000000000}\right)\right)} \cdot \cos \left(\frac{-314159265359}{50000000000} \cdot z0\right) - \left(\mathsf{neg}\left(\sin \left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot \cos \frac{-7853981852531433}{5000000000000000}\right)\right) \]

   12. lower-*.f64 N/A

       \[\leadsto \color{blue}{\sin \left(\mathsf{neg}\left(\frac{-7853981852531433}{5000000000000000}\right)\right) \cdot \cos \left(\frac{-314159265359}{50000000000} \cdot z0\right)} - \left(\mathsf{neg}\left(\sin \left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot \cos \frac{-7853981852531433}{5000000000000000}\right)\right) \]

   13. lower-sin.f64 N/A

       \[\leadsto \color{blue}{\sin \left(\mathsf{neg}\left(\frac{-7853981852531433}{5000000000000000}\right)\right)} \cdot \cos \left(\frac{-314159265359}{50000000000} \cdot z0\right) - \left(\mathsf{neg}\left(\sin \left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot \cos \frac{-7853981852531433}{5000000000000000}\right)\right) \]

   14. metadata-eval N/A

       \[\leadsto \sin \color{blue}{\frac{7853981852531433}{5000000000000000}} \cdot \cos \left(\frac{-314159265359}{50000000000} \cdot z0\right) - \left(\mathsf{neg}\left(\sin \left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot \cos \frac{-7853981852531433}{5000000000000000}\right)\right) \]

   15. cos-neg-rev N/A

       \[\leadsto \sin \frac{7853981852531433}{5000000000000000} \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\frac{-314159265359}{50000000000} \cdot z0\right)\right)} - \left(\mathsf{neg}\left(\sin \left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot \cos \frac{-7853981852531433}{5000000000000000}\right)\right) \]

   16. lower-cos.f64 N/A

       \[\leadsto \sin \frac{7853981852531433}{5000000000000000} \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\frac{-314159265359}{50000000000} \cdot z0\right)\right)} - \left(\mathsf{neg}\left(\sin \left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot \cos \frac{-7853981852531433}{5000000000000000}\right)\right) \]

   17. lift-*.f64 N/A

       \[\leadsto \sin \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\mathsf{neg}\left(\color{blue}{\frac{-314159265359}{50000000000} \cdot z0}\right)\right) - \left(\mathsf{neg}\left(\sin \left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot \cos \frac{-7853981852531433}{5000000000000000}\right)\right) \]

   18. distribute-lft-neg-out N/A

       \[\leadsto \sin \frac{7853981852531433}{5000000000000000} \cdot \cos \color{blue}{\left(\left(\mathsf{neg}\left(\frac{-314159265359}{50000000000}\right)\right) \cdot z0\right)} - \left(\mathsf{neg}\left(\sin \left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot \cos \frac{-7853981852531433}{5000000000000000}\right)\right) \]

   19. lower-*.f64 N/A

       \[\leadsto \sin \frac{7853981852531433}{5000000000000000} \cdot \cos \color{blue}{\left(\left(\mathsf{neg}\left(\frac{-314159265359}{50000000000}\right)\right) \cdot z0\right)} - \left(\mathsf{neg}\left(\sin \left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot \cos \frac{-7853981852531433}{5000000000000000}\right)\right) \]

   20. metadata-eval N/A

       \[\leadsto \sin \frac{7853981852531433}{5000000000000000} \cdot \cos \left(\color{blue}{\frac{314159265359}{50000000000}} \cdot z0\right) - \left(\mathsf{neg}\left(\sin \left(\frac{-314159265359}{50000000000} \cdot z0\right) \cdot \cos \frac{-7853981852531433}{5000000000000000}\right)\right) \]

3. Applied rewrites 57.3%

   \[\leadsto \color{blue}{\sin 1.5707963705062866 \cdot \cos \left(6.28318530718 \cdot z0\right) - \sin \left(6.28318530718 \cdot z0\right) \cdot \cos 1.5707963705062866} \]

5. Applied rewrites 57.3%

   \[\leadsto \color{blue}{\cos \left(\left(-1.5707963705062866 - -6.28318530718 \cdot z0\right) - \left(\pi \cdot 0.5 + \pi\right)\right)} \]
6. Add Preprocessing

Alternative 9: 57.2% accurate, 0.9× speedup

Math

\[\cos \left(\left(0.5 \cdot \pi - 1.5707963705062866\right) - z0 \cdot -6.28318530718\right) \]

FPCore

(FPCore (z0)
  :precision binary64
  (cos (- (- (* 0.5 PI) 1.5707963705062866) (* z0 -6.28318530718))))

C

double code(double z0) {
	return cos((((0.5 * ((double) M_PI)) - 1.5707963705062866) - (z0 * -6.28318530718)));
}

Java

public static double code(double z0) {
	return Math.cos((((0.5 * Math.PI) - 1.5707963705062866) - (z0 * -6.28318530718)));
}

Python

def code(z0):
	return math.cos((((0.5 * math.pi) - 1.5707963705062866) - (z0 * -6.28318530718)))

Julia

function code(z0)
	return cos(Float64(Float64(Float64(0.5 * pi) - 1.5707963705062866) - Float64(z0 * -6.28318530718)))
end

MATLAB

function tmp = code(z0)
	tmp = cos((((0.5 * pi) - 1.5707963705062866) - (z0 * -6.28318530718)));
end

Wolfram

code[z0_] := N[Cos[N[(N[(N[(0.5 * Pi), $MachinePrecision] - 1.5707963705062866), $MachinePrecision] - N[(z0 * -6.28318530718), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 57.2%

   \[\sin \left(-6.28318530718 \cdot z0 - -1.5707963705062866\right) \]
2. Step-by-step derivation

   1. lift-sin.f64 N/A

      \[\leadsto \color{blue}{\sin \left(\frac{-314159265359}{50000000000} \cdot z0 - \frac{-7853981852531433}{5000000000000000}\right)} \]

   2. lift--.f64 N/A

      \[\leadsto \sin \color{blue}{\left(\frac{-314159265359}{50000000000} \cdot z0 - \frac{-7853981852531433}{5000000000000000}\right)} \]

   3. sub-negate-rev N/A

      \[\leadsto \sin \color{blue}{\left(\mathsf{neg}\left(\left(\frac{-7853981852531433}{5000000000000000} - \frac{-314159265359}{50000000000} \cdot z0\right)\right)\right)} \]

   4. sin-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\sin \left(\frac{-7853981852531433}{5000000000000000} - \frac{-314159265359}{50000000000} \cdot z0\right)\right)} \]

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

      \[\leadsto \color{blue}{\cos \left(\left(\frac{-7853981852531433}{5000000000000000} - \frac{-314159265359}{50000000000} \cdot z0\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

   6. +-commutative N/A

      \[\leadsto \cos \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{2} + \left(\frac{-7853981852531433}{5000000000000000} - \frac{-314159265359}{50000000000} \cdot z0\right)\right)} \]

   7. associate--l+ N/A

      \[\leadsto \cos \color{blue}{\left(\left(\frac{\mathsf{PI}\left(\right)}{2} + \frac{-7853981852531433}{5000000000000000}\right) - \frac{-314159265359}{50000000000} \cdot z0\right)} \]

   8. +-commutative N/A

      \[\leadsto \cos \left(\color{blue}{\left(\frac{-7853981852531433}{5000000000000000} + \frac{\mathsf{PI}\left(\right)}{2}\right)} - \frac{-314159265359}{50000000000} \cdot z0\right) \]

   9. lower-cos.f64 N/A

      \[\leadsto \color{blue}{\cos \left(\left(\frac{-7853981852531433}{5000000000000000} + \frac{\mathsf{PI}\left(\right)}{2}\right) - \frac{-314159265359}{50000000000} \cdot z0\right)} \]

   10. lower--.f64 N/A

       \[\leadsto \cos \color{blue}{\left(\left(\frac{-7853981852531433}{5000000000000000} + \frac{\mathsf{PI}\left(\right)}{2}\right) - \frac{-314159265359}{50000000000} \cdot z0\right)} \]

   11. +-commutative N/A

       \[\leadsto \cos \left(\color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{2} + \frac{-7853981852531433}{5000000000000000}\right)} - \frac{-314159265359}{50000000000} \cdot z0\right) \]

   12. add-flip N/A

       \[\leadsto \cos \left(\color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{2} - \left(\mathsf{neg}\left(\frac{-7853981852531433}{5000000000000000}\right)\right)\right)} - \frac{-314159265359}{50000000000} \cdot z0\right) \]

   13. lower--.f64 N/A

       \[\leadsto \cos \left(\color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{2} - \left(\mathsf{neg}\left(\frac{-7853981852531433}{5000000000000000}\right)\right)\right)} - \frac{-314159265359}{50000000000} \cdot z0\right) \]

   14. mult-flip N/A

       \[\leadsto \cos \left(\left(\color{blue}{\mathsf{PI}\left(\right) \cdot \frac{1}{2}} - \left(\mathsf{neg}\left(\frac{-7853981852531433}{5000000000000000}\right)\right)\right) - \frac{-314159265359}{50000000000} \cdot z0\right) \]

   15. *-commutative N/A

       \[\leadsto \cos \left(\left(\color{blue}{\frac{1}{2} \cdot \mathsf{PI}\left(\right)} - \left(\mathsf{neg}\left(\frac{-7853981852531433}{5000000000000000}\right)\right)\right) - \frac{-314159265359}{50000000000} \cdot z0\right) \]

   16. lower-*.f64 N/A

       \[\leadsto \cos \left(\left(\color{blue}{\frac{1}{2} \cdot \mathsf{PI}\left(\right)} - \left(\mathsf{neg}\left(\frac{-7853981852531433}{5000000000000000}\right)\right)\right) - \frac{-314159265359}{50000000000} \cdot z0\right) \]

   17. metadata-eval N/A

       \[\leadsto \cos \left(\left(\color{blue}{\frac{1}{2}} \cdot \mathsf{PI}\left(\right) - \left(\mathsf{neg}\left(\frac{-7853981852531433}{5000000000000000}\right)\right)\right) - \frac{-314159265359}{50000000000} \cdot z0\right) \]

   18. lower-PI.f64 N/A

       \[\leadsto \cos \left(\left(\frac{1}{2} \cdot \color{blue}{\pi} - \left(\mathsf{neg}\left(\frac{-7853981852531433}{5000000000000000}\right)\right)\right) - \frac{-314159265359}{50000000000} \cdot z0\right) \]

   19. metadata-eval 57.3%

       \[\leadsto \cos \left(\left(0.5 \cdot \pi - \color{blue}{1.5707963705062866}\right) - -6.28318530718 \cdot z0\right) \]

   20. lift-*.f64 N/A

       \[\leadsto \cos \left(\left(\frac{1}{2} \cdot \pi - \frac{7853981852531433}{5000000000000000}\right) - \color{blue}{\frac{-314159265359}{50000000000} \cdot z0}\right) \]

   21. *-commutative N/A

       \[\leadsto \cos \left(\left(\frac{1}{2} \cdot \pi - \frac{7853981852531433}{5000000000000000}\right) - \color{blue}{z0 \cdot \frac{-314159265359}{50000000000}}\right) \]

   22. lower-*.f64 57.3%

       \[\leadsto \cos \left(\left(0.5 \cdot \pi - 1.5707963705062866\right) - \color{blue}{z0 \cdot -6.28318530718}\right) \]

3. Applied rewrites 57.3%

   \[\leadsto \color{blue}{\cos \left(\left(0.5 \cdot \pi - 1.5707963705062866\right) - z0 \cdot -6.28318530718\right)} \]
4. Add Preprocessing

Alternative 10: 55.7% accurate, 1.1× speedup

Math

\[\sin 1.5707963705062866 \]

FPCore

(FPCore (z0)
  :precision binary64
  (sin 1.5707963705062866))

C

double code(double z0) {
	return sin(1.5707963705062866);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(z0)
use fmin_fmax_functions
    real(8), intent (in) :: z0
    code = sin(1.5707963705062866d0)
end function

Java

public static double code(double z0) {
	return Math.sin(1.5707963705062866);
}

Python

def code(z0):
	return math.sin(1.5707963705062866)

Julia

function code(z0)
	return sin(1.5707963705062866)
end

MATLAB

function tmp = code(z0)
	tmp = sin(1.5707963705062866);
end

Wolfram

code[z0_] := N[Sin[1.5707963705062866], $MachinePrecision]

Derivation

1. Initial program 57.2%

   \[\sin \left(-6.28318530718 \cdot z0 - -1.5707963705062866\right) \]

2. Taylor expanded in z0 around 0

   \[\leadsto \sin \color{blue}{\frac{7853981852531433}{5000000000000000}} \]
3. Step-by-step derivation

4. Applied rewrites 55.7%

   \[\leadsto \sin \color{blue}{1.5707963705062866} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025250 
(FPCore (z0)
  :name "(sin (- (* -314159265359/50000000000 z0) -7853981852531433/5000000000000000))"
  :precision binary64
  (sin (- (* -6.28318530718 z0) -1.5707963705062866)))