UniformSampleCone, z

Percentage Accurate: 99.9% → 99.9%

Time: 1.8s

Alternatives: 4

Speedup: 1.0×

Specification

\[\left(\left(2.328306437 \cdot 10^{-10} \leq ux \land ux \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq uy \land uy \leq 1\right)\right) \land \left(0 \leq maxCos \land maxCos \leq 1\right)\]

Math

\[\left(1 - ux\right) + ux \cdot maxCos \]

FPCore

(FPCore (ux uy maxCos)
  :precision binary32
  (+ (- 1.0 ux) (* ux maxCos)))

C

float code(float ux, float uy, float maxCos) {
	return (1.0f - ux) + (ux * maxCos);
}

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(4) function code(ux, uy, maxcos)
use fmin_fmax_functions
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = (1.0e0 - ux) + (ux * maxcos)
end function

Julia

function code(ux, uy, maxCos)
	return Float32(Float32(Float32(1.0) - ux) + Float32(ux * maxCos))
end

MATLAB

function tmp = code(ux, uy, maxCos)
	tmp = (single(1.0) - ux) + (ux * maxCos);
end

Initial Program: 99.9% accurate, 1.0× speedup

Math

\[\left(1 - ux\right) + ux \cdot maxCos \]

FPCore

(FPCore (ux uy maxCos)
  :precision binary32
  (+ (- 1.0 ux) (* ux maxCos)))

C

float code(float ux, float uy, float maxCos) {
	return (1.0f - ux) + (ux * maxCos);
}

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(4) function code(ux, uy, maxcos)
use fmin_fmax_functions
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = (1.0e0 - ux) + (ux * maxcos)
end function

Julia

function code(ux, uy, maxCos)
	return Float32(Float32(Float32(1.0) - ux) + Float32(ux * maxCos))
end

MATLAB

function tmp = code(ux, uy, maxCos)
	tmp = (single(1.0) - ux) + (ux * maxCos);
end

Alternative 1: 99.9% accurate, 1.0× speedup

Math

\[\left(maxCos \cdot ux - ux\right) - -1 \]

FPCore

(FPCore (ux uy maxCos)
  :precision binary32
  (- (- (* maxCos ux) ux) -1.0))

C

float code(float ux, float uy, float maxCos) {
	return ((maxCos * ux) - ux) - -1.0f;
}

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(4) function code(ux, uy, maxcos)
use fmin_fmax_functions
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = ((maxcos * ux) - ux) - (-1.0e0)
end function

Julia

function code(ux, uy, maxCos)
	return Float32(Float32(Float32(maxCos * ux) - ux) - Float32(-1.0))
end

MATLAB

function tmp = code(ux, uy, maxCos)
	tmp = ((maxCos * ux) - ux) - single(-1.0);
end

Derivation

1. Initial program 99.9%

   \[\left(1 - ux\right) + ux \cdot maxCos \]
2. Step-by-step derivation

   1. lift-+.f32 N/A

      \[\leadsto \color{blue}{\left(1 - ux\right) + ux \cdot maxCos} \]

   2. lift-*.f32 N/A

      \[\leadsto \left(1 - ux\right) + \color{blue}{ux \cdot maxCos} \]

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

      \[\leadsto \color{blue}{\left(1 - ux\right) - \left(\mathsf{neg}\left(ux\right)\right) \cdot maxCos} \]

   4. lift--.f32 N/A

      \[\leadsto \color{blue}{\left(1 - ux\right)} - \left(\mathsf{neg}\left(ux\right)\right) \cdot maxCos \]

   5. sub-flip N/A

      \[\leadsto \color{blue}{\left(1 + \left(\mathsf{neg}\left(ux\right)\right)\right)} - \left(\mathsf{neg}\left(ux\right)\right) \cdot maxCos \]

   6. associate--l+ N/A

      \[\leadsto \color{blue}{1 + \left(\left(\mathsf{neg}\left(ux\right)\right) - \left(\mathsf{neg}\left(ux\right)\right) \cdot maxCos\right)} \]

   7. +-commutative N/A

      \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(ux\right)\right) - \left(\mathsf{neg}\left(ux\right)\right) \cdot maxCos\right) + 1} \]

   8. add-flip-rev N/A

      \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(ux\right)\right) - \left(\mathsf{neg}\left(ux\right)\right) \cdot maxCos\right) - \left(\mathsf{neg}\left(1\right)\right)} \]

   9. lower--.f32 N/A

      \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(ux\right)\right) - \left(\mathsf{neg}\left(ux\right)\right) \cdot maxCos\right) - \left(\mathsf{neg}\left(1\right)\right)} \]

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

       \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(ux\right)\right) + ux \cdot maxCos\right)} - \left(\mathsf{neg}\left(1\right)\right) \]

   11. lift-*.f32 N/A

       \[\leadsto \left(\left(\mathsf{neg}\left(ux\right)\right) + \color{blue}{ux \cdot maxCos}\right) - \left(\mathsf{neg}\left(1\right)\right) \]

   12. +-commutative N/A

       \[\leadsto \color{blue}{\left(ux \cdot maxCos + \left(\mathsf{neg}\left(ux\right)\right)\right)} - \left(\mathsf{neg}\left(1\right)\right) \]

   13. sub-flip N/A

       \[\leadsto \color{blue}{\left(ux \cdot maxCos - ux\right)} - \left(\mathsf{neg}\left(1\right)\right) \]

   14. lower--.f32 N/A

       \[\leadsto \color{blue}{\left(ux \cdot maxCos - ux\right)} - \left(\mathsf{neg}\left(1\right)\right) \]

   15. lift-*.f32 N/A

       \[\leadsto \left(\color{blue}{ux \cdot maxCos} - ux\right) - \left(\mathsf{neg}\left(1\right)\right) \]

   16. *-commutative N/A

       \[\leadsto \left(\color{blue}{maxCos \cdot ux} - ux\right) - \left(\mathsf{neg}\left(1\right)\right) \]

   17. lower-*.f32 N/A

       \[\leadsto \left(\color{blue}{maxCos \cdot ux} - ux\right) - \left(\mathsf{neg}\left(1\right)\right) \]

   18. metadata-eval 99.9%

       \[\leadsto \left(maxCos \cdot ux - ux\right) - \color{blue}{-1} \]

3. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\left(maxCos \cdot ux - ux\right) - -1} \]
4. Add Preprocessing

Alternative 2: 99.9% accurate, 1.1× speedup

Math

\[\mathsf{fma}\left(ux, maxCos, 1 - ux\right) \]

FPCore

(FPCore (ux uy maxCos)
  :precision binary32
  (fma ux maxCos (- 1.0 ux)))

C

float code(float ux, float uy, float maxCos) {
	return fmaf(ux, maxCos, (1.0f - ux));
}

Julia

function code(ux, uy, maxCos)
	return fma(ux, maxCos, Float32(Float32(1.0) - ux))
end

Derivation

1. Initial program 99.9%

   \[\left(1 - ux\right) + ux \cdot maxCos \]
2. Step-by-step derivation

   1. lift-+.f32 N/A

      \[\leadsto \color{blue}{\left(1 - ux\right) + ux \cdot maxCos} \]

   2. +-commutative N/A

      \[\leadsto \color{blue}{ux \cdot maxCos + \left(1 - ux\right)} \]

   3. lift-*.f32 N/A

      \[\leadsto \color{blue}{ux \cdot maxCos} + \left(1 - ux\right) \]

   4. lower-fma.f32 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(ux, maxCos, 1 - ux\right)} \]

3. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(ux, maxCos, 1 - ux\right)} \]
4. Add Preprocessing

Alternative 3: 98.0% accurate, 2.6× speedup

Math

\[1 - ux \]

FPCore

(FPCore (ux uy maxCos)
  :precision binary32
  (- 1.0 ux))

C

float code(float ux, float uy, float maxCos) {
	return 1.0f - ux;
}

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(4) function code(ux, uy, maxcos)
use fmin_fmax_functions
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = 1.0e0 - ux
end function

Julia

function code(ux, uy, maxCos)
	return Float32(Float32(1.0) - ux)
end

MATLAB

function tmp = code(ux, uy, maxCos)
	tmp = single(1.0) - ux;
end

Derivation

1. Initial program 99.9%

   \[\left(1 - ux\right) + ux \cdot maxCos \]

2. Taylor expanded in maxCos around 0

   \[\leadsto \color{blue}{1 - ux} \]
3. Step-by-step derivation

   1. lower--.f32 98.0%

      \[\leadsto 1 - \color{blue}{ux} \]

4. Applied rewrites 98.0%

   \[\leadsto \color{blue}{1 - ux} \]
5. Add Preprocessing

Alternative 4: 71.3% accurate, 9.2× speedup

Math

\[1 \]

FPCore

(FPCore (ux uy maxCos)
  :precision binary32
  1.0)

C

float code(float ux, float uy, float maxCos) {
	return 1.0f;
}

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(4) function code(ux, uy, maxcos)
use fmin_fmax_functions
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = 1.0e0
end function

Julia

function code(ux, uy, maxCos)
	return Float32(1.0)
end

MATLAB

function tmp = code(ux, uy, maxCos)
	tmp = single(1.0);
end

Derivation

1. Initial program 99.9%

   \[\left(1 - ux\right) + ux \cdot maxCos \]
2. Step-by-step derivation

   1. lift-+.f32 N/A

      \[\leadsto \color{blue}{\left(1 - ux\right) + ux \cdot maxCos} \]

   2. lift-*.f32 N/A

      \[\leadsto \left(1 - ux\right) + \color{blue}{ux \cdot maxCos} \]

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

      \[\leadsto \color{blue}{\left(1 - ux\right) - \left(\mathsf{neg}\left(ux\right)\right) \cdot maxCos} \]

   4. lift--.f32 N/A

      \[\leadsto \color{blue}{\left(1 - ux\right)} - \left(\mathsf{neg}\left(ux\right)\right) \cdot maxCos \]

   5. sub-flip N/A

      \[\leadsto \color{blue}{\left(1 + \left(\mathsf{neg}\left(ux\right)\right)\right)} - \left(\mathsf{neg}\left(ux\right)\right) \cdot maxCos \]

   6. associate--l+ N/A

      \[\leadsto \color{blue}{1 + \left(\left(\mathsf{neg}\left(ux\right)\right) - \left(\mathsf{neg}\left(ux\right)\right) \cdot maxCos\right)} \]

   7. +-commutative N/A

      \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(ux\right)\right) - \left(\mathsf{neg}\left(ux\right)\right) \cdot maxCos\right) + 1} \]

   8. add-flip-rev N/A

      \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(ux\right)\right) - \left(\mathsf{neg}\left(ux\right)\right) \cdot maxCos\right) - \left(\mathsf{neg}\left(1\right)\right)} \]

   9. lower--.f32 N/A

      \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(ux\right)\right) - \left(\mathsf{neg}\left(ux\right)\right) \cdot maxCos\right) - \left(\mathsf{neg}\left(1\right)\right)} \]

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

       \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(ux\right)\right) + ux \cdot maxCos\right)} - \left(\mathsf{neg}\left(1\right)\right) \]

   11. lift-*.f32 N/A

       \[\leadsto \left(\left(\mathsf{neg}\left(ux\right)\right) + \color{blue}{ux \cdot maxCos}\right) - \left(\mathsf{neg}\left(1\right)\right) \]

   12. +-commutative N/A

       \[\leadsto \color{blue}{\left(ux \cdot maxCos + \left(\mathsf{neg}\left(ux\right)\right)\right)} - \left(\mathsf{neg}\left(1\right)\right) \]

   13. sub-flip N/A

       \[\leadsto \color{blue}{\left(ux \cdot maxCos - ux\right)} - \left(\mathsf{neg}\left(1\right)\right) \]

   14. lower--.f32 N/A

       \[\leadsto \color{blue}{\left(ux \cdot maxCos - ux\right)} - \left(\mathsf{neg}\left(1\right)\right) \]

   15. lift-*.f32 N/A

       \[\leadsto \left(\color{blue}{ux \cdot maxCos} - ux\right) - \left(\mathsf{neg}\left(1\right)\right) \]

   16. *-commutative N/A

       \[\leadsto \left(\color{blue}{maxCos \cdot ux} - ux\right) - \left(\mathsf{neg}\left(1\right)\right) \]

   17. lower-*.f32 N/A

       \[\leadsto \left(\color{blue}{maxCos \cdot ux} - ux\right) - \left(\mathsf{neg}\left(1\right)\right) \]

   18. metadata-eval 99.9%

       \[\leadsto \left(maxCos \cdot ux - ux\right) - \color{blue}{-1} \]

3. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\left(maxCos \cdot ux - ux\right) - -1} \]

4. Taylor expanded in ux around 0

   \[\leadsto \color{blue}{1} \]
5. Step-by-step derivation

6. Applied rewrites 71.3%

   \[\leadsto \color{blue}{1} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2025227 
(FPCore (ux uy maxCos)
  :name "UniformSampleCone, z"
  :precision binary32
  :pre (and (and (and (<= 2.328306437e-10 ux) (<= ux 1.0)) (and (<= 2.328306437e-10 uy) (<= uy 1.0))) (and (<= 0.0 maxCos) (<= maxCos 1.0)))
  (+ (- 1.0 ux) (* ux maxCos)))