Disney BSSRDF, sample scattering profile, upper

Percentage Accurate: 96.0% → 96.5%

Time: 5.0s

Alternatives: 16

Speedup: 1.2×

Specification

\[\left(0 \leq s \land s \leq 256\right) \land \left(0.25 \leq u \land u \leq 1\right)\]

Math

\[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]

FPCore

(FPCore (s u)
  :precision binary32
  (* (* 3.0 s) (log (/ 1.0 (- 1.0 (/ (- u 0.25) 0.75))))))

C

float code(float s, float u) {
	return (3.0f * s) * logf((1.0f / (1.0f - ((u - 0.25f) / 0.75f))));
}

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(s, u)
use fmin_fmax_functions
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = (3.0e0 * s) * log((1.0e0 / (1.0e0 - ((u - 0.25e0) / 0.75e0))))
end function

Julia

function code(s, u)
	return Float32(Float32(Float32(3.0) * s) * log(Float32(Float32(1.0) / Float32(Float32(1.0) - Float32(Float32(u - Float32(0.25)) / Float32(0.75))))))
end

MATLAB

function tmp = code(s, u)
	tmp = (single(3.0) * s) * log((single(1.0) / (single(1.0) - ((u - single(0.25)) / single(0.75)))));
end

Initial Program: 96.0% accurate, 1.0× speedup

Math

\[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]

FPCore

(FPCore (s u)
  :precision binary32
  (* (* 3.0 s) (log (/ 1.0 (- 1.0 (/ (- u 0.25) 0.75))))))

C

float code(float s, float u) {
	return (3.0f * s) * logf((1.0f / (1.0f - ((u - 0.25f) / 0.75f))));
}

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(s, u)
use fmin_fmax_functions
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = (3.0e0 * s) * log((1.0e0 / (1.0e0 - ((u - 0.25e0) / 0.75e0))))
end function

Julia

function code(s, u)
	return Float32(Float32(Float32(3.0) * s) * log(Float32(Float32(1.0) / Float32(Float32(1.0) - Float32(Float32(u - Float32(0.25)) / Float32(0.75))))))
end

MATLAB

function tmp = code(s, u)
	tmp = (single(3.0) * s) * log((single(1.0) / (single(1.0) - ((u - single(0.25)) / single(0.75)))));
end

Alternative 1: 96.5% accurate, 1.0× speedup

Math

\[\left(1 + \frac{\left(\left(-\log \left(1 - u\right)\right) \cdot s\right) \cdot 3}{-0.8630462288856506 \cdot s}\right) \cdot \left(-0.8630462288856506 \cdot s\right) \]

FPCore

(FPCore (s u)
  :precision binary32
  (*
 (+
  1.0
  (/ (* (* (- (log (- 1.0 u))) s) 3.0) (* -0.8630462288856506 s)))
 (* -0.8630462288856506 s)))

C

float code(float s, float u) {
	return (1.0f + (((-logf((1.0f - u)) * s) * 3.0f) / (-0.8630462288856506f * s))) * (-0.8630462288856506f * s);
}

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(s, u)
use fmin_fmax_functions
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = (1.0e0 + (((-log((1.0e0 - u)) * s) * 3.0e0) / ((-0.8630462288856506e0) * s))) * ((-0.8630462288856506e0) * s)
end function

Julia

function code(s, u)
	return Float32(Float32(Float32(1.0) + Float32(Float32(Float32(Float32(-log(Float32(Float32(1.0) - u))) * s) * Float32(3.0)) / Float32(Float32(-0.8630462288856506) * s))) * Float32(Float32(-0.8630462288856506) * s))
end

MATLAB

function tmp = code(s, u)
	tmp = (single(1.0) + (((-log((single(1.0) - u)) * s) * single(3.0)) / (single(-0.8630462288856506) * s))) * (single(-0.8630462288856506) * s);
end

Derivation

1. Initial program 96.0%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]
2. Step-by-step derivation

   1. lift-log.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)} \]

   2. lift-/.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \color{blue}{\left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)} \]

   3. lift--.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}}\right) \]

   4. lift-/.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \color{blue}{\frac{u - \frac{1}{4}}{\frac{3}{4}}}}\right) \]

   5. sub-to-fraction N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{1 \cdot \frac{3}{4} - \left(u - \frac{1}{4}\right)}{\frac{3}{4}}}}\right) \]

   6. div-flip-rev N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \color{blue}{\left(\frac{\frac{3}{4}}{1 \cdot \frac{3}{4} - \left(u - \frac{1}{4}\right)}\right)} \]

   7. log-div N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\left(\log \frac{3}{4} - \log \left(1 \cdot \frac{3}{4} - \left(u - \frac{1}{4}\right)\right)\right)} \]

   8. lower-unsound--.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\left(\log \frac{3}{4} - \log \left(1 \cdot \frac{3}{4} - \left(u - \frac{1}{4}\right)\right)\right)} \]

   9. lower-unsound-log.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\color{blue}{\log \frac{3}{4}} - \log \left(1 \cdot \frac{3}{4} - \left(u - \frac{1}{4}\right)\right)\right) \]

   10. lower-unsound-log.f32 N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\log \frac{3}{4} - \color{blue}{\log \left(1 \cdot \frac{3}{4} - \left(u - \frac{1}{4}\right)\right)}\right) \]

   11. metadata-eval N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\log \frac{3}{4} - \log \left(\color{blue}{\frac{3}{4}} - \left(u - \frac{1}{4}\right)\right)\right) \]

   12. lift--.f32 N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\log \frac{3}{4} - \log \left(\frac{3}{4} - \color{blue}{\left(u - \frac{1}{4}\right)}\right)\right) \]

   13. sub-negate-rev N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\log \frac{3}{4} - \log \left(\frac{3}{4} - \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{4} - u\right)\right)\right)}\right)\right) \]

   14. add-flip-rev N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\log \frac{3}{4} - \log \color{blue}{\left(\frac{3}{4} + \left(\frac{1}{4} - u\right)\right)}\right) \]

   15. associate-+r- N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\log \frac{3}{4} - \log \color{blue}{\left(\left(\frac{3}{4} + \frac{1}{4}\right) - u\right)}\right) \]

   16. metadata-eval N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\log \frac{3}{4} - \log \left(\color{blue}{1} - u\right)\right) \]

   17. lower--.f32 96.5%

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\log 0.75 - \log \color{blue}{\left(1 - u\right)}\right) \]

3. Applied rewrites 96.5%

   \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\left(\log 0.75 - \log \left(1 - u\right)\right)} \]
4. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \color{blue}{\left(3 \cdot s\right) \cdot \left(\log \frac{3}{4} - \log \left(1 - u\right)\right)} \]

   2. lift--.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\left(\log \frac{3}{4} - \log \left(1 - u\right)\right)} \]

   3. lift-log.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\color{blue}{\log \frac{3}{4}} - \log \left(1 - u\right)\right) \]

   4. lift-log.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\log \frac{3}{4} - \color{blue}{\log \left(1 - u\right)}\right) \]

   5. diff-log N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\log \left(\frac{\frac{3}{4}}{1 - u}\right)} \]

   6. lift-/.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \color{blue}{\left(\frac{\frac{3}{4}}{1 - u}\right)} \]

   7. lift-log.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\log \left(\frac{\frac{3}{4}}{1 - u}\right)} \]

   8. lift-log.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\log \left(\frac{\frac{3}{4}}{1 - u}\right)} \]

   9. lift-/.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \color{blue}{\left(\frac{\frac{3}{4}}{1 - u}\right)} \]

   10. diff-log N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\left(\log \frac{3}{4} - \log \left(1 - u\right)\right)} \]

   11. lift-log.f32 N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\color{blue}{\log \frac{3}{4}} - \log \left(1 - u\right)\right) \]

   12. lift-log.f32 N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\log \frac{3}{4} - \color{blue}{\log \left(1 - u\right)}\right) \]

   13. sub-flip N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\left(\log \frac{3}{4} + \left(\mathsf{neg}\left(\log \left(1 - u\right)\right)\right)\right)} \]

   14. distribute-lft-in N/A

       \[\leadsto \color{blue}{\left(3 \cdot s\right) \cdot \log \frac{3}{4} + \left(3 \cdot s\right) \cdot \left(\mathsf{neg}\left(\log \left(1 - u\right)\right)\right)} \]

   15. lift-*.f32 N/A

       \[\leadsto \color{blue}{\left(3 \cdot s\right)} \cdot \log \frac{3}{4} + \left(3 \cdot s\right) \cdot \left(\mathsf{neg}\left(\log \left(1 - u\right)\right)\right) \]

   16. associate-*r* N/A

       \[\leadsto \color{blue}{3 \cdot \left(s \cdot \log \frac{3}{4}\right)} + \left(3 \cdot s\right) \cdot \left(\mathsf{neg}\left(\log \left(1 - u\right)\right)\right) \]

   17. lift-*.f32 N/A

       \[\leadsto 3 \cdot \color{blue}{\left(s \cdot \log \frac{3}{4}\right)} + \left(3 \cdot s\right) \cdot \left(\mathsf{neg}\left(\log \left(1 - u\right)\right)\right) \]

   18. lift-*.f32 N/A

       \[\leadsto \color{blue}{3 \cdot \left(s \cdot \log \frac{3}{4}\right)} + \left(3 \cdot s\right) \cdot \left(\mathsf{neg}\left(\log \left(1 - u\right)\right)\right) \]

   19. sum-to-mult N/A

       \[\leadsto \color{blue}{\left(1 + \frac{\left(3 \cdot s\right) \cdot \left(\mathsf{neg}\left(\log \left(1 - u\right)\right)\right)}{3 \cdot \left(s \cdot \log \frac{3}{4}\right)}\right) \cdot \left(3 \cdot \left(s \cdot \log \frac{3}{4}\right)\right)} \]

5. Applied rewrites 96.2%

   \[\leadsto \color{blue}{\left(1 + \frac{\left(\left(-\log \left(1 - u\right)\right) \cdot s\right) \cdot 3}{\log 0.421875 \cdot s}\right) \cdot \left(\log 0.421875 \cdot s\right)} \]

6. Evaluated real constant 96.2%

   \[\leadsto \left(1 + \frac{\left(\left(-\log \left(1 - u\right)\right) \cdot s\right) \cdot 3}{\color{blue}{-0.8630462288856506} \cdot s}\right) \cdot \left(\log 0.421875 \cdot s\right) \]

7. Evaluated real constant 96.2%

   \[\leadsto \left(1 + \frac{\left(\left(-\log \left(1 - u\right)\right) \cdot s\right) \cdot 3}{-0.8630462288856506 \cdot s}\right) \cdot \left(\color{blue}{-0.8630462288856506} \cdot s\right) \]
8. Add Preprocessing

Alternative 2: 96.5% accurate, 1.1× speedup

Math

\[-0.8630462288856506 \cdot s + -3 \cdot \left(s \cdot \log \left(1 - u\right)\right) \]

FPCore

(FPCore (s u)
  :precision binary32
  (+ (* -0.8630462288856506 s) (* -3.0 (* s (log (- 1.0 u))))))

C

float code(float s, float u) {
	return (-0.8630462288856506f * s) + (-3.0f * (s * logf((1.0f - u))));
}

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(s, u)
use fmin_fmax_functions
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = ((-0.8630462288856506e0) * s) + ((-3.0e0) * (s * log((1.0e0 - u))))
end function

Julia

function code(s, u)
	return Float32(Float32(Float32(-0.8630462288856506) * s) + Float32(Float32(-3.0) * Float32(s * log(Float32(Float32(1.0) - u)))))
end

MATLAB

function tmp = code(s, u)
	tmp = (single(-0.8630462288856506) * s) + (single(-3.0) * (s * log((single(1.0) - u))));
end

Derivation

1. Initial program 96.0%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]
2. Step-by-step derivation

   1. lift-log.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)} \]

   2. lift-/.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \color{blue}{\left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)} \]

   3. lift--.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}}\right) \]

   4. lift-/.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \color{blue}{\frac{u - \frac{1}{4}}{\frac{3}{4}}}}\right) \]

   5. sub-to-fraction N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{1 \cdot \frac{3}{4} - \left(u - \frac{1}{4}\right)}{\frac{3}{4}}}}\right) \]

   6. div-flip-rev N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \color{blue}{\left(\frac{\frac{3}{4}}{1 \cdot \frac{3}{4} - \left(u - \frac{1}{4}\right)}\right)} \]

   7. log-div N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\left(\log \frac{3}{4} - \log \left(1 \cdot \frac{3}{4} - \left(u - \frac{1}{4}\right)\right)\right)} \]

   8. lower-unsound--.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\left(\log \frac{3}{4} - \log \left(1 \cdot \frac{3}{4} - \left(u - \frac{1}{4}\right)\right)\right)} \]

   9. lower-unsound-log.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\color{blue}{\log \frac{3}{4}} - \log \left(1 \cdot \frac{3}{4} - \left(u - \frac{1}{4}\right)\right)\right) \]

   10. lower-unsound-log.f32 N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\log \frac{3}{4} - \color{blue}{\log \left(1 \cdot \frac{3}{4} - \left(u - \frac{1}{4}\right)\right)}\right) \]

   11. metadata-eval N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\log \frac{3}{4} - \log \left(\color{blue}{\frac{3}{4}} - \left(u - \frac{1}{4}\right)\right)\right) \]

   12. lift--.f32 N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\log \frac{3}{4} - \log \left(\frac{3}{4} - \color{blue}{\left(u - \frac{1}{4}\right)}\right)\right) \]

   13. sub-negate-rev N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\log \frac{3}{4} - \log \left(\frac{3}{4} - \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{4} - u\right)\right)\right)}\right)\right) \]

   14. add-flip-rev N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\log \frac{3}{4} - \log \color{blue}{\left(\frac{3}{4} + \left(\frac{1}{4} - u\right)\right)}\right) \]

   15. associate-+r- N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\log \frac{3}{4} - \log \color{blue}{\left(\left(\frac{3}{4} + \frac{1}{4}\right) - u\right)}\right) \]

   16. metadata-eval N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\log \frac{3}{4} - \log \left(\color{blue}{1} - u\right)\right) \]

   17. lower--.f32 96.5%

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\log 0.75 - \log \color{blue}{\left(1 - u\right)}\right) \]

3. Applied rewrites 96.5%

   \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\left(\log 0.75 - \log \left(1 - u\right)\right)} \]
4. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \color{blue}{\left(3 \cdot s\right) \cdot \left(\log \frac{3}{4} - \log \left(1 - u\right)\right)} \]

   2. lift--.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\left(\log \frac{3}{4} - \log \left(1 - u\right)\right)} \]

   3. lift-log.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\color{blue}{\log \frac{3}{4}} - \log \left(1 - u\right)\right) \]

   4. lift-log.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\log \frac{3}{4} - \color{blue}{\log \left(1 - u\right)}\right) \]

   5. diff-log N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\log \left(\frac{\frac{3}{4}}{1 - u}\right)} \]

   6. lift-/.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \color{blue}{\left(\frac{\frac{3}{4}}{1 - u}\right)} \]

   7. lift-log.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\log \left(\frac{\frac{3}{4}}{1 - u}\right)} \]

   8. lift-log.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\log \left(\frac{\frac{3}{4}}{1 - u}\right)} \]

   9. lift-/.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \color{blue}{\left(\frac{\frac{3}{4}}{1 - u}\right)} \]

   10. diff-log N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\left(\log \frac{3}{4} - \log \left(1 - u\right)\right)} \]

   11. lift-log.f32 N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\color{blue}{\log \frac{3}{4}} - \log \left(1 - u\right)\right) \]

   12. lift-log.f32 N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\log \frac{3}{4} - \color{blue}{\log \left(1 - u\right)}\right) \]

   13. sub-flip N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\left(\log \frac{3}{4} + \left(\mathsf{neg}\left(\log \left(1 - u\right)\right)\right)\right)} \]

   14. distribute-lft-in N/A

       \[\leadsto \color{blue}{\left(3 \cdot s\right) \cdot \log \frac{3}{4} + \left(3 \cdot s\right) \cdot \left(\mathsf{neg}\left(\log \left(1 - u\right)\right)\right)} \]

   15. lift-*.f32 N/A

       \[\leadsto \color{blue}{\left(3 \cdot s\right)} \cdot \log \frac{3}{4} + \left(3 \cdot s\right) \cdot \left(\mathsf{neg}\left(\log \left(1 - u\right)\right)\right) \]

   16. associate-*r* N/A

       \[\leadsto \color{blue}{3 \cdot \left(s \cdot \log \frac{3}{4}\right)} + \left(3 \cdot s\right) \cdot \left(\mathsf{neg}\left(\log \left(1 - u\right)\right)\right) \]

   17. lift-*.f32 N/A

       \[\leadsto 3 \cdot \color{blue}{\left(s \cdot \log \frac{3}{4}\right)} + \left(3 \cdot s\right) \cdot \left(\mathsf{neg}\left(\log \left(1 - u\right)\right)\right) \]

   18. lift-*.f32 N/A

       \[\leadsto \color{blue}{3 \cdot \left(s \cdot \log \frac{3}{4}\right)} + \left(3 \cdot s\right) \cdot \left(\mathsf{neg}\left(\log \left(1 - u\right)\right)\right) \]

   19. sum-to-mult N/A

       \[\leadsto \color{blue}{\left(1 + \frac{\left(3 \cdot s\right) \cdot \left(\mathsf{neg}\left(\log \left(1 - u\right)\right)\right)}{3 \cdot \left(s \cdot \log \frac{3}{4}\right)}\right) \cdot \left(3 \cdot \left(s \cdot \log \frac{3}{4}\right)\right)} \]

5. Applied rewrites 96.2%

   \[\leadsto \color{blue}{\left(1 + \frac{\left(\left(-\log \left(1 - u\right)\right) \cdot s\right) \cdot 3}{\log 0.421875 \cdot s}\right) \cdot \left(\log 0.421875 \cdot s\right)} \]

6. Evaluated real constant 96.2%

   \[\leadsto \left(1 + \frac{\left(\left(-\log \left(1 - u\right)\right) \cdot s\right) \cdot 3}{\color{blue}{-0.8630462288856506} \cdot s}\right) \cdot \left(\log 0.421875 \cdot s\right) \]

7. Evaluated real constant 96.2%

   \[\leadsto \left(1 + \frac{\left(\left(-\log \left(1 - u\right)\right) \cdot s\right) \cdot 3}{-0.8630462288856506 \cdot s}\right) \cdot \left(\color{blue}{-0.8630462288856506} \cdot s\right) \]
8. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \color{blue}{\left(1 + \frac{\left(\left(-\log \left(1 - u\right)\right) \cdot s\right) \cdot 3}{\frac{-14479513}{16777216} \cdot s}\right) \cdot \left(\frac{-14479513}{16777216} \cdot s\right)} \]

   2. lift-+.f32 N/A

      \[\leadsto \color{blue}{\left(1 + \frac{\left(\left(-\log \left(1 - u\right)\right) \cdot s\right) \cdot 3}{\frac{-14479513}{16777216} \cdot s}\right)} \cdot \left(\frac{-14479513}{16777216} \cdot s\right) \]

   3. lift-/.f32 N/A

      \[\leadsto \left(1 + \color{blue}{\frac{\left(\left(-\log \left(1 - u\right)\right) \cdot s\right) \cdot 3}{\frac{-14479513}{16777216} \cdot s}}\right) \cdot \left(\frac{-14479513}{16777216} \cdot s\right) \]

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

      \[\leadsto \color{blue}{\frac{-14479513}{16777216} \cdot s + \left(\left(-\log \left(1 - u\right)\right) \cdot s\right) \cdot 3} \]

   5. lower-+.f32 96.3%

      \[\leadsto \color{blue}{-0.8630462288856506 \cdot s + \left(\left(-\log \left(1 - u\right)\right) \cdot s\right) \cdot 3} \]

   6. lift-*.f32 N/A

      \[\leadsto \frac{-14479513}{16777216} \cdot s + \color{blue}{\left(\left(-\log \left(1 - u\right)\right) \cdot s\right) \cdot 3} \]

   7. *-commutative N/A

      \[\leadsto \frac{-14479513}{16777216} \cdot s + \color{blue}{3 \cdot \left(\left(-\log \left(1 - u\right)\right) \cdot s\right)} \]

   8. lift-*.f32 N/A

      \[\leadsto \frac{-14479513}{16777216} \cdot s + 3 \cdot \color{blue}{\left(\left(-\log \left(1 - u\right)\right) \cdot s\right)} \]

   9. lift-neg.f32 N/A

      \[\leadsto \frac{-14479513}{16777216} \cdot s + 3 \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\log \left(1 - u\right)\right)\right)} \cdot s\right) \]

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

       \[\leadsto \frac{-14479513}{16777216} \cdot s + 3 \cdot \color{blue}{\left(\mathsf{neg}\left(\log \left(1 - u\right) \cdot s\right)\right)} \]

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

       \[\leadsto \frac{-14479513}{16777216} \cdot s + \color{blue}{\left(\mathsf{neg}\left(3 \cdot \left(\log \left(1 - u\right) \cdot s\right)\right)\right)} \]

   12. *-commutative N/A

       \[\leadsto \frac{-14479513}{16777216} \cdot s + \left(\mathsf{neg}\left(3 \cdot \color{blue}{\left(s \cdot \log \left(1 - u\right)\right)}\right)\right) \]

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

       \[\leadsto \frac{-14479513}{16777216} \cdot s + \color{blue}{\left(\mathsf{neg}\left(3\right)\right) \cdot \left(s \cdot \log \left(1 - u\right)\right)} \]

   14. lower-*.f32 N/A

       \[\leadsto \frac{-14479513}{16777216} \cdot s + \color{blue}{\left(\mathsf{neg}\left(3\right)\right) \cdot \left(s \cdot \log \left(1 - u\right)\right)} \]

   15. metadata-eval N/A

       \[\leadsto \frac{-14479513}{16777216} \cdot s + \color{blue}{-3} \cdot \left(s \cdot \log \left(1 - u\right)\right) \]

   16. lower-*.f32 96.3%

       \[\leadsto -0.8630462288856506 \cdot s + -3 \cdot \color{blue}{\left(s \cdot \log \left(1 - u\right)\right)} \]

9. Applied rewrites 96.3%

   \[\leadsto \color{blue}{-0.8630462288856506 \cdot s + -3 \cdot \left(s \cdot \log \left(1 - u\right)\right)} \]
10. Add Preprocessing

Alternative 3: 96.3% accurate, 1.1× speedup

Math

\[\left(1 + 3.476059450342011 \cdot \log \left(1 - u\right)\right) \cdot \left(-0.8630462288856506 \cdot s\right) \]

FPCore

(FPCore (s u)
  :precision binary32
  (*
 (+ 1.0 (* 3.476059450342011 (log (- 1.0 u))))
 (* -0.8630462288856506 s)))

C

float code(float s, float u) {
	return (1.0f + (3.476059450342011f * logf((1.0f - u)))) * (-0.8630462288856506f * s);
}

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(s, u)
use fmin_fmax_functions
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = (1.0e0 + (3.476059450342011e0 * log((1.0e0 - u)))) * ((-0.8630462288856506e0) * s)
end function

Julia

function code(s, u)
	return Float32(Float32(Float32(1.0) + Float32(Float32(3.476059450342011) * log(Float32(Float32(1.0) - u)))) * Float32(Float32(-0.8630462288856506) * s))
end

MATLAB

function tmp = code(s, u)
	tmp = (single(1.0) + (single(3.476059450342011) * log((single(1.0) - u)))) * (single(-0.8630462288856506) * s);
end

Derivation

1. Initial program 96.0%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]
2. Step-by-step derivation

   1. lift-log.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)} \]

   2. lift-/.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \color{blue}{\left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)} \]

   3. lift--.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}}\right) \]

   4. lift-/.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \color{blue}{\frac{u - \frac{1}{4}}{\frac{3}{4}}}}\right) \]

   5. sub-to-fraction N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{1 \cdot \frac{3}{4} - \left(u - \frac{1}{4}\right)}{\frac{3}{4}}}}\right) \]

   6. div-flip-rev N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \color{blue}{\left(\frac{\frac{3}{4}}{1 \cdot \frac{3}{4} - \left(u - \frac{1}{4}\right)}\right)} \]

   7. log-div N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\left(\log \frac{3}{4} - \log \left(1 \cdot \frac{3}{4} - \left(u - \frac{1}{4}\right)\right)\right)} \]

   8. lower-unsound--.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\left(\log \frac{3}{4} - \log \left(1 \cdot \frac{3}{4} - \left(u - \frac{1}{4}\right)\right)\right)} \]

   9. lower-unsound-log.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\color{blue}{\log \frac{3}{4}} - \log \left(1 \cdot \frac{3}{4} - \left(u - \frac{1}{4}\right)\right)\right) \]

   10. lower-unsound-log.f32 N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\log \frac{3}{4} - \color{blue}{\log \left(1 \cdot \frac{3}{4} - \left(u - \frac{1}{4}\right)\right)}\right) \]

   11. metadata-eval N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\log \frac{3}{4} - \log \left(\color{blue}{\frac{3}{4}} - \left(u - \frac{1}{4}\right)\right)\right) \]

   12. lift--.f32 N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\log \frac{3}{4} - \log \left(\frac{3}{4} - \color{blue}{\left(u - \frac{1}{4}\right)}\right)\right) \]

   13. sub-negate-rev N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\log \frac{3}{4} - \log \left(\frac{3}{4} - \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{4} - u\right)\right)\right)}\right)\right) \]

   14. add-flip-rev N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\log \frac{3}{4} - \log \color{blue}{\left(\frac{3}{4} + \left(\frac{1}{4} - u\right)\right)}\right) \]

   15. associate-+r- N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\log \frac{3}{4} - \log \color{blue}{\left(\left(\frac{3}{4} + \frac{1}{4}\right) - u\right)}\right) \]

   16. metadata-eval N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\log \frac{3}{4} - \log \left(\color{blue}{1} - u\right)\right) \]

   17. lower--.f32 96.5%

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\log 0.75 - \log \color{blue}{\left(1 - u\right)}\right) \]

3. Applied rewrites 96.5%

   \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\left(\log 0.75 - \log \left(1 - u\right)\right)} \]
4. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \color{blue}{\left(3 \cdot s\right) \cdot \left(\log \frac{3}{4} - \log \left(1 - u\right)\right)} \]

   2. lift--.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\left(\log \frac{3}{4} - \log \left(1 - u\right)\right)} \]

   3. lift-log.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\color{blue}{\log \frac{3}{4}} - \log \left(1 - u\right)\right) \]

   4. lift-log.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\log \frac{3}{4} - \color{blue}{\log \left(1 - u\right)}\right) \]

   5. diff-log N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\log \left(\frac{\frac{3}{4}}{1 - u}\right)} \]

   6. lift-/.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \color{blue}{\left(\frac{\frac{3}{4}}{1 - u}\right)} \]

   7. lift-log.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\log \left(\frac{\frac{3}{4}}{1 - u}\right)} \]

   8. lift-log.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\log \left(\frac{\frac{3}{4}}{1 - u}\right)} \]

   9. lift-/.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \color{blue}{\left(\frac{\frac{3}{4}}{1 - u}\right)} \]

   10. diff-log N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\left(\log \frac{3}{4} - \log \left(1 - u\right)\right)} \]

   11. lift-log.f32 N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\color{blue}{\log \frac{3}{4}} - \log \left(1 - u\right)\right) \]

   12. lift-log.f32 N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\log \frac{3}{4} - \color{blue}{\log \left(1 - u\right)}\right) \]

   13. sub-flip N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\left(\log \frac{3}{4} + \left(\mathsf{neg}\left(\log \left(1 - u\right)\right)\right)\right)} \]

   14. distribute-lft-in N/A

       \[\leadsto \color{blue}{\left(3 \cdot s\right) \cdot \log \frac{3}{4} + \left(3 \cdot s\right) \cdot \left(\mathsf{neg}\left(\log \left(1 - u\right)\right)\right)} \]

   15. lift-*.f32 N/A

       \[\leadsto \color{blue}{\left(3 \cdot s\right)} \cdot \log \frac{3}{4} + \left(3 \cdot s\right) \cdot \left(\mathsf{neg}\left(\log \left(1 - u\right)\right)\right) \]

   16. associate-*r* N/A

       \[\leadsto \color{blue}{3 \cdot \left(s \cdot \log \frac{3}{4}\right)} + \left(3 \cdot s\right) \cdot \left(\mathsf{neg}\left(\log \left(1 - u\right)\right)\right) \]

   17. lift-*.f32 N/A

       \[\leadsto 3 \cdot \color{blue}{\left(s \cdot \log \frac{3}{4}\right)} + \left(3 \cdot s\right) \cdot \left(\mathsf{neg}\left(\log \left(1 - u\right)\right)\right) \]

   18. lift-*.f32 N/A

       \[\leadsto \color{blue}{3 \cdot \left(s \cdot \log \frac{3}{4}\right)} + \left(3 \cdot s\right) \cdot \left(\mathsf{neg}\left(\log \left(1 - u\right)\right)\right) \]

   19. sum-to-mult N/A

       \[\leadsto \color{blue}{\left(1 + \frac{\left(3 \cdot s\right) \cdot \left(\mathsf{neg}\left(\log \left(1 - u\right)\right)\right)}{3 \cdot \left(s \cdot \log \frac{3}{4}\right)}\right) \cdot \left(3 \cdot \left(s \cdot \log \frac{3}{4}\right)\right)} \]

5. Applied rewrites 96.2%

   \[\leadsto \color{blue}{\left(1 + \frac{\left(\left(-\log \left(1 - u\right)\right) \cdot s\right) \cdot 3}{\log 0.421875 \cdot s}\right) \cdot \left(\log 0.421875 \cdot s\right)} \]

6. Evaluated real constant 96.2%

   \[\leadsto \left(1 + \frac{\left(\left(-\log \left(1 - u\right)\right) \cdot s\right) \cdot 3}{\color{blue}{-0.8630462288856506} \cdot s}\right) \cdot \left(\log 0.421875 \cdot s\right) \]

7. Evaluated real constant 96.2%

   \[\leadsto \left(1 + \frac{\left(\left(-\log \left(1 - u\right)\right) \cdot s\right) \cdot 3}{-0.8630462288856506 \cdot s}\right) \cdot \left(\color{blue}{-0.8630462288856506} \cdot s\right) \]

8. Taylor expanded in s around 0

   \[\leadsto \left(1 + \color{blue}{\frac{50331648}{14479513} \cdot \log \left(1 - u\right)}\right) \cdot \left(-0.8630462288856506 \cdot s\right) \]
9. Step-by-step derivation

   1. lower-*.f32 N/A

      \[\leadsto \left(1 + \frac{50331648}{14479513} \cdot \color{blue}{\log \left(1 - u\right)}\right) \cdot \left(\frac{-14479513}{16777216} \cdot s\right) \]

   2. lower-log.f32 N/A

      \[\leadsto \left(1 + \frac{50331648}{14479513} \cdot \log \left(1 - u\right)\right) \cdot \left(\frac{-14479513}{16777216} \cdot s\right) \]

   3. lower--.f32 96.5%

      \[\leadsto \left(1 + 3.476059450342011 \cdot \log \left(1 - u\right)\right) \cdot \left(-0.8630462288856506 \cdot s\right) \]

10. Applied rewrites 96.5%

    \[\leadsto \left(1 + \color{blue}{3.476059450342011 \cdot \log \left(1 - u\right)}\right) \cdot \left(-0.8630462288856506 \cdot s\right) \]
11. Add Preprocessing

Alternative 4: 96.2% accurate, 1.1× speedup

Math

\[\left(\left(-\log \left(-1.3333333333333333 \cdot u - -1.3333333333333333\right)\right) \cdot 3\right) \cdot s \]

FPCore

(FPCore (s u)
  :precision binary32
  (*
 (* (- (log (- (* -1.3333333333333333 u) -1.3333333333333333))) 3.0)
 s))

C

float code(float s, float u) {
	return (-logf(((-1.3333333333333333f * u) - -1.3333333333333333f)) * 3.0f) * s;
}

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(s, u)
use fmin_fmax_functions
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = (-log((((-1.3333333333333333e0) * u) - (-1.3333333333333333e0))) * 3.0e0) * s
end function

Julia

function code(s, u)
	return Float32(Float32(Float32(-log(Float32(Float32(Float32(-1.3333333333333333) * u) - Float32(-1.3333333333333333)))) * Float32(3.0)) * s)
end

MATLAB

function tmp = code(s, u)
	tmp = (-log(((single(-1.3333333333333333) * u) - single(-1.3333333333333333))) * single(3.0)) * s;
end

Derivation

1. Initial program 96.0%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]
2. Step-by-step derivation

   1. lift--.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}}\right) \]

   2. sub-flip N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{1 + \left(\mathsf{neg}\left(\frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right)}}\right) \]

   3. +-commutative N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{\left(\mathsf{neg}\left(\frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right) + 1}}\right) \]

   4. add-flip N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{\left(\mathsf{neg}\left(\frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right) - \left(\mathsf{neg}\left(1\right)\right)}}\right) \]

   5. metadata-eval N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\left(\mathsf{neg}\left(\frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right) - \color{blue}{-1}}\right) \]

   6. lower--.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{\left(\mathsf{neg}\left(\frac{u - \frac{1}{4}}{\frac{3}{4}}\right)\right) - -1}}\right) \]

   7. lift-/.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\left(\mathsf{neg}\left(\color{blue}{\frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)\right) - -1}\right) \]

   8. distribute-neg-frac2 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{u - \frac{1}{4}}{\mathsf{neg}\left(\frac{3}{4}\right)}} - -1}\right) \]

   9. mult-flip-rev N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{\left(u - \frac{1}{4}\right) \cdot \frac{1}{\mathsf{neg}\left(\frac{3}{4}\right)}} - -1}\right) \]

   10. *-commutative N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{1}{\mathsf{neg}\left(\frac{3}{4}\right)} \cdot \left(u - \frac{1}{4}\right)} - -1}\right) \]

   11. lower-*.f32 N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{1}{\mathsf{neg}\left(\frac{3}{4}\right)} \cdot \left(u - \frac{1}{4}\right)} - -1}\right) \]

   12. metadata-eval N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\frac{1}{\color{blue}{\frac{-3}{4}}} \cdot \left(u - \frac{1}{4}\right) - -1}\right) \]

   13. metadata-eval 95.8%

       \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{-1.3333333333333333} \cdot \left(u - 0.25\right) - -1}\right) \]

3. Applied rewrites 95.8%

   \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{-1.3333333333333333 \cdot \left(u - 0.25\right) - -1}}\right) \]
4. Step-by-step derivation

   1. lift--.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{-4}{3} \cdot \left(u - \frac{1}{4}\right) - -1}}\right) \]

   2. sub-flip N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{-4}{3} \cdot \left(u - \frac{1}{4}\right) + \left(\mathsf{neg}\left(-1\right)\right)}}\right) \]

   3. lift-*.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{-4}{3} \cdot \left(u - \frac{1}{4}\right)} + \left(\mathsf{neg}\left(-1\right)\right)}\right) \]

   4. lift--.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\frac{-4}{3} \cdot \color{blue}{\left(u - \frac{1}{4}\right)} + \left(\mathsf{neg}\left(-1\right)\right)}\right) \]

   5. sub-flip N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\frac{-4}{3} \cdot \color{blue}{\left(u + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)\right)} + \left(\mathsf{neg}\left(-1\right)\right)}\right) \]

   6. distribute-rgt-in N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{\left(u \cdot \frac{-4}{3} + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right) \cdot \frac{-4}{3}\right)} + \left(\mathsf{neg}\left(-1\right)\right)}\right) \]

   7. metadata-eval N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\left(u \cdot \frac{-4}{3} + \color{blue}{\frac{-1}{4}} \cdot \frac{-4}{3}\right) + \left(\mathsf{neg}\left(-1\right)\right)}\right) \]

   8. metadata-eval N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\left(u \cdot \frac{-4}{3} + \color{blue}{\frac{1}{3}}\right) + \left(\mathsf{neg}\left(-1\right)\right)}\right) \]

   9. metadata-eval N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\left(u \cdot \frac{-4}{3} + \color{blue}{\frac{\frac{1}{4}}{\frac{3}{4}}}\right) + \left(\mathsf{neg}\left(-1\right)\right)}\right) \]

   10. metadata-eval N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\left(u \cdot \frac{-4}{3} + \frac{\frac{1}{4}}{\frac{3}{4}}\right) + \color{blue}{1}}\right) \]

   11. associate-+l+ N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{u \cdot \frac{-4}{3} + \left(\frac{\frac{1}{4}}{\frac{3}{4}} + 1\right)}}\right) \]

   12. metadata-eval N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{u \cdot \frac{-4}{3} + \left(\color{blue}{\frac{1}{3}} + 1\right)}\right) \]

   13. metadata-eval N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{u \cdot \frac{-4}{3} + \color{blue}{\frac{4}{3}}}\right) \]

   14. metadata-eval N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{u \cdot \frac{-4}{3} + \color{blue}{\left(\mathsf{neg}\left(\frac{-4}{3}\right)\right)}}\right) \]

   15. sub-flip-reverse N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{u \cdot \frac{-4}{3} - \frac{-4}{3}}}\right) \]

   16. lower--.f32 N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{u \cdot \frac{-4}{3} - \frac{-4}{3}}}\right) \]

   17. *-commutative N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{-4}{3} \cdot u} - \frac{-4}{3}}\right) \]

   18. lower-*.f32 95.6%

       \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{-1.3333333333333333 \cdot u} - -1.3333333333333333}\right) \]

5. Applied rewrites 95.6%

   \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{-1.3333333333333333 \cdot u - -1.3333333333333333}}\right) \]
6. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \color{blue}{\left(3 \cdot s\right) \cdot \log \left(\frac{1}{\frac{-4}{3} \cdot u - \frac{-4}{3}}\right)} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\log \left(\frac{1}{\frac{-4}{3} \cdot u - \frac{-4}{3}}\right) \cdot \left(3 \cdot s\right)} \]

   3. lift-*.f32 N/A

      \[\leadsto \log \left(\frac{1}{\frac{-4}{3} \cdot u - \frac{-4}{3}}\right) \cdot \color{blue}{\left(3 \cdot s\right)} \]

   4. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\log \left(\frac{1}{\frac{-4}{3} \cdot u - \frac{-4}{3}}\right) \cdot 3\right) \cdot s} \]

   5. lower-*.f32 N/A

      \[\leadsto \color{blue}{\left(\log \left(\frac{1}{\frac{-4}{3} \cdot u - \frac{-4}{3}}\right) \cdot 3\right) \cdot s} \]

   6. lower-*.f32 95.5%

      \[\leadsto \color{blue}{\left(\log \left(\frac{1}{-1.3333333333333333 \cdot u - -1.3333333333333333}\right) \cdot 3\right)} \cdot s \]

   7. lift-log.f32 N/A

      \[\leadsto \left(\color{blue}{\log \left(\frac{1}{\frac{-4}{3} \cdot u - \frac{-4}{3}}\right)} \cdot 3\right) \cdot s \]

   8. lift-/.f32 N/A

      \[\leadsto \left(\log \color{blue}{\left(\frac{1}{\frac{-4}{3} \cdot u - \frac{-4}{3}}\right)} \cdot 3\right) \cdot s \]

   9. log-rec N/A

      \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(\log \left(\frac{-4}{3} \cdot u - \frac{-4}{3}\right)\right)\right)} \cdot 3\right) \cdot s \]

   10. lower-neg.f32 N/A

       \[\leadsto \left(\color{blue}{\left(-\log \left(\frac{-4}{3} \cdot u - \frac{-4}{3}\right)\right)} \cdot 3\right) \cdot s \]

   11. lower-log.f32 96.2%

       \[\leadsto \left(\left(-\color{blue}{\log \left(-1.3333333333333333 \cdot u - -1.3333333333333333\right)}\right) \cdot 3\right) \cdot s \]

7. Applied rewrites 96.2%

   \[\leadsto \color{blue}{\left(\left(-\log \left(-1.3333333333333333 \cdot u - -1.3333333333333333\right)\right) \cdot 3\right) \cdot s} \]
8. Add Preprocessing

Alternative 5: 96.2% accurate, 1.2× speedup

Math

\[\left(3 \cdot s\right) \cdot \left(-0.28768208622932434 - \log \left(1 - u\right)\right) \]

FPCore

(FPCore (s u)
  :precision binary32
  (* (* 3.0 s) (- -0.28768208622932434 (log (- 1.0 u)))))

C

float code(float s, float u) {
	return (3.0f * s) * (-0.28768208622932434f - logf((1.0f - u)));
}

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(s, u)
use fmin_fmax_functions
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = (3.0e0 * s) * ((-0.28768208622932434e0) - log((1.0e0 - u)))
end function

Julia

function code(s, u)
	return Float32(Float32(Float32(3.0) * s) * Float32(Float32(-0.28768208622932434) - log(Float32(Float32(1.0) - u))))
end

MATLAB

function tmp = code(s, u)
	tmp = (single(3.0) * s) * (single(-0.28768208622932434) - log((single(1.0) - u)));
end

Derivation

1. Initial program 96.0%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]
2. Step-by-step derivation

   1. lift-log.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)} \]

   2. lift-/.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \color{blue}{\left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)} \]

   3. lift--.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}}\right) \]

   4. lift-/.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \color{blue}{\frac{u - \frac{1}{4}}{\frac{3}{4}}}}\right) \]

   5. sub-to-fraction N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{1 \cdot \frac{3}{4} - \left(u - \frac{1}{4}\right)}{\frac{3}{4}}}}\right) \]

   6. div-flip-rev N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \color{blue}{\left(\frac{\frac{3}{4}}{1 \cdot \frac{3}{4} - \left(u - \frac{1}{4}\right)}\right)} \]

   7. log-div N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\left(\log \frac{3}{4} - \log \left(1 \cdot \frac{3}{4} - \left(u - \frac{1}{4}\right)\right)\right)} \]

   8. lower-unsound--.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\left(\log \frac{3}{4} - \log \left(1 \cdot \frac{3}{4} - \left(u - \frac{1}{4}\right)\right)\right)} \]

   9. lower-unsound-log.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\color{blue}{\log \frac{3}{4}} - \log \left(1 \cdot \frac{3}{4} - \left(u - \frac{1}{4}\right)\right)\right) \]

   10. lower-unsound-log.f32 N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\log \frac{3}{4} - \color{blue}{\log \left(1 \cdot \frac{3}{4} - \left(u - \frac{1}{4}\right)\right)}\right) \]

   11. metadata-eval N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\log \frac{3}{4} - \log \left(\color{blue}{\frac{3}{4}} - \left(u - \frac{1}{4}\right)\right)\right) \]

   12. lift--.f32 N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\log \frac{3}{4} - \log \left(\frac{3}{4} - \color{blue}{\left(u - \frac{1}{4}\right)}\right)\right) \]

   13. sub-negate-rev N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\log \frac{3}{4} - \log \left(\frac{3}{4} - \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{4} - u\right)\right)\right)}\right)\right) \]

   14. add-flip-rev N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\log \frac{3}{4} - \log \color{blue}{\left(\frac{3}{4} + \left(\frac{1}{4} - u\right)\right)}\right) \]

   15. associate-+r- N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\log \frac{3}{4} - \log \color{blue}{\left(\left(\frac{3}{4} + \frac{1}{4}\right) - u\right)}\right) \]

   16. metadata-eval N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\log \frac{3}{4} - \log \left(\color{blue}{1} - u\right)\right) \]

   17. lower--.f32 96.5%

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\log 0.75 - \log \color{blue}{\left(1 - u\right)}\right) \]

3. Applied rewrites 96.5%

   \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\left(\log 0.75 - \log \left(1 - u\right)\right)} \]

4. Evaluated real constant 96.5%

   \[\leadsto \left(3 \cdot s\right) \cdot \left(\color{blue}{-0.28768208622932434} - \log \left(1 - u\right)\right) \]
5. Add Preprocessing

Alternative 6: 40.6% accurate, 1.9× speedup

Math

\[\left(1 + \frac{u \cdot \left(3 \cdot s + u \cdot \left(1.5 \cdot s + u \cdot \left(s + 0.75 \cdot \left(s \cdot u\right)\right)\right)\right)}{-0.8630462288856506 \cdot s}\right) \cdot \left(-0.8630462288856506 \cdot s\right) \]

FPCore

(FPCore (s u)
  :precision binary32
  (*
 (+
  1.0
  (/
   (*
    u
    (+ (* 3.0 s) (* u (+ (* 1.5 s) (* u (+ s (* 0.75 (* s u))))))))
   (* -0.8630462288856506 s)))
 (* -0.8630462288856506 s)))

C

float code(float s, float u) {
	return (1.0f + ((u * ((3.0f * s) + (u * ((1.5f * s) + (u * (s + (0.75f * (s * u)))))))) / (-0.8630462288856506f * s))) * (-0.8630462288856506f * s);
}

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(s, u)
use fmin_fmax_functions
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = (1.0e0 + ((u * ((3.0e0 * s) + (u * ((1.5e0 * s) + (u * (s + (0.75e0 * (s * u)))))))) / ((-0.8630462288856506e0) * s))) * ((-0.8630462288856506e0) * s)
end function

Julia

function code(s, u)
	return Float32(Float32(Float32(1.0) + Float32(Float32(u * Float32(Float32(Float32(3.0) * s) + Float32(u * Float32(Float32(Float32(1.5) * s) + Float32(u * Float32(s + Float32(Float32(0.75) * Float32(s * u)))))))) / Float32(Float32(-0.8630462288856506) * s))) * Float32(Float32(-0.8630462288856506) * s))
end

MATLAB

function tmp = code(s, u)
	tmp = (single(1.0) + ((u * ((single(3.0) * s) + (u * ((single(1.5) * s) + (u * (s + (single(0.75) * (s * u)))))))) / (single(-0.8630462288856506) * s))) * (single(-0.8630462288856506) * s);
end

Derivation

1. Initial program 96.0%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]
2. Step-by-step derivation

   1. lift-log.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)} \]

   2. lift-/.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \color{blue}{\left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)} \]

   3. lift--.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}}\right) \]

   4. lift-/.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \color{blue}{\frac{u - \frac{1}{4}}{\frac{3}{4}}}}\right) \]

   5. sub-to-fraction N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{1 \cdot \frac{3}{4} - \left(u - \frac{1}{4}\right)}{\frac{3}{4}}}}\right) \]

   6. div-flip-rev N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \color{blue}{\left(\frac{\frac{3}{4}}{1 \cdot \frac{3}{4} - \left(u - \frac{1}{4}\right)}\right)} \]

   7. log-div N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\left(\log \frac{3}{4} - \log \left(1 \cdot \frac{3}{4} - \left(u - \frac{1}{4}\right)\right)\right)} \]

   8. lower-unsound--.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\left(\log \frac{3}{4} - \log \left(1 \cdot \frac{3}{4} - \left(u - \frac{1}{4}\right)\right)\right)} \]

   9. lower-unsound-log.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\color{blue}{\log \frac{3}{4}} - \log \left(1 \cdot \frac{3}{4} - \left(u - \frac{1}{4}\right)\right)\right) \]

   10. lower-unsound-log.f32 N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\log \frac{3}{4} - \color{blue}{\log \left(1 \cdot \frac{3}{4} - \left(u - \frac{1}{4}\right)\right)}\right) \]

   11. metadata-eval N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\log \frac{3}{4} - \log \left(\color{blue}{\frac{3}{4}} - \left(u - \frac{1}{4}\right)\right)\right) \]

   12. lift--.f32 N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\log \frac{3}{4} - \log \left(\frac{3}{4} - \color{blue}{\left(u - \frac{1}{4}\right)}\right)\right) \]

   13. sub-negate-rev N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\log \frac{3}{4} - \log \left(\frac{3}{4} - \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{4} - u\right)\right)\right)}\right)\right) \]

   14. add-flip-rev N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\log \frac{3}{4} - \log \color{blue}{\left(\frac{3}{4} + \left(\frac{1}{4} - u\right)\right)}\right) \]

   15. associate-+r- N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\log \frac{3}{4} - \log \color{blue}{\left(\left(\frac{3}{4} + \frac{1}{4}\right) - u\right)}\right) \]

   16. metadata-eval N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\log \frac{3}{4} - \log \left(\color{blue}{1} - u\right)\right) \]

   17. lower--.f32 96.5%

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\log 0.75 - \log \color{blue}{\left(1 - u\right)}\right) \]

3. Applied rewrites 96.5%

   \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\left(\log 0.75 - \log \left(1 - u\right)\right)} \]
4. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \color{blue}{\left(3 \cdot s\right) \cdot \left(\log \frac{3}{4} - \log \left(1 - u\right)\right)} \]

   2. lift--.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\left(\log \frac{3}{4} - \log \left(1 - u\right)\right)} \]

   3. lift-log.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\color{blue}{\log \frac{3}{4}} - \log \left(1 - u\right)\right) \]

   4. lift-log.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\log \frac{3}{4} - \color{blue}{\log \left(1 - u\right)}\right) \]

   5. diff-log N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\log \left(\frac{\frac{3}{4}}{1 - u}\right)} \]

   6. lift-/.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \color{blue}{\left(\frac{\frac{3}{4}}{1 - u}\right)} \]

   7. lift-log.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\log \left(\frac{\frac{3}{4}}{1 - u}\right)} \]

   8. lift-log.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\log \left(\frac{\frac{3}{4}}{1 - u}\right)} \]

   9. lift-/.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \color{blue}{\left(\frac{\frac{3}{4}}{1 - u}\right)} \]

   10. diff-log N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\left(\log \frac{3}{4} - \log \left(1 - u\right)\right)} \]

   11. lift-log.f32 N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\color{blue}{\log \frac{3}{4}} - \log \left(1 - u\right)\right) \]

   12. lift-log.f32 N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\log \frac{3}{4} - \color{blue}{\log \left(1 - u\right)}\right) \]

   13. sub-flip N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\left(\log \frac{3}{4} + \left(\mathsf{neg}\left(\log \left(1 - u\right)\right)\right)\right)} \]

   14. distribute-lft-in N/A

       \[\leadsto \color{blue}{\left(3 \cdot s\right) \cdot \log \frac{3}{4} + \left(3 \cdot s\right) \cdot \left(\mathsf{neg}\left(\log \left(1 - u\right)\right)\right)} \]

   15. lift-*.f32 N/A

       \[\leadsto \color{blue}{\left(3 \cdot s\right)} \cdot \log \frac{3}{4} + \left(3 \cdot s\right) \cdot \left(\mathsf{neg}\left(\log \left(1 - u\right)\right)\right) \]

   16. associate-*r* N/A

       \[\leadsto \color{blue}{3 \cdot \left(s \cdot \log \frac{3}{4}\right)} + \left(3 \cdot s\right) \cdot \left(\mathsf{neg}\left(\log \left(1 - u\right)\right)\right) \]

   17. lift-*.f32 N/A

       \[\leadsto 3 \cdot \color{blue}{\left(s \cdot \log \frac{3}{4}\right)} + \left(3 \cdot s\right) \cdot \left(\mathsf{neg}\left(\log \left(1 - u\right)\right)\right) \]

   18. lift-*.f32 N/A

       \[\leadsto \color{blue}{3 \cdot \left(s \cdot \log \frac{3}{4}\right)} + \left(3 \cdot s\right) \cdot \left(\mathsf{neg}\left(\log \left(1 - u\right)\right)\right) \]

   19. sum-to-mult N/A

       \[\leadsto \color{blue}{\left(1 + \frac{\left(3 \cdot s\right) \cdot \left(\mathsf{neg}\left(\log \left(1 - u\right)\right)\right)}{3 \cdot \left(s \cdot \log \frac{3}{4}\right)}\right) \cdot \left(3 \cdot \left(s \cdot \log \frac{3}{4}\right)\right)} \]

5. Applied rewrites 96.2%

   \[\leadsto \color{blue}{\left(1 + \frac{\left(\left(-\log \left(1 - u\right)\right) \cdot s\right) \cdot 3}{\log 0.421875 \cdot s}\right) \cdot \left(\log 0.421875 \cdot s\right)} \]

6. Evaluated real constant 96.2%

   \[\leadsto \left(1 + \frac{\left(\left(-\log \left(1 - u\right)\right) \cdot s\right) \cdot 3}{\color{blue}{-0.8630462288856506} \cdot s}\right) \cdot \left(\log 0.421875 \cdot s\right) \]

7. Evaluated real constant 96.2%

   \[\leadsto \left(1 + \frac{\left(\left(-\log \left(1 - u\right)\right) \cdot s\right) \cdot 3}{-0.8630462288856506 \cdot s}\right) \cdot \left(\color{blue}{-0.8630462288856506} \cdot s\right) \]

8. Taylor expanded in u around 0

   \[\leadsto \left(1 + \frac{\color{blue}{u \cdot \left(3 \cdot s + u \cdot \left(\frac{3}{2} \cdot s + u \cdot \left(s + \frac{3}{4} \cdot \left(s \cdot u\right)\right)\right)\right)}}{-0.8630462288856506 \cdot s}\right) \cdot \left(-0.8630462288856506 \cdot s\right) \]
9. Step-by-step derivation

   1. lower-*.f32 N/A

      \[\leadsto \left(1 + \frac{u \cdot \color{blue}{\left(3 \cdot s + u \cdot \left(\frac{3}{2} \cdot s + u \cdot \left(s + \frac{3}{4} \cdot \left(s \cdot u\right)\right)\right)\right)}}{\frac{-14479513}{16777216} \cdot s}\right) \cdot \left(\frac{-14479513}{16777216} \cdot s\right) \]

   2. lower-+.f32 N/A

      \[\leadsto \left(1 + \frac{u \cdot \left(3 \cdot s + \color{blue}{u \cdot \left(\frac{3}{2} \cdot s + u \cdot \left(s + \frac{3}{4} \cdot \left(s \cdot u\right)\right)\right)}\right)}{\frac{-14479513}{16777216} \cdot s}\right) \cdot \left(\frac{-14479513}{16777216} \cdot s\right) \]

   3. lower-*.f32 N/A

      \[\leadsto \left(1 + \frac{u \cdot \left(3 \cdot s + \color{blue}{u} \cdot \left(\frac{3}{2} \cdot s + u \cdot \left(s + \frac{3}{4} \cdot \left(s \cdot u\right)\right)\right)\right)}{\frac{-14479513}{16777216} \cdot s}\right) \cdot \left(\frac{-14479513}{16777216} \cdot s\right) \]

   4. lower-*.f32 N/A

      \[\leadsto \left(1 + \frac{u \cdot \left(3 \cdot s + u \cdot \color{blue}{\left(\frac{3}{2} \cdot s + u \cdot \left(s + \frac{3}{4} \cdot \left(s \cdot u\right)\right)\right)}\right)}{\frac{-14479513}{16777216} \cdot s}\right) \cdot \left(\frac{-14479513}{16777216} \cdot s\right) \]

   5. lower-+.f32 N/A

      \[\leadsto \left(1 + \frac{u \cdot \left(3 \cdot s + u \cdot \left(\frac{3}{2} \cdot s + \color{blue}{u \cdot \left(s + \frac{3}{4} \cdot \left(s \cdot u\right)\right)}\right)\right)}{\frac{-14479513}{16777216} \cdot s}\right) \cdot \left(\frac{-14479513}{16777216} \cdot s\right) \]

   6. lower-*.f32 N/A

      \[\leadsto \left(1 + \frac{u \cdot \left(3 \cdot s + u \cdot \left(\frac{3}{2} \cdot s + \color{blue}{u} \cdot \left(s + \frac{3}{4} \cdot \left(s \cdot u\right)\right)\right)\right)}{\frac{-14479513}{16777216} \cdot s}\right) \cdot \left(\frac{-14479513}{16777216} \cdot s\right) \]

   7. lower-*.f32 N/A

      \[\leadsto \left(1 + \frac{u \cdot \left(3 \cdot s + u \cdot \left(\frac{3}{2} \cdot s + u \cdot \color{blue}{\left(s + \frac{3}{4} \cdot \left(s \cdot u\right)\right)}\right)\right)}{\frac{-14479513}{16777216} \cdot s}\right) \cdot \left(\frac{-14479513}{16777216} \cdot s\right) \]

   8. lower-+.f32 N/A

      \[\leadsto \left(1 + \frac{u \cdot \left(3 \cdot s + u \cdot \left(\frac{3}{2} \cdot s + u \cdot \left(s + \color{blue}{\frac{3}{4} \cdot \left(s \cdot u\right)}\right)\right)\right)}{\frac{-14479513}{16777216} \cdot s}\right) \cdot \left(\frac{-14479513}{16777216} \cdot s\right) \]

   9. lower-*.f32 N/A

      \[\leadsto \left(1 + \frac{u \cdot \left(3 \cdot s + u \cdot \left(\frac{3}{2} \cdot s + u \cdot \left(s + \frac{3}{4} \cdot \color{blue}{\left(s \cdot u\right)}\right)\right)\right)}{\frac{-14479513}{16777216} \cdot s}\right) \cdot \left(\frac{-14479513}{16777216} \cdot s\right) \]

   10. lower-*.f32 40.6%

       \[\leadsto \left(1 + \frac{u \cdot \left(3 \cdot s + u \cdot \left(1.5 \cdot s + u \cdot \left(s + 0.75 \cdot \left(s \cdot \color{blue}{u}\right)\right)\right)\right)}{-0.8630462288856506 \cdot s}\right) \cdot \left(-0.8630462288856506 \cdot s\right) \]

10. Applied rewrites 40.6%

    \[\leadsto \left(1 + \frac{\color{blue}{u \cdot \left(3 \cdot s + u \cdot \left(1.5 \cdot s + u \cdot \left(s + 0.75 \cdot \left(s \cdot u\right)\right)\right)\right)}}{-0.8630462288856506 \cdot s}\right) \cdot \left(-0.8630462288856506 \cdot s\right) \]
11. Add Preprocessing

Alternative 7: 40.6% accurate, 2.0× speedup

Math

\[\left(1 + \frac{\left(\left(u \cdot \left(1 + u \cdot \left(0.5 + u \cdot \left(0.3333333333333333 + 0.25 \cdot u\right)\right)\right)\right) \cdot s\right) \cdot 3}{-0.8630462288856506 \cdot s}\right) \cdot \left(-0.8630462288856506 \cdot s\right) \]

FPCore

(FPCore (s u)
  :precision binary32
  (*
 (+
  1.0
  (/
   (*
    (*
     (*
      u
      (+ 1.0 (* u (+ 0.5 (* u (+ 0.3333333333333333 (* 0.25 u)))))))
     s)
    3.0)
   (* -0.8630462288856506 s)))
 (* -0.8630462288856506 s)))

C

float code(float s, float u) {
	return (1.0f + ((((u * (1.0f + (u * (0.5f + (u * (0.3333333333333333f + (0.25f * u))))))) * s) * 3.0f) / (-0.8630462288856506f * s))) * (-0.8630462288856506f * s);
}

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(s, u)
use fmin_fmax_functions
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = (1.0e0 + ((((u * (1.0e0 + (u * (0.5e0 + (u * (0.3333333333333333e0 + (0.25e0 * u))))))) * s) * 3.0e0) / ((-0.8630462288856506e0) * s))) * ((-0.8630462288856506e0) * s)
end function

Julia

function code(s, u)
	return Float32(Float32(Float32(1.0) + Float32(Float32(Float32(Float32(u * Float32(Float32(1.0) + Float32(u * Float32(Float32(0.5) + Float32(u * Float32(Float32(0.3333333333333333) + Float32(Float32(0.25) * u))))))) * s) * Float32(3.0)) / Float32(Float32(-0.8630462288856506) * s))) * Float32(Float32(-0.8630462288856506) * s))
end

MATLAB

function tmp = code(s, u)
	tmp = (single(1.0) + ((((u * (single(1.0) + (u * (single(0.5) + (u * (single(0.3333333333333333) + (single(0.25) * u))))))) * s) * single(3.0)) / (single(-0.8630462288856506) * s))) * (single(-0.8630462288856506) * s);
end

Derivation

1. Initial program 96.0%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]
2. Step-by-step derivation

   1. lift-log.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)} \]

   2. lift-/.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \color{blue}{\left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)} \]

   3. lift--.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}}\right) \]

   4. lift-/.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \color{blue}{\frac{u - \frac{1}{4}}{\frac{3}{4}}}}\right) \]

   5. sub-to-fraction N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{1 \cdot \frac{3}{4} - \left(u - \frac{1}{4}\right)}{\frac{3}{4}}}}\right) \]

   6. div-flip-rev N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \color{blue}{\left(\frac{\frac{3}{4}}{1 \cdot \frac{3}{4} - \left(u - \frac{1}{4}\right)}\right)} \]

   7. log-div N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\left(\log \frac{3}{4} - \log \left(1 \cdot \frac{3}{4} - \left(u - \frac{1}{4}\right)\right)\right)} \]

   8. lower-unsound--.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\left(\log \frac{3}{4} - \log \left(1 \cdot \frac{3}{4} - \left(u - \frac{1}{4}\right)\right)\right)} \]

   9. lower-unsound-log.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\color{blue}{\log \frac{3}{4}} - \log \left(1 \cdot \frac{3}{4} - \left(u - \frac{1}{4}\right)\right)\right) \]

   10. lower-unsound-log.f32 N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\log \frac{3}{4} - \color{blue}{\log \left(1 \cdot \frac{3}{4} - \left(u - \frac{1}{4}\right)\right)}\right) \]

   11. metadata-eval N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\log \frac{3}{4} - \log \left(\color{blue}{\frac{3}{4}} - \left(u - \frac{1}{4}\right)\right)\right) \]

   12. lift--.f32 N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\log \frac{3}{4} - \log \left(\frac{3}{4} - \color{blue}{\left(u - \frac{1}{4}\right)}\right)\right) \]

   13. sub-negate-rev N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\log \frac{3}{4} - \log \left(\frac{3}{4} - \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{4} - u\right)\right)\right)}\right)\right) \]

   14. add-flip-rev N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\log \frac{3}{4} - \log \color{blue}{\left(\frac{3}{4} + \left(\frac{1}{4} - u\right)\right)}\right) \]

   15. associate-+r- N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\log \frac{3}{4} - \log \color{blue}{\left(\left(\frac{3}{4} + \frac{1}{4}\right) - u\right)}\right) \]

   16. metadata-eval N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\log \frac{3}{4} - \log \left(\color{blue}{1} - u\right)\right) \]

   17. lower--.f32 96.5%

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\log 0.75 - \log \color{blue}{\left(1 - u\right)}\right) \]

3. Applied rewrites 96.5%

   \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\left(\log 0.75 - \log \left(1 - u\right)\right)} \]
4. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \color{blue}{\left(3 \cdot s\right) \cdot \left(\log \frac{3}{4} - \log \left(1 - u\right)\right)} \]

   2. lift--.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\left(\log \frac{3}{4} - \log \left(1 - u\right)\right)} \]

   3. lift-log.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\color{blue}{\log \frac{3}{4}} - \log \left(1 - u\right)\right) \]

   4. lift-log.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\log \frac{3}{4} - \color{blue}{\log \left(1 - u\right)}\right) \]

   5. diff-log N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\log \left(\frac{\frac{3}{4}}{1 - u}\right)} \]

   6. lift-/.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \color{blue}{\left(\frac{\frac{3}{4}}{1 - u}\right)} \]

   7. lift-log.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\log \left(\frac{\frac{3}{4}}{1 - u}\right)} \]

   8. lift-log.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\log \left(\frac{\frac{3}{4}}{1 - u}\right)} \]

   9. lift-/.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \color{blue}{\left(\frac{\frac{3}{4}}{1 - u}\right)} \]

   10. diff-log N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\left(\log \frac{3}{4} - \log \left(1 - u\right)\right)} \]

   11. lift-log.f32 N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\color{blue}{\log \frac{3}{4}} - \log \left(1 - u\right)\right) \]

   12. lift-log.f32 N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\log \frac{3}{4} - \color{blue}{\log \left(1 - u\right)}\right) \]

   13. sub-flip N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\left(\log \frac{3}{4} + \left(\mathsf{neg}\left(\log \left(1 - u\right)\right)\right)\right)} \]

   14. distribute-lft-in N/A

       \[\leadsto \color{blue}{\left(3 \cdot s\right) \cdot \log \frac{3}{4} + \left(3 \cdot s\right) \cdot \left(\mathsf{neg}\left(\log \left(1 - u\right)\right)\right)} \]

   15. lift-*.f32 N/A

       \[\leadsto \color{blue}{\left(3 \cdot s\right)} \cdot \log \frac{3}{4} + \left(3 \cdot s\right) \cdot \left(\mathsf{neg}\left(\log \left(1 - u\right)\right)\right) \]

   16. associate-*r* N/A

       \[\leadsto \color{blue}{3 \cdot \left(s \cdot \log \frac{3}{4}\right)} + \left(3 \cdot s\right) \cdot \left(\mathsf{neg}\left(\log \left(1 - u\right)\right)\right) \]

   17. lift-*.f32 N/A

       \[\leadsto 3 \cdot \color{blue}{\left(s \cdot \log \frac{3}{4}\right)} + \left(3 \cdot s\right) \cdot \left(\mathsf{neg}\left(\log \left(1 - u\right)\right)\right) \]

   18. lift-*.f32 N/A

       \[\leadsto \color{blue}{3 \cdot \left(s \cdot \log \frac{3}{4}\right)} + \left(3 \cdot s\right) \cdot \left(\mathsf{neg}\left(\log \left(1 - u\right)\right)\right) \]

   19. sum-to-mult N/A

       \[\leadsto \color{blue}{\left(1 + \frac{\left(3 \cdot s\right) \cdot \left(\mathsf{neg}\left(\log \left(1 - u\right)\right)\right)}{3 \cdot \left(s \cdot \log \frac{3}{4}\right)}\right) \cdot \left(3 \cdot \left(s \cdot \log \frac{3}{4}\right)\right)} \]

5. Applied rewrites 96.2%

   \[\leadsto \color{blue}{\left(1 + \frac{\left(\left(-\log \left(1 - u\right)\right) \cdot s\right) \cdot 3}{\log 0.421875 \cdot s}\right) \cdot \left(\log 0.421875 \cdot s\right)} \]

6. Evaluated real constant 96.2%

   \[\leadsto \left(1 + \frac{\left(\left(-\log \left(1 - u\right)\right) \cdot s\right) \cdot 3}{\color{blue}{-0.8630462288856506} \cdot s}\right) \cdot \left(\log 0.421875 \cdot s\right) \]

7. Evaluated real constant 96.2%

   \[\leadsto \left(1 + \frac{\left(\left(-\log \left(1 - u\right)\right) \cdot s\right) \cdot 3}{-0.8630462288856506 \cdot s}\right) \cdot \left(\color{blue}{-0.8630462288856506} \cdot s\right) \]

8. Taylor expanded in u around 0

   \[\leadsto \left(1 + \frac{\left(\color{blue}{\left(u \cdot \left(1 + u \cdot \left(\frac{1}{2} + u \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u\right)\right)\right)\right)} \cdot s\right) \cdot 3}{-0.8630462288856506 \cdot s}\right) \cdot \left(-0.8630462288856506 \cdot s\right) \]
9. Step-by-step derivation

   1. lower-*.f32 N/A

      \[\leadsto \left(1 + \frac{\left(\left(u \cdot \color{blue}{\left(1 + u \cdot \left(\frac{1}{2} + u \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u\right)\right)\right)}\right) \cdot s\right) \cdot 3}{\frac{-14479513}{16777216} \cdot s}\right) \cdot \left(\frac{-14479513}{16777216} \cdot s\right) \]

   2. lower-+.f32 N/A

      \[\leadsto \left(1 + \frac{\left(\left(u \cdot \left(1 + \color{blue}{u \cdot \left(\frac{1}{2} + u \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u\right)\right)}\right)\right) \cdot s\right) \cdot 3}{\frac{-14479513}{16777216} \cdot s}\right) \cdot \left(\frac{-14479513}{16777216} \cdot s\right) \]

   3. lower-*.f32 N/A

      \[\leadsto \left(1 + \frac{\left(\left(u \cdot \left(1 + u \cdot \color{blue}{\left(\frac{1}{2} + u \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u\right)\right)}\right)\right) \cdot s\right) \cdot 3}{\frac{-14479513}{16777216} \cdot s}\right) \cdot \left(\frac{-14479513}{16777216} \cdot s\right) \]

   4. lower-+.f32 N/A

      \[\leadsto \left(1 + \frac{\left(\left(u \cdot \left(1 + u \cdot \left(\frac{1}{2} + \color{blue}{u \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u\right)}\right)\right)\right) \cdot s\right) \cdot 3}{\frac{-14479513}{16777216} \cdot s}\right) \cdot \left(\frac{-14479513}{16777216} \cdot s\right) \]

   5. lower-*.f32 N/A

      \[\leadsto \left(1 + \frac{\left(\left(u \cdot \left(1 + u \cdot \left(\frac{1}{2} + u \cdot \color{blue}{\left(\frac{1}{3} + \frac{1}{4} \cdot u\right)}\right)\right)\right) \cdot s\right) \cdot 3}{\frac{-14479513}{16777216} \cdot s}\right) \cdot \left(\frac{-14479513}{16777216} \cdot s\right) \]

   6. lower-+.f32 N/A

      \[\leadsto \left(1 + \frac{\left(\left(u \cdot \left(1 + u \cdot \left(\frac{1}{2} + u \cdot \left(\frac{1}{3} + \color{blue}{\frac{1}{4} \cdot u}\right)\right)\right)\right) \cdot s\right) \cdot 3}{\frac{-14479513}{16777216} \cdot s}\right) \cdot \left(\frac{-14479513}{16777216} \cdot s\right) \]

   7. lower-*.f32 40.6%

      \[\leadsto \left(1 + \frac{\left(\left(u \cdot \left(1 + u \cdot \left(0.5 + u \cdot \left(0.3333333333333333 + 0.25 \cdot \color{blue}{u}\right)\right)\right)\right) \cdot s\right) \cdot 3}{-0.8630462288856506 \cdot s}\right) \cdot \left(-0.8630462288856506 \cdot s\right) \]

10. Applied rewrites 40.6%

    \[\leadsto \left(1 + \frac{\left(\color{blue}{\left(u \cdot \left(1 + u \cdot \left(0.5 + u \cdot \left(0.3333333333333333 + 0.25 \cdot u\right)\right)\right)\right)} \cdot s\right) \cdot 3}{-0.8630462288856506 \cdot s}\right) \cdot \left(-0.8630462288856506 \cdot s\right) \]
11. Add Preprocessing

Alternative 8: 40.6% accurate, 3.2× speedup

Math

\[\left(1 + u \cdot \left(u \cdot \left(u \cdot \left(-0.8690148625855028 \cdot u - 1.158686483447337\right) - 1.7380297251710055\right) - 3.476059450342011\right)\right) \cdot \left(-0.8630462288856506 \cdot s\right) \]

FPCore

(FPCore (s u)
  :precision binary32
  (*
 (+
  1.0
  (*
   u
   (-
    (*
     u
     (-
      (* u (- (* -0.8690148625855028 u) 1.158686483447337))
      1.7380297251710055))
    3.476059450342011)))
 (* -0.8630462288856506 s)))

C

float code(float s, float u) {
	return (1.0f + (u * ((u * ((u * ((-0.8690148625855028f * u) - 1.158686483447337f)) - 1.7380297251710055f)) - 3.476059450342011f))) * (-0.8630462288856506f * s);
}

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(s, u)
use fmin_fmax_functions
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = (1.0e0 + (u * ((u * ((u * (((-0.8690148625855028e0) * u) - 1.158686483447337e0)) - 1.7380297251710055e0)) - 3.476059450342011e0))) * ((-0.8630462288856506e0) * s)
end function

Julia

function code(s, u)
	return Float32(Float32(Float32(1.0) + Float32(u * Float32(Float32(u * Float32(Float32(u * Float32(Float32(Float32(-0.8690148625855028) * u) - Float32(1.158686483447337))) - Float32(1.7380297251710055))) - Float32(3.476059450342011)))) * Float32(Float32(-0.8630462288856506) * s))
end

MATLAB

function tmp = code(s, u)
	tmp = (single(1.0) + (u * ((u * ((u * ((single(-0.8690148625855028) * u) - single(1.158686483447337))) - single(1.7380297251710055))) - single(3.476059450342011)))) * (single(-0.8630462288856506) * s);
end

Derivation

1. Initial program 96.0%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]
2. Step-by-step derivation

   1. lift-log.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)} \]

   2. lift-/.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \color{blue}{\left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)} \]

   3. lift--.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}}\right) \]

   4. lift-/.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \color{blue}{\frac{u - \frac{1}{4}}{\frac{3}{4}}}}\right) \]

   5. sub-to-fraction N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{1 \cdot \frac{3}{4} - \left(u - \frac{1}{4}\right)}{\frac{3}{4}}}}\right) \]

   6. div-flip-rev N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \color{blue}{\left(\frac{\frac{3}{4}}{1 \cdot \frac{3}{4} - \left(u - \frac{1}{4}\right)}\right)} \]

   7. log-div N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\left(\log \frac{3}{4} - \log \left(1 \cdot \frac{3}{4} - \left(u - \frac{1}{4}\right)\right)\right)} \]

   8. lower-unsound--.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\left(\log \frac{3}{4} - \log \left(1 \cdot \frac{3}{4} - \left(u - \frac{1}{4}\right)\right)\right)} \]

   9. lower-unsound-log.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\color{blue}{\log \frac{3}{4}} - \log \left(1 \cdot \frac{3}{4} - \left(u - \frac{1}{4}\right)\right)\right) \]

   10. lower-unsound-log.f32 N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\log \frac{3}{4} - \color{blue}{\log \left(1 \cdot \frac{3}{4} - \left(u - \frac{1}{4}\right)\right)}\right) \]

   11. metadata-eval N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\log \frac{3}{4} - \log \left(\color{blue}{\frac{3}{4}} - \left(u - \frac{1}{4}\right)\right)\right) \]

   12. lift--.f32 N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\log \frac{3}{4} - \log \left(\frac{3}{4} - \color{blue}{\left(u - \frac{1}{4}\right)}\right)\right) \]

   13. sub-negate-rev N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\log \frac{3}{4} - \log \left(\frac{3}{4} - \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{4} - u\right)\right)\right)}\right)\right) \]

   14. add-flip-rev N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\log \frac{3}{4} - \log \color{blue}{\left(\frac{3}{4} + \left(\frac{1}{4} - u\right)\right)}\right) \]

   15. associate-+r- N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\log \frac{3}{4} - \log \color{blue}{\left(\left(\frac{3}{4} + \frac{1}{4}\right) - u\right)}\right) \]

   16. metadata-eval N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\log \frac{3}{4} - \log \left(\color{blue}{1} - u\right)\right) \]

   17. lower--.f32 96.5%

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\log 0.75 - \log \color{blue}{\left(1 - u\right)}\right) \]

3. Applied rewrites 96.5%

   \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\left(\log 0.75 - \log \left(1 - u\right)\right)} \]
4. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \color{blue}{\left(3 \cdot s\right) \cdot \left(\log \frac{3}{4} - \log \left(1 - u\right)\right)} \]

   2. lift--.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\left(\log \frac{3}{4} - \log \left(1 - u\right)\right)} \]

   3. lift-log.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\color{blue}{\log \frac{3}{4}} - \log \left(1 - u\right)\right) \]

   4. lift-log.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\log \frac{3}{4} - \color{blue}{\log \left(1 - u\right)}\right) \]

   5. diff-log N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\log \left(\frac{\frac{3}{4}}{1 - u}\right)} \]

   6. lift-/.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \color{blue}{\left(\frac{\frac{3}{4}}{1 - u}\right)} \]

   7. lift-log.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\log \left(\frac{\frac{3}{4}}{1 - u}\right)} \]

   8. lift-log.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\log \left(\frac{\frac{3}{4}}{1 - u}\right)} \]

   9. lift-/.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \color{blue}{\left(\frac{\frac{3}{4}}{1 - u}\right)} \]

   10. diff-log N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\left(\log \frac{3}{4} - \log \left(1 - u\right)\right)} \]

   11. lift-log.f32 N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\color{blue}{\log \frac{3}{4}} - \log \left(1 - u\right)\right) \]

   12. lift-log.f32 N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\log \frac{3}{4} - \color{blue}{\log \left(1 - u\right)}\right) \]

   13. sub-flip N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\left(\log \frac{3}{4} + \left(\mathsf{neg}\left(\log \left(1 - u\right)\right)\right)\right)} \]

   14. distribute-lft-in N/A

       \[\leadsto \color{blue}{\left(3 \cdot s\right) \cdot \log \frac{3}{4} + \left(3 \cdot s\right) \cdot \left(\mathsf{neg}\left(\log \left(1 - u\right)\right)\right)} \]

   15. lift-*.f32 N/A

       \[\leadsto \color{blue}{\left(3 \cdot s\right)} \cdot \log \frac{3}{4} + \left(3 \cdot s\right) \cdot \left(\mathsf{neg}\left(\log \left(1 - u\right)\right)\right) \]

   16. associate-*r* N/A

       \[\leadsto \color{blue}{3 \cdot \left(s \cdot \log \frac{3}{4}\right)} + \left(3 \cdot s\right) \cdot \left(\mathsf{neg}\left(\log \left(1 - u\right)\right)\right) \]

   17. lift-*.f32 N/A

       \[\leadsto 3 \cdot \color{blue}{\left(s \cdot \log \frac{3}{4}\right)} + \left(3 \cdot s\right) \cdot \left(\mathsf{neg}\left(\log \left(1 - u\right)\right)\right) \]

   18. lift-*.f32 N/A

       \[\leadsto \color{blue}{3 \cdot \left(s \cdot \log \frac{3}{4}\right)} + \left(3 \cdot s\right) \cdot \left(\mathsf{neg}\left(\log \left(1 - u\right)\right)\right) \]

   19. sum-to-mult N/A

       \[\leadsto \color{blue}{\left(1 + \frac{\left(3 \cdot s\right) \cdot \left(\mathsf{neg}\left(\log \left(1 - u\right)\right)\right)}{3 \cdot \left(s \cdot \log \frac{3}{4}\right)}\right) \cdot \left(3 \cdot \left(s \cdot \log \frac{3}{4}\right)\right)} \]

5. Applied rewrites 96.2%

   \[\leadsto \color{blue}{\left(1 + \frac{\left(\left(-\log \left(1 - u\right)\right) \cdot s\right) \cdot 3}{\log 0.421875 \cdot s}\right) \cdot \left(\log 0.421875 \cdot s\right)} \]

6. Evaluated real constant 96.2%

   \[\leadsto \left(1 + \frac{\left(\left(-\log \left(1 - u\right)\right) \cdot s\right) \cdot 3}{\color{blue}{-0.8630462288856506} \cdot s}\right) \cdot \left(\log 0.421875 \cdot s\right) \]

7. Evaluated real constant 96.2%

   \[\leadsto \left(1 + \frac{\left(\left(-\log \left(1 - u\right)\right) \cdot s\right) \cdot 3}{-0.8630462288856506 \cdot s}\right) \cdot \left(\color{blue}{-0.8630462288856506} \cdot s\right) \]

8. Taylor expanded in u around 0

   \[\leadsto \left(1 + \color{blue}{u \cdot \left(u \cdot \left(u \cdot \left(\frac{-12582912}{14479513} \cdot u - \frac{16777216}{14479513}\right) - \frac{25165824}{14479513}\right) - \frac{50331648}{14479513}\right)}\right) \cdot \left(-0.8630462288856506 \cdot s\right) \]
9. Step-by-step derivation

   1. lower-*.f32 N/A

      \[\leadsto \left(1 + u \cdot \color{blue}{\left(u \cdot \left(u \cdot \left(\frac{-12582912}{14479513} \cdot u - \frac{16777216}{14479513}\right) - \frac{25165824}{14479513}\right) - \frac{50331648}{14479513}\right)}\right) \cdot \left(\frac{-14479513}{16777216} \cdot s\right) \]

   2. lower--.f32 N/A

      \[\leadsto \left(1 + u \cdot \left(u \cdot \left(u \cdot \left(\frac{-12582912}{14479513} \cdot u - \frac{16777216}{14479513}\right) - \frac{25165824}{14479513}\right) - \color{blue}{\frac{50331648}{14479513}}\right)\right) \cdot \left(\frac{-14479513}{16777216} \cdot s\right) \]

   3. lower-*.f32 N/A

      \[\leadsto \left(1 + u \cdot \left(u \cdot \left(u \cdot \left(\frac{-12582912}{14479513} \cdot u - \frac{16777216}{14479513}\right) - \frac{25165824}{14479513}\right) - \frac{50331648}{14479513}\right)\right) \cdot \left(\frac{-14479513}{16777216} \cdot s\right) \]

   4. lower--.f32 N/A

      \[\leadsto \left(1 + u \cdot \left(u \cdot \left(u \cdot \left(\frac{-12582912}{14479513} \cdot u - \frac{16777216}{14479513}\right) - \frac{25165824}{14479513}\right) - \frac{50331648}{14479513}\right)\right) \cdot \left(\frac{-14479513}{16777216} \cdot s\right) \]

   5. lower-*.f32 N/A

      \[\leadsto \left(1 + u \cdot \left(u \cdot \left(u \cdot \left(\frac{-12582912}{14479513} \cdot u - \frac{16777216}{14479513}\right) - \frac{25165824}{14479513}\right) - \frac{50331648}{14479513}\right)\right) \cdot \left(\frac{-14479513}{16777216} \cdot s\right) \]

   6. lower--.f32 N/A

      \[\leadsto \left(1 + u \cdot \left(u \cdot \left(u \cdot \left(\frac{-12582912}{14479513} \cdot u - \frac{16777216}{14479513}\right) - \frac{25165824}{14479513}\right) - \frac{50331648}{14479513}\right)\right) \cdot \left(\frac{-14479513}{16777216} \cdot s\right) \]

   7. lower-*.f32 40.6%

      \[\leadsto \left(1 + u \cdot \left(u \cdot \left(u \cdot \left(-0.8690148625855028 \cdot u - 1.158686483447337\right) - 1.7380297251710055\right) - 3.476059450342011\right)\right) \cdot \left(-0.8630462288856506 \cdot s\right) \]

10. Applied rewrites 40.6%

    \[\leadsto \left(1 + \color{blue}{u \cdot \left(u \cdot \left(u \cdot \left(-0.8690148625855028 \cdot u - 1.158686483447337\right) - 1.7380297251710055\right) - 3.476059450342011\right)}\right) \cdot \left(-0.8630462288856506 \cdot s\right) \]
11. Add Preprocessing

Alternative 9: 36.7% accurate, 3.5× speedup

Math

\[-0.8630462288856506 \cdot s + u \cdot \left(3 \cdot s + u \cdot \left(1.5 \cdot s + s \cdot u\right)\right) \]

FPCore

(FPCore (s u)
  :precision binary32
  (+
 (* -0.8630462288856506 s)
 (* u (+ (* 3.0 s) (* u (+ (* 1.5 s) (* s u)))))))

C

float code(float s, float u) {
	return (-0.8630462288856506f * s) + (u * ((3.0f * s) + (u * ((1.5f * s) + (s * u)))));
}

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(s, u)
use fmin_fmax_functions
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = ((-0.8630462288856506e0) * s) + (u * ((3.0e0 * s) + (u * ((1.5e0 * s) + (s * u)))))
end function

Julia

function code(s, u)
	return Float32(Float32(Float32(-0.8630462288856506) * s) + Float32(u * Float32(Float32(Float32(3.0) * s) + Float32(u * Float32(Float32(Float32(1.5) * s) + Float32(s * u))))))
end

MATLAB

function tmp = code(s, u)
	tmp = (single(-0.8630462288856506) * s) + (u * ((single(3.0) * s) + (u * ((single(1.5) * s) + (s * u)))));
end

Derivation

1. Initial program 96.0%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]
2. Step-by-step derivation

   1. lift-log.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)} \]

   2. lift-/.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \color{blue}{\left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)} \]

   3. lift--.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}}\right) \]

   4. lift-/.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \color{blue}{\frac{u - \frac{1}{4}}{\frac{3}{4}}}}\right) \]

   5. sub-to-fraction N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{1 \cdot \frac{3}{4} - \left(u - \frac{1}{4}\right)}{\frac{3}{4}}}}\right) \]

   6. div-flip-rev N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \color{blue}{\left(\frac{\frac{3}{4}}{1 \cdot \frac{3}{4} - \left(u - \frac{1}{4}\right)}\right)} \]

   7. log-div N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\left(\log \frac{3}{4} - \log \left(1 \cdot \frac{3}{4} - \left(u - \frac{1}{4}\right)\right)\right)} \]

   8. lower-unsound--.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\left(\log \frac{3}{4} - \log \left(1 \cdot \frac{3}{4} - \left(u - \frac{1}{4}\right)\right)\right)} \]

   9. lower-unsound-log.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\color{blue}{\log \frac{3}{4}} - \log \left(1 \cdot \frac{3}{4} - \left(u - \frac{1}{4}\right)\right)\right) \]

   10. lower-unsound-log.f32 N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\log \frac{3}{4} - \color{blue}{\log \left(1 \cdot \frac{3}{4} - \left(u - \frac{1}{4}\right)\right)}\right) \]

   11. metadata-eval N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\log \frac{3}{4} - \log \left(\color{blue}{\frac{3}{4}} - \left(u - \frac{1}{4}\right)\right)\right) \]

   12. lift--.f32 N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\log \frac{3}{4} - \log \left(\frac{3}{4} - \color{blue}{\left(u - \frac{1}{4}\right)}\right)\right) \]

   13. sub-negate-rev N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\log \frac{3}{4} - \log \left(\frac{3}{4} - \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{4} - u\right)\right)\right)}\right)\right) \]

   14. add-flip-rev N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\log \frac{3}{4} - \log \color{blue}{\left(\frac{3}{4} + \left(\frac{1}{4} - u\right)\right)}\right) \]

   15. associate-+r- N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\log \frac{3}{4} - \log \color{blue}{\left(\left(\frac{3}{4} + \frac{1}{4}\right) - u\right)}\right) \]

   16. metadata-eval N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\log \frac{3}{4} - \log \left(\color{blue}{1} - u\right)\right) \]

   17. lower--.f32 96.5%

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\log 0.75 - \log \color{blue}{\left(1 - u\right)}\right) \]

3. Applied rewrites 96.5%

   \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\left(\log 0.75 - \log \left(1 - u\right)\right)} \]
4. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \color{blue}{\left(3 \cdot s\right) \cdot \left(\log \frac{3}{4} - \log \left(1 - u\right)\right)} \]

   2. lift--.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\left(\log \frac{3}{4} - \log \left(1 - u\right)\right)} \]

   3. lift-log.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\color{blue}{\log \frac{3}{4}} - \log \left(1 - u\right)\right) \]

   4. lift-log.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\log \frac{3}{4} - \color{blue}{\log \left(1 - u\right)}\right) \]

   5. diff-log N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\log \left(\frac{\frac{3}{4}}{1 - u}\right)} \]

   6. lift-/.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \color{blue}{\left(\frac{\frac{3}{4}}{1 - u}\right)} \]

   7. lift-log.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\log \left(\frac{\frac{3}{4}}{1 - u}\right)} \]

   8. lift-log.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\log \left(\frac{\frac{3}{4}}{1 - u}\right)} \]

   9. lift-/.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \color{blue}{\left(\frac{\frac{3}{4}}{1 - u}\right)} \]

   10. diff-log N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\left(\log \frac{3}{4} - \log \left(1 - u\right)\right)} \]

   11. lift-log.f32 N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\color{blue}{\log \frac{3}{4}} - \log \left(1 - u\right)\right) \]

   12. lift-log.f32 N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\log \frac{3}{4} - \color{blue}{\log \left(1 - u\right)}\right) \]

   13. sub-flip N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\left(\log \frac{3}{4} + \left(\mathsf{neg}\left(\log \left(1 - u\right)\right)\right)\right)} \]

   14. distribute-lft-in N/A

       \[\leadsto \color{blue}{\left(3 \cdot s\right) \cdot \log \frac{3}{4} + \left(3 \cdot s\right) \cdot \left(\mathsf{neg}\left(\log \left(1 - u\right)\right)\right)} \]

   15. lift-*.f32 N/A

       \[\leadsto \color{blue}{\left(3 \cdot s\right)} \cdot \log \frac{3}{4} + \left(3 \cdot s\right) \cdot \left(\mathsf{neg}\left(\log \left(1 - u\right)\right)\right) \]

   16. associate-*r* N/A

       \[\leadsto \color{blue}{3 \cdot \left(s \cdot \log \frac{3}{4}\right)} + \left(3 \cdot s\right) \cdot \left(\mathsf{neg}\left(\log \left(1 - u\right)\right)\right) \]

   17. lift-*.f32 N/A

       \[\leadsto 3 \cdot \color{blue}{\left(s \cdot \log \frac{3}{4}\right)} + \left(3 \cdot s\right) \cdot \left(\mathsf{neg}\left(\log \left(1 - u\right)\right)\right) \]

   18. lift-*.f32 N/A

       \[\leadsto \color{blue}{3 \cdot \left(s \cdot \log \frac{3}{4}\right)} + \left(3 \cdot s\right) \cdot \left(\mathsf{neg}\left(\log \left(1 - u\right)\right)\right) \]

   19. sum-to-mult N/A

       \[\leadsto \color{blue}{\left(1 + \frac{\left(3 \cdot s\right) \cdot \left(\mathsf{neg}\left(\log \left(1 - u\right)\right)\right)}{3 \cdot \left(s \cdot \log \frac{3}{4}\right)}\right) \cdot \left(3 \cdot \left(s \cdot \log \frac{3}{4}\right)\right)} \]

5. Applied rewrites 96.2%

   \[\leadsto \color{blue}{\left(1 + \frac{\left(\left(-\log \left(1 - u\right)\right) \cdot s\right) \cdot 3}{\log 0.421875 \cdot s}\right) \cdot \left(\log 0.421875 \cdot s\right)} \]

6. Evaluated real constant 96.2%

   \[\leadsto \left(1 + \frac{\left(\left(-\log \left(1 - u\right)\right) \cdot s\right) \cdot 3}{\color{blue}{-0.8630462288856506} \cdot s}\right) \cdot \left(\log 0.421875 \cdot s\right) \]

7. Evaluated real constant 96.2%

   \[\leadsto \left(1 + \frac{\left(\left(-\log \left(1 - u\right)\right) \cdot s\right) \cdot 3}{-0.8630462288856506 \cdot s}\right) \cdot \left(\color{blue}{-0.8630462288856506} \cdot s\right) \]

8. Taylor expanded in u around 0

   \[\leadsto \color{blue}{\frac{-14479513}{16777216} \cdot s + u \cdot \left(3 \cdot s + u \cdot \left(\frac{3}{2} \cdot s + s \cdot u\right)\right)} \]
9. Step-by-step derivation

   1. lower-+.f32 N/A

      \[\leadsto \frac{-14479513}{16777216} \cdot s + \color{blue}{u \cdot \left(3 \cdot s + u \cdot \left(\frac{3}{2} \cdot s + s \cdot u\right)\right)} \]

   2. lower-*.f32 N/A

      \[\leadsto \frac{-14479513}{16777216} \cdot s + \color{blue}{u} \cdot \left(3 \cdot s + u \cdot \left(\frac{3}{2} \cdot s + s \cdot u\right)\right) \]

   3. lower-*.f32 N/A

      \[\leadsto \frac{-14479513}{16777216} \cdot s + u \cdot \color{blue}{\left(3 \cdot s + u \cdot \left(\frac{3}{2} \cdot s + s \cdot u\right)\right)} \]

   4. lower-+.f32 N/A

      \[\leadsto \frac{-14479513}{16777216} \cdot s + u \cdot \left(3 \cdot s + \color{blue}{u \cdot \left(\frac{3}{2} \cdot s + s \cdot u\right)}\right) \]

   5. lower-*.f32 N/A

      \[\leadsto \frac{-14479513}{16777216} \cdot s + u \cdot \left(3 \cdot s + \color{blue}{u} \cdot \left(\frac{3}{2} \cdot s + s \cdot u\right)\right) \]

   6. lower-*.f32 N/A

      \[\leadsto \frac{-14479513}{16777216} \cdot s + u \cdot \left(3 \cdot s + u \cdot \color{blue}{\left(\frac{3}{2} \cdot s + s \cdot u\right)}\right) \]

   7. lower-+.f32 N/A

      \[\leadsto \frac{-14479513}{16777216} \cdot s + u \cdot \left(3 \cdot s + u \cdot \left(\frac{3}{2} \cdot s + \color{blue}{s \cdot u}\right)\right) \]

   8. lower-*.f32 N/A

      \[\leadsto \frac{-14479513}{16777216} \cdot s + u \cdot \left(3 \cdot s + u \cdot \left(\frac{3}{2} \cdot s + \color{blue}{s} \cdot u\right)\right) \]

   9. lower-*.f32 36.7%

      \[\leadsto -0.8630462288856506 \cdot s + u \cdot \left(3 \cdot s + u \cdot \left(1.5 \cdot s + s \cdot \color{blue}{u}\right)\right) \]

10. Applied rewrites 36.7%

    \[\leadsto \color{blue}{-0.8630462288856506 \cdot s + u \cdot \left(3 \cdot s + u \cdot \left(1.5 \cdot s + s \cdot u\right)\right)} \]
11. Add Preprocessing

Alternative 10: 36.7% accurate, 4.0× speedup

Math

\[\left(1 + u \cdot \left(u \cdot \left(-1.158686483447337 \cdot u - 1.7380297251710055\right) - 3.476059450342011\right)\right) \cdot \left(-0.8630462288856506 \cdot s\right) \]

FPCore

(FPCore (s u)
  :precision binary32
  (*
 (+
  1.0
  (*
   u
   (-
    (* u (- (* -1.158686483447337 u) 1.7380297251710055))
    3.476059450342011)))
 (* -0.8630462288856506 s)))

C

float code(float s, float u) {
	return (1.0f + (u * ((u * ((-1.158686483447337f * u) - 1.7380297251710055f)) - 3.476059450342011f))) * (-0.8630462288856506f * s);
}

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(s, u)
use fmin_fmax_functions
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = (1.0e0 + (u * ((u * (((-1.158686483447337e0) * u) - 1.7380297251710055e0)) - 3.476059450342011e0))) * ((-0.8630462288856506e0) * s)
end function

Julia

function code(s, u)
	return Float32(Float32(Float32(1.0) + Float32(u * Float32(Float32(u * Float32(Float32(Float32(-1.158686483447337) * u) - Float32(1.7380297251710055))) - Float32(3.476059450342011)))) * Float32(Float32(-0.8630462288856506) * s))
end

MATLAB

function tmp = code(s, u)
	tmp = (single(1.0) + (u * ((u * ((single(-1.158686483447337) * u) - single(1.7380297251710055))) - single(3.476059450342011)))) * (single(-0.8630462288856506) * s);
end

Derivation

1. Initial program 96.0%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]
2. Step-by-step derivation

   1. lift-log.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)} \]

   2. lift-/.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \color{blue}{\left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)} \]

   3. lift--.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}}\right) \]

   4. lift-/.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \color{blue}{\frac{u - \frac{1}{4}}{\frac{3}{4}}}}\right) \]

   5. sub-to-fraction N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{1 \cdot \frac{3}{4} - \left(u - \frac{1}{4}\right)}{\frac{3}{4}}}}\right) \]

   6. div-flip-rev N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \color{blue}{\left(\frac{\frac{3}{4}}{1 \cdot \frac{3}{4} - \left(u - \frac{1}{4}\right)}\right)} \]

   7. log-div N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\left(\log \frac{3}{4} - \log \left(1 \cdot \frac{3}{4} - \left(u - \frac{1}{4}\right)\right)\right)} \]

   8. lower-unsound--.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\left(\log \frac{3}{4} - \log \left(1 \cdot \frac{3}{4} - \left(u - \frac{1}{4}\right)\right)\right)} \]

   9. lower-unsound-log.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\color{blue}{\log \frac{3}{4}} - \log \left(1 \cdot \frac{3}{4} - \left(u - \frac{1}{4}\right)\right)\right) \]

   10. lower-unsound-log.f32 N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\log \frac{3}{4} - \color{blue}{\log \left(1 \cdot \frac{3}{4} - \left(u - \frac{1}{4}\right)\right)}\right) \]

   11. metadata-eval N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\log \frac{3}{4} - \log \left(\color{blue}{\frac{3}{4}} - \left(u - \frac{1}{4}\right)\right)\right) \]

   12. lift--.f32 N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\log \frac{3}{4} - \log \left(\frac{3}{4} - \color{blue}{\left(u - \frac{1}{4}\right)}\right)\right) \]

   13. sub-negate-rev N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\log \frac{3}{4} - \log \left(\frac{3}{4} - \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{4} - u\right)\right)\right)}\right)\right) \]

   14. add-flip-rev N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\log \frac{3}{4} - \log \color{blue}{\left(\frac{3}{4} + \left(\frac{1}{4} - u\right)\right)}\right) \]

   15. associate-+r- N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\log \frac{3}{4} - \log \color{blue}{\left(\left(\frac{3}{4} + \frac{1}{4}\right) - u\right)}\right) \]

   16. metadata-eval N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\log \frac{3}{4} - \log \left(\color{blue}{1} - u\right)\right) \]

   17. lower--.f32 96.5%

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\log 0.75 - \log \color{blue}{\left(1 - u\right)}\right) \]

3. Applied rewrites 96.5%

   \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\left(\log 0.75 - \log \left(1 - u\right)\right)} \]
4. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \color{blue}{\left(3 \cdot s\right) \cdot \left(\log \frac{3}{4} - \log \left(1 - u\right)\right)} \]

   2. lift--.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\left(\log \frac{3}{4} - \log \left(1 - u\right)\right)} \]

   3. lift-log.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\color{blue}{\log \frac{3}{4}} - \log \left(1 - u\right)\right) \]

   4. lift-log.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\log \frac{3}{4} - \color{blue}{\log \left(1 - u\right)}\right) \]

   5. diff-log N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\log \left(\frac{\frac{3}{4}}{1 - u}\right)} \]

   6. lift-/.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \color{blue}{\left(\frac{\frac{3}{4}}{1 - u}\right)} \]

   7. lift-log.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\log \left(\frac{\frac{3}{4}}{1 - u}\right)} \]

   8. lift-log.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\log \left(\frac{\frac{3}{4}}{1 - u}\right)} \]

   9. lift-/.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \color{blue}{\left(\frac{\frac{3}{4}}{1 - u}\right)} \]

   10. diff-log N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\left(\log \frac{3}{4} - \log \left(1 - u\right)\right)} \]

   11. lift-log.f32 N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\color{blue}{\log \frac{3}{4}} - \log \left(1 - u\right)\right) \]

   12. lift-log.f32 N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\log \frac{3}{4} - \color{blue}{\log \left(1 - u\right)}\right) \]

   13. sub-flip N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\left(\log \frac{3}{4} + \left(\mathsf{neg}\left(\log \left(1 - u\right)\right)\right)\right)} \]

   14. distribute-lft-in N/A

       \[\leadsto \color{blue}{\left(3 \cdot s\right) \cdot \log \frac{3}{4} + \left(3 \cdot s\right) \cdot \left(\mathsf{neg}\left(\log \left(1 - u\right)\right)\right)} \]

   15. lift-*.f32 N/A

       \[\leadsto \color{blue}{\left(3 \cdot s\right)} \cdot \log \frac{3}{4} + \left(3 \cdot s\right) \cdot \left(\mathsf{neg}\left(\log \left(1 - u\right)\right)\right) \]

   16. associate-*r* N/A

       \[\leadsto \color{blue}{3 \cdot \left(s \cdot \log \frac{3}{4}\right)} + \left(3 \cdot s\right) \cdot \left(\mathsf{neg}\left(\log \left(1 - u\right)\right)\right) \]

   17. lift-*.f32 N/A

       \[\leadsto 3 \cdot \color{blue}{\left(s \cdot \log \frac{3}{4}\right)} + \left(3 \cdot s\right) \cdot \left(\mathsf{neg}\left(\log \left(1 - u\right)\right)\right) \]

   18. lift-*.f32 N/A

       \[\leadsto \color{blue}{3 \cdot \left(s \cdot \log \frac{3}{4}\right)} + \left(3 \cdot s\right) \cdot \left(\mathsf{neg}\left(\log \left(1 - u\right)\right)\right) \]

   19. sum-to-mult N/A

       \[\leadsto \color{blue}{\left(1 + \frac{\left(3 \cdot s\right) \cdot \left(\mathsf{neg}\left(\log \left(1 - u\right)\right)\right)}{3 \cdot \left(s \cdot \log \frac{3}{4}\right)}\right) \cdot \left(3 \cdot \left(s \cdot \log \frac{3}{4}\right)\right)} \]

5. Applied rewrites 96.2%

   \[\leadsto \color{blue}{\left(1 + \frac{\left(\left(-\log \left(1 - u\right)\right) \cdot s\right) \cdot 3}{\log 0.421875 \cdot s}\right) \cdot \left(\log 0.421875 \cdot s\right)} \]

6. Evaluated real constant 96.2%

   \[\leadsto \left(1 + \frac{\left(\left(-\log \left(1 - u\right)\right) \cdot s\right) \cdot 3}{\color{blue}{-0.8630462288856506} \cdot s}\right) \cdot \left(\log 0.421875 \cdot s\right) \]

7. Evaluated real constant 96.2%

   \[\leadsto \left(1 + \frac{\left(\left(-\log \left(1 - u\right)\right) \cdot s\right) \cdot 3}{-0.8630462288856506 \cdot s}\right) \cdot \left(\color{blue}{-0.8630462288856506} \cdot s\right) \]

8. Taylor expanded in u around 0

   \[\leadsto \left(1 + \color{blue}{u \cdot \left(u \cdot \left(\frac{-16777216}{14479513} \cdot u - \frac{25165824}{14479513}\right) - \frac{50331648}{14479513}\right)}\right) \cdot \left(-0.8630462288856506 \cdot s\right) \]
9. Step-by-step derivation

   1. lower-*.f32 N/A

      \[\leadsto \left(1 + u \cdot \color{blue}{\left(u \cdot \left(\frac{-16777216}{14479513} \cdot u - \frac{25165824}{14479513}\right) - \frac{50331648}{14479513}\right)}\right) \cdot \left(\frac{-14479513}{16777216} \cdot s\right) \]

   2. lower--.f32 N/A

      \[\leadsto \left(1 + u \cdot \left(u \cdot \left(\frac{-16777216}{14479513} \cdot u - \frac{25165824}{14479513}\right) - \color{blue}{\frac{50331648}{14479513}}\right)\right) \cdot \left(\frac{-14479513}{16777216} \cdot s\right) \]

   3. lower-*.f32 N/A

      \[\leadsto \left(1 + u \cdot \left(u \cdot \left(\frac{-16777216}{14479513} \cdot u - \frac{25165824}{14479513}\right) - \frac{50331648}{14479513}\right)\right) \cdot \left(\frac{-14479513}{16777216} \cdot s\right) \]

   4. lower--.f32 N/A

      \[\leadsto \left(1 + u \cdot \left(u \cdot \left(\frac{-16777216}{14479513} \cdot u - \frac{25165824}{14479513}\right) - \frac{50331648}{14479513}\right)\right) \cdot \left(\frac{-14479513}{16777216} \cdot s\right) \]

   5. lower-*.f32 36.7%

      \[\leadsto \left(1 + u \cdot \left(u \cdot \left(-1.158686483447337 \cdot u - 1.7380297251710055\right) - 3.476059450342011\right)\right) \cdot \left(-0.8630462288856506 \cdot s\right) \]

10. Applied rewrites 36.7%

    \[\leadsto \left(1 + \color{blue}{u \cdot \left(u \cdot \left(-1.158686483447337 \cdot u - 1.7380297251710055\right) - 3.476059450342011\right)}\right) \cdot \left(-0.8630462288856506 \cdot s\right) \]
11. Add Preprocessing

Alternative 11: 32.2% accurate, 4.3× speedup

Math

\[-0.8630462288856506 \cdot s + u \cdot \left(1.5 \cdot \left(s \cdot u\right) + 3 \cdot s\right) \]

FPCore

(FPCore (s u)
  :precision binary32
  (+ (* -0.8630462288856506 s) (* u (+ (* 1.5 (* s u)) (* 3.0 s)))))

C

float code(float s, float u) {
	return (-0.8630462288856506f * s) + (u * ((1.5f * (s * u)) + (3.0f * s)));
}

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(s, u)
use fmin_fmax_functions
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = ((-0.8630462288856506e0) * s) + (u * ((1.5e0 * (s * u)) + (3.0e0 * s)))
end function

Julia

function code(s, u)
	return Float32(Float32(Float32(-0.8630462288856506) * s) + Float32(u * Float32(Float32(Float32(1.5) * Float32(s * u)) + Float32(Float32(3.0) * s))))
end

MATLAB

function tmp = code(s, u)
	tmp = (single(-0.8630462288856506) * s) + (u * ((single(1.5) * (s * u)) + (single(3.0) * s)));
end

Derivation

1. Initial program 96.0%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]
2. Step-by-step derivation

   1. lift-log.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)} \]

   2. lift-/.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \color{blue}{\left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)} \]

   3. lift--.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}}\right) \]

   4. lift-/.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \color{blue}{\frac{u - \frac{1}{4}}{\frac{3}{4}}}}\right) \]

   5. sub-to-fraction N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{1 \cdot \frac{3}{4} - \left(u - \frac{1}{4}\right)}{\frac{3}{4}}}}\right) \]

   6. div-flip-rev N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \color{blue}{\left(\frac{\frac{3}{4}}{1 \cdot \frac{3}{4} - \left(u - \frac{1}{4}\right)}\right)} \]

   7. log-div N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\left(\log \frac{3}{4} - \log \left(1 \cdot \frac{3}{4} - \left(u - \frac{1}{4}\right)\right)\right)} \]

   8. lower-unsound--.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\left(\log \frac{3}{4} - \log \left(1 \cdot \frac{3}{4} - \left(u - \frac{1}{4}\right)\right)\right)} \]

   9. lower-unsound-log.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\color{blue}{\log \frac{3}{4}} - \log \left(1 \cdot \frac{3}{4} - \left(u - \frac{1}{4}\right)\right)\right) \]

   10. lower-unsound-log.f32 N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\log \frac{3}{4} - \color{blue}{\log \left(1 \cdot \frac{3}{4} - \left(u - \frac{1}{4}\right)\right)}\right) \]

   11. metadata-eval N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\log \frac{3}{4} - \log \left(\color{blue}{\frac{3}{4}} - \left(u - \frac{1}{4}\right)\right)\right) \]

   12. lift--.f32 N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\log \frac{3}{4} - \log \left(\frac{3}{4} - \color{blue}{\left(u - \frac{1}{4}\right)}\right)\right) \]

   13. sub-negate-rev N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\log \frac{3}{4} - \log \left(\frac{3}{4} - \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{4} - u\right)\right)\right)}\right)\right) \]

   14. add-flip-rev N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\log \frac{3}{4} - \log \color{blue}{\left(\frac{3}{4} + \left(\frac{1}{4} - u\right)\right)}\right) \]

   15. associate-+r- N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\log \frac{3}{4} - \log \color{blue}{\left(\left(\frac{3}{4} + \frac{1}{4}\right) - u\right)}\right) \]

   16. metadata-eval N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\log \frac{3}{4} - \log \left(\color{blue}{1} - u\right)\right) \]

   17. lower--.f32 96.5%

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\log 0.75 - \log \color{blue}{\left(1 - u\right)}\right) \]

3. Applied rewrites 96.5%

   \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\left(\log 0.75 - \log \left(1 - u\right)\right)} \]
4. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \color{blue}{\left(3 \cdot s\right) \cdot \left(\log \frac{3}{4} - \log \left(1 - u\right)\right)} \]

   2. lift--.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\left(\log \frac{3}{4} - \log \left(1 - u\right)\right)} \]

   3. lift-log.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\color{blue}{\log \frac{3}{4}} - \log \left(1 - u\right)\right) \]

   4. lift-log.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\log \frac{3}{4} - \color{blue}{\log \left(1 - u\right)}\right) \]

   5. diff-log N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\log \left(\frac{\frac{3}{4}}{1 - u}\right)} \]

   6. lift-/.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \color{blue}{\left(\frac{\frac{3}{4}}{1 - u}\right)} \]

   7. lift-log.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\log \left(\frac{\frac{3}{4}}{1 - u}\right)} \]

   8. lift-log.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\log \left(\frac{\frac{3}{4}}{1 - u}\right)} \]

   9. lift-/.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \color{blue}{\left(\frac{\frac{3}{4}}{1 - u}\right)} \]

   10. diff-log N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\left(\log \frac{3}{4} - \log \left(1 - u\right)\right)} \]

   11. lift-log.f32 N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\color{blue}{\log \frac{3}{4}} - \log \left(1 - u\right)\right) \]

   12. lift-log.f32 N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\log \frac{3}{4} - \color{blue}{\log \left(1 - u\right)}\right) \]

   13. sub-flip N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\left(\log \frac{3}{4} + \left(\mathsf{neg}\left(\log \left(1 - u\right)\right)\right)\right)} \]

   14. distribute-lft-in N/A

       \[\leadsto \color{blue}{\left(3 \cdot s\right) \cdot \log \frac{3}{4} + \left(3 \cdot s\right) \cdot \left(\mathsf{neg}\left(\log \left(1 - u\right)\right)\right)} \]

   15. lift-*.f32 N/A

       \[\leadsto \color{blue}{\left(3 \cdot s\right)} \cdot \log \frac{3}{4} + \left(3 \cdot s\right) \cdot \left(\mathsf{neg}\left(\log \left(1 - u\right)\right)\right) \]

   16. associate-*r* N/A

       \[\leadsto \color{blue}{3 \cdot \left(s \cdot \log \frac{3}{4}\right)} + \left(3 \cdot s\right) \cdot \left(\mathsf{neg}\left(\log \left(1 - u\right)\right)\right) \]

   17. lift-*.f32 N/A

       \[\leadsto 3 \cdot \color{blue}{\left(s \cdot \log \frac{3}{4}\right)} + \left(3 \cdot s\right) \cdot \left(\mathsf{neg}\left(\log \left(1 - u\right)\right)\right) \]

   18. lift-*.f32 N/A

       \[\leadsto \color{blue}{3 \cdot \left(s \cdot \log \frac{3}{4}\right)} + \left(3 \cdot s\right) \cdot \left(\mathsf{neg}\left(\log \left(1 - u\right)\right)\right) \]

   19. sum-to-mult N/A

       \[\leadsto \color{blue}{\left(1 + \frac{\left(3 \cdot s\right) \cdot \left(\mathsf{neg}\left(\log \left(1 - u\right)\right)\right)}{3 \cdot \left(s \cdot \log \frac{3}{4}\right)}\right) \cdot \left(3 \cdot \left(s \cdot \log \frac{3}{4}\right)\right)} \]

5. Applied rewrites 96.2%

   \[\leadsto \color{blue}{\left(1 + \frac{\left(\left(-\log \left(1 - u\right)\right) \cdot s\right) \cdot 3}{\log 0.421875 \cdot s}\right) \cdot \left(\log 0.421875 \cdot s\right)} \]

6. Evaluated real constant 96.2%

   \[\leadsto \left(1 + \frac{\left(\left(-\log \left(1 - u\right)\right) \cdot s\right) \cdot 3}{\color{blue}{-0.8630462288856506} \cdot s}\right) \cdot \left(\log 0.421875 \cdot s\right) \]

7. Evaluated real constant 96.2%

   \[\leadsto \left(1 + \frac{\left(\left(-\log \left(1 - u\right)\right) \cdot s\right) \cdot 3}{-0.8630462288856506 \cdot s}\right) \cdot \left(\color{blue}{-0.8630462288856506} \cdot s\right) \]

8. Taylor expanded in u around 0

   \[\leadsto \color{blue}{\frac{-14479513}{16777216} \cdot s + u \cdot \left(\frac{3}{2} \cdot \left(s \cdot u\right) + 3 \cdot s\right)} \]
9. Step-by-step derivation

   1. lower-+.f32 N/A

      \[\leadsto \frac{-14479513}{16777216} \cdot s + \color{blue}{u \cdot \left(\frac{3}{2} \cdot \left(s \cdot u\right) + 3 \cdot s\right)} \]

   2. lower-*.f32 N/A

      \[\leadsto \frac{-14479513}{16777216} \cdot s + \color{blue}{u} \cdot \left(\frac{3}{2} \cdot \left(s \cdot u\right) + 3 \cdot s\right) \]

   3. lower-*.f32 N/A

      \[\leadsto \frac{-14479513}{16777216} \cdot s + u \cdot \color{blue}{\left(\frac{3}{2} \cdot \left(s \cdot u\right) + 3 \cdot s\right)} \]

   4. lower-+.f32 N/A

      \[\leadsto \frac{-14479513}{16777216} \cdot s + u \cdot \left(\frac{3}{2} \cdot \left(s \cdot u\right) + \color{blue}{3 \cdot s}\right) \]

   5. lower-*.f32 N/A

      \[\leadsto \frac{-14479513}{16777216} \cdot s + u \cdot \left(\frac{3}{2} \cdot \left(s \cdot u\right) + \color{blue}{3} \cdot s\right) \]

   6. lower-*.f32 N/A

      \[\leadsto \frac{-14479513}{16777216} \cdot s + u \cdot \left(\frac{3}{2} \cdot \left(s \cdot u\right) + 3 \cdot s\right) \]

   7. lower-*.f32 32.2%

      \[\leadsto -0.8630462288856506 \cdot s + u \cdot \left(1.5 \cdot \left(s \cdot u\right) + 3 \cdot \color{blue}{s}\right) \]

10. Applied rewrites 32.2%

    \[\leadsto \color{blue}{-0.8630462288856506 \cdot s + u \cdot \left(1.5 \cdot \left(s \cdot u\right) + 3 \cdot s\right)} \]
11. Add Preprocessing

Alternative 12: 32.2% accurate, 5.1× speedup

Math

\[\left(1 + u \cdot \left(-1.7380297251710055 \cdot u - 3.476059450342011\right)\right) \cdot \left(-0.8630462288856506 \cdot s\right) \]

FPCore

(FPCore (s u)
  :precision binary32
  (*
 (+ 1.0 (* u (- (* -1.7380297251710055 u) 3.476059450342011)))
 (* -0.8630462288856506 s)))

C

float code(float s, float u) {
	return (1.0f + (u * ((-1.7380297251710055f * u) - 3.476059450342011f))) * (-0.8630462288856506f * s);
}

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(s, u)
use fmin_fmax_functions
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = (1.0e0 + (u * (((-1.7380297251710055e0) * u) - 3.476059450342011e0))) * ((-0.8630462288856506e0) * s)
end function

Julia

function code(s, u)
	return Float32(Float32(Float32(1.0) + Float32(u * Float32(Float32(Float32(-1.7380297251710055) * u) - Float32(3.476059450342011)))) * Float32(Float32(-0.8630462288856506) * s))
end

MATLAB

function tmp = code(s, u)
	tmp = (single(1.0) + (u * ((single(-1.7380297251710055) * u) - single(3.476059450342011)))) * (single(-0.8630462288856506) * s);
end

Derivation

1. Initial program 96.0%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]
2. Step-by-step derivation

   1. lift-log.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)} \]

   2. lift-/.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \color{blue}{\left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)} \]

   3. lift--.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}}\right) \]

   4. lift-/.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \color{blue}{\frac{u - \frac{1}{4}}{\frac{3}{4}}}}\right) \]

   5. sub-to-fraction N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{1 \cdot \frac{3}{4} - \left(u - \frac{1}{4}\right)}{\frac{3}{4}}}}\right) \]

   6. div-flip-rev N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \color{blue}{\left(\frac{\frac{3}{4}}{1 \cdot \frac{3}{4} - \left(u - \frac{1}{4}\right)}\right)} \]

   7. log-div N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\left(\log \frac{3}{4} - \log \left(1 \cdot \frac{3}{4} - \left(u - \frac{1}{4}\right)\right)\right)} \]

   8. lower-unsound--.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\left(\log \frac{3}{4} - \log \left(1 \cdot \frac{3}{4} - \left(u - \frac{1}{4}\right)\right)\right)} \]

   9. lower-unsound-log.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\color{blue}{\log \frac{3}{4}} - \log \left(1 \cdot \frac{3}{4} - \left(u - \frac{1}{4}\right)\right)\right) \]

   10. lower-unsound-log.f32 N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\log \frac{3}{4} - \color{blue}{\log \left(1 \cdot \frac{3}{4} - \left(u - \frac{1}{4}\right)\right)}\right) \]

   11. metadata-eval N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\log \frac{3}{4} - \log \left(\color{blue}{\frac{3}{4}} - \left(u - \frac{1}{4}\right)\right)\right) \]

   12. lift--.f32 N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\log \frac{3}{4} - \log \left(\frac{3}{4} - \color{blue}{\left(u - \frac{1}{4}\right)}\right)\right) \]

   13. sub-negate-rev N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\log \frac{3}{4} - \log \left(\frac{3}{4} - \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{4} - u\right)\right)\right)}\right)\right) \]

   14. add-flip-rev N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\log \frac{3}{4} - \log \color{blue}{\left(\frac{3}{4} + \left(\frac{1}{4} - u\right)\right)}\right) \]

   15. associate-+r- N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\log \frac{3}{4} - \log \color{blue}{\left(\left(\frac{3}{4} + \frac{1}{4}\right) - u\right)}\right) \]

   16. metadata-eval N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\log \frac{3}{4} - \log \left(\color{blue}{1} - u\right)\right) \]

   17. lower--.f32 96.5%

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\log 0.75 - \log \color{blue}{\left(1 - u\right)}\right) \]

3. Applied rewrites 96.5%

   \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\left(\log 0.75 - \log \left(1 - u\right)\right)} \]
4. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \color{blue}{\left(3 \cdot s\right) \cdot \left(\log \frac{3}{4} - \log \left(1 - u\right)\right)} \]

   2. lift--.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\left(\log \frac{3}{4} - \log \left(1 - u\right)\right)} \]

   3. lift-log.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\color{blue}{\log \frac{3}{4}} - \log \left(1 - u\right)\right) \]

   4. lift-log.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\log \frac{3}{4} - \color{blue}{\log \left(1 - u\right)}\right) \]

   5. diff-log N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\log \left(\frac{\frac{3}{4}}{1 - u}\right)} \]

   6. lift-/.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \color{blue}{\left(\frac{\frac{3}{4}}{1 - u}\right)} \]

   7. lift-log.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\log \left(\frac{\frac{3}{4}}{1 - u}\right)} \]

   8. lift-log.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\log \left(\frac{\frac{3}{4}}{1 - u}\right)} \]

   9. lift-/.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \color{blue}{\left(\frac{\frac{3}{4}}{1 - u}\right)} \]

   10. diff-log N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\left(\log \frac{3}{4} - \log \left(1 - u\right)\right)} \]

   11. lift-log.f32 N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\color{blue}{\log \frac{3}{4}} - \log \left(1 - u\right)\right) \]

   12. lift-log.f32 N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\log \frac{3}{4} - \color{blue}{\log \left(1 - u\right)}\right) \]

   13. sub-flip N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\left(\log \frac{3}{4} + \left(\mathsf{neg}\left(\log \left(1 - u\right)\right)\right)\right)} \]

   14. distribute-lft-in N/A

       \[\leadsto \color{blue}{\left(3 \cdot s\right) \cdot \log \frac{3}{4} + \left(3 \cdot s\right) \cdot \left(\mathsf{neg}\left(\log \left(1 - u\right)\right)\right)} \]

   15. lift-*.f32 N/A

       \[\leadsto \color{blue}{\left(3 \cdot s\right)} \cdot \log \frac{3}{4} + \left(3 \cdot s\right) \cdot \left(\mathsf{neg}\left(\log \left(1 - u\right)\right)\right) \]

   16. associate-*r* N/A

       \[\leadsto \color{blue}{3 \cdot \left(s \cdot \log \frac{3}{4}\right)} + \left(3 \cdot s\right) \cdot \left(\mathsf{neg}\left(\log \left(1 - u\right)\right)\right) \]

   17. lift-*.f32 N/A

       \[\leadsto 3 \cdot \color{blue}{\left(s \cdot \log \frac{3}{4}\right)} + \left(3 \cdot s\right) \cdot \left(\mathsf{neg}\left(\log \left(1 - u\right)\right)\right) \]

   18. lift-*.f32 N/A

       \[\leadsto \color{blue}{3 \cdot \left(s \cdot \log \frac{3}{4}\right)} + \left(3 \cdot s\right) \cdot \left(\mathsf{neg}\left(\log \left(1 - u\right)\right)\right) \]

   19. sum-to-mult N/A

       \[\leadsto \color{blue}{\left(1 + \frac{\left(3 \cdot s\right) \cdot \left(\mathsf{neg}\left(\log \left(1 - u\right)\right)\right)}{3 \cdot \left(s \cdot \log \frac{3}{4}\right)}\right) \cdot \left(3 \cdot \left(s \cdot \log \frac{3}{4}\right)\right)} \]

5. Applied rewrites 96.2%

   \[\leadsto \color{blue}{\left(1 + \frac{\left(\left(-\log \left(1 - u\right)\right) \cdot s\right) \cdot 3}{\log 0.421875 \cdot s}\right) \cdot \left(\log 0.421875 \cdot s\right)} \]

6. Evaluated real constant 96.2%

   \[\leadsto \left(1 + \frac{\left(\left(-\log \left(1 - u\right)\right) \cdot s\right) \cdot 3}{\color{blue}{-0.8630462288856506} \cdot s}\right) \cdot \left(\log 0.421875 \cdot s\right) \]

7. Evaluated real constant 96.2%

   \[\leadsto \left(1 + \frac{\left(\left(-\log \left(1 - u\right)\right) \cdot s\right) \cdot 3}{-0.8630462288856506 \cdot s}\right) \cdot \left(\color{blue}{-0.8630462288856506} \cdot s\right) \]

8. Taylor expanded in u around 0

   \[\leadsto \left(1 + \color{blue}{u \cdot \left(\frac{-25165824}{14479513} \cdot u - \frac{50331648}{14479513}\right)}\right) \cdot \left(-0.8630462288856506 \cdot s\right) \]
9. Step-by-step derivation

   1. lower-*.f32 N/A

      \[\leadsto \left(1 + u \cdot \color{blue}{\left(\frac{-25165824}{14479513} \cdot u - \frac{50331648}{14479513}\right)}\right) \cdot \left(\frac{-14479513}{16777216} \cdot s\right) \]

   2. lower--.f32 N/A

      \[\leadsto \left(1 + u \cdot \left(\frac{-25165824}{14479513} \cdot u - \color{blue}{\frac{50331648}{14479513}}\right)\right) \cdot \left(\frac{-14479513}{16777216} \cdot s\right) \]

   3. lower-*.f32 32.2%

      \[\leadsto \left(1 + u \cdot \left(-1.7380297251710055 \cdot u - 3.476059450342011\right)\right) \cdot \left(-0.8630462288856506 \cdot s\right) \]

10. Applied rewrites 32.2%

    \[\leadsto \left(1 + \color{blue}{u \cdot \left(-1.7380297251710055 \cdot u - 3.476059450342011\right)}\right) \cdot \left(-0.8630462288856506 \cdot s\right) \]
11. Add Preprocessing

Alternative 13: 25.8% accurate, 7.3× speedup

Math

\[-0.8630462288856506 \cdot s + 3 \cdot \left(s \cdot u\right) \]

FPCore

(FPCore (s u)
  :precision binary32
  (+ (* -0.8630462288856506 s) (* 3.0 (* s u))))

C

float code(float s, float u) {
	return (-0.8630462288856506f * s) + (3.0f * (s * u));
}

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(s, u)
use fmin_fmax_functions
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = ((-0.8630462288856506e0) * s) + (3.0e0 * (s * u))
end function

Julia

function code(s, u)
	return Float32(Float32(Float32(-0.8630462288856506) * s) + Float32(Float32(3.0) * Float32(s * u)))
end

MATLAB

function tmp = code(s, u)
	tmp = (single(-0.8630462288856506) * s) + (single(3.0) * (s * u));
end

Derivation

1. Initial program 96.0%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]
2. Step-by-step derivation

   1. lift-log.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\log \left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)} \]

   2. lift-/.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \color{blue}{\left(\frac{1}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}\right)} \]

   3. lift--.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{1 - \frac{u - \frac{1}{4}}{\frac{3}{4}}}}\right) \]

   4. lift-/.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \color{blue}{\frac{u - \frac{1}{4}}{\frac{3}{4}}}}\right) \]

   5. sub-to-fraction N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \left(\frac{1}{\color{blue}{\frac{1 \cdot \frac{3}{4} - \left(u - \frac{1}{4}\right)}{\frac{3}{4}}}}\right) \]

   6. div-flip-rev N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \color{blue}{\left(\frac{\frac{3}{4}}{1 \cdot \frac{3}{4} - \left(u - \frac{1}{4}\right)}\right)} \]

   7. log-div N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\left(\log \frac{3}{4} - \log \left(1 \cdot \frac{3}{4} - \left(u - \frac{1}{4}\right)\right)\right)} \]

   8. lower-unsound--.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\left(\log \frac{3}{4} - \log \left(1 \cdot \frac{3}{4} - \left(u - \frac{1}{4}\right)\right)\right)} \]

   9. lower-unsound-log.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\color{blue}{\log \frac{3}{4}} - \log \left(1 \cdot \frac{3}{4} - \left(u - \frac{1}{4}\right)\right)\right) \]

   10. lower-unsound-log.f32 N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\log \frac{3}{4} - \color{blue}{\log \left(1 \cdot \frac{3}{4} - \left(u - \frac{1}{4}\right)\right)}\right) \]

   11. metadata-eval N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\log \frac{3}{4} - \log \left(\color{blue}{\frac{3}{4}} - \left(u - \frac{1}{4}\right)\right)\right) \]

   12. lift--.f32 N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\log \frac{3}{4} - \log \left(\frac{3}{4} - \color{blue}{\left(u - \frac{1}{4}\right)}\right)\right) \]

   13. sub-negate-rev N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\log \frac{3}{4} - \log \left(\frac{3}{4} - \color{blue}{\left(\mathsf{neg}\left(\left(\frac{1}{4} - u\right)\right)\right)}\right)\right) \]

   14. add-flip-rev N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\log \frac{3}{4} - \log \color{blue}{\left(\frac{3}{4} + \left(\frac{1}{4} - u\right)\right)}\right) \]

   15. associate-+r- N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\log \frac{3}{4} - \log \color{blue}{\left(\left(\frac{3}{4} + \frac{1}{4}\right) - u\right)}\right) \]

   16. metadata-eval N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\log \frac{3}{4} - \log \left(\color{blue}{1} - u\right)\right) \]

   17. lower--.f32 96.5%

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\log 0.75 - \log \color{blue}{\left(1 - u\right)}\right) \]

3. Applied rewrites 96.5%

   \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\left(\log 0.75 - \log \left(1 - u\right)\right)} \]
4. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \color{blue}{\left(3 \cdot s\right) \cdot \left(\log \frac{3}{4} - \log \left(1 - u\right)\right)} \]

   2. lift--.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\left(\log \frac{3}{4} - \log \left(1 - u\right)\right)} \]

   3. lift-log.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\color{blue}{\log \frac{3}{4}} - \log \left(1 - u\right)\right) \]

   4. lift-log.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \left(\log \frac{3}{4} - \color{blue}{\log \left(1 - u\right)}\right) \]

   5. diff-log N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\log \left(\frac{\frac{3}{4}}{1 - u}\right)} \]

   6. lift-/.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \color{blue}{\left(\frac{\frac{3}{4}}{1 - u}\right)} \]

   7. lift-log.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\log \left(\frac{\frac{3}{4}}{1 - u}\right)} \]

   8. lift-log.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\log \left(\frac{\frac{3}{4}}{1 - u}\right)} \]

   9. lift-/.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \log \color{blue}{\left(\frac{\frac{3}{4}}{1 - u}\right)} \]

   10. diff-log N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\left(\log \frac{3}{4} - \log \left(1 - u\right)\right)} \]

   11. lift-log.f32 N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\color{blue}{\log \frac{3}{4}} - \log \left(1 - u\right)\right) \]

   12. lift-log.f32 N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \left(\log \frac{3}{4} - \color{blue}{\log \left(1 - u\right)}\right) \]

   13. sub-flip N/A

       \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\left(\log \frac{3}{4} + \left(\mathsf{neg}\left(\log \left(1 - u\right)\right)\right)\right)} \]

   14. distribute-lft-in N/A

       \[\leadsto \color{blue}{\left(3 \cdot s\right) \cdot \log \frac{3}{4} + \left(3 \cdot s\right) \cdot \left(\mathsf{neg}\left(\log \left(1 - u\right)\right)\right)} \]

   15. lift-*.f32 N/A

       \[\leadsto \color{blue}{\left(3 \cdot s\right)} \cdot \log \frac{3}{4} + \left(3 \cdot s\right) \cdot \left(\mathsf{neg}\left(\log \left(1 - u\right)\right)\right) \]

   16. associate-*r* N/A

       \[\leadsto \color{blue}{3 \cdot \left(s \cdot \log \frac{3}{4}\right)} + \left(3 \cdot s\right) \cdot \left(\mathsf{neg}\left(\log \left(1 - u\right)\right)\right) \]

   17. lift-*.f32 N/A

       \[\leadsto 3 \cdot \color{blue}{\left(s \cdot \log \frac{3}{4}\right)} + \left(3 \cdot s\right) \cdot \left(\mathsf{neg}\left(\log \left(1 - u\right)\right)\right) \]

   18. lift-*.f32 N/A

       \[\leadsto \color{blue}{3 \cdot \left(s \cdot \log \frac{3}{4}\right)} + \left(3 \cdot s\right) \cdot \left(\mathsf{neg}\left(\log \left(1 - u\right)\right)\right) \]

   19. sum-to-mult N/A

       \[\leadsto \color{blue}{\left(1 + \frac{\left(3 \cdot s\right) \cdot \left(\mathsf{neg}\left(\log \left(1 - u\right)\right)\right)}{3 \cdot \left(s \cdot \log \frac{3}{4}\right)}\right) \cdot \left(3 \cdot \left(s \cdot \log \frac{3}{4}\right)\right)} \]

5. Applied rewrites 96.2%

   \[\leadsto \color{blue}{\left(1 + \frac{\left(\left(-\log \left(1 - u\right)\right) \cdot s\right) \cdot 3}{\log 0.421875 \cdot s}\right) \cdot \left(\log 0.421875 \cdot s\right)} \]

6. Evaluated real constant 96.2%

   \[\leadsto \left(1 + \frac{\left(\left(-\log \left(1 - u\right)\right) \cdot s\right) \cdot 3}{\color{blue}{-0.8630462288856506} \cdot s}\right) \cdot \left(\log 0.421875 \cdot s\right) \]

7. Evaluated real constant 96.2%

   \[\leadsto \left(1 + \frac{\left(\left(-\log \left(1 - u\right)\right) \cdot s\right) \cdot 3}{-0.8630462288856506 \cdot s}\right) \cdot \left(\color{blue}{-0.8630462288856506} \cdot s\right) \]

8. Taylor expanded in u around 0

   \[\leadsto \color{blue}{\frac{-14479513}{16777216} \cdot s + 3 \cdot \left(s \cdot u\right)} \]
9. Step-by-step derivation

   1. lower-+.f32 N/A

      \[\leadsto \frac{-14479513}{16777216} \cdot s + \color{blue}{3 \cdot \left(s \cdot u\right)} \]

   2. lower-*.f32 N/A

      \[\leadsto \frac{-14479513}{16777216} \cdot s + \color{blue}{3} \cdot \left(s \cdot u\right) \]

   3. lower-*.f32 N/A

      \[\leadsto \frac{-14479513}{16777216} \cdot s + 3 \cdot \color{blue}{\left(s \cdot u\right)} \]

   4. lower-*.f32 25.8%

      \[\leadsto -0.8630462288856506 \cdot s + 3 \cdot \left(s \cdot \color{blue}{u}\right) \]

10. Applied rewrites 25.8%

    \[\leadsto \color{blue}{-0.8630462288856506 \cdot s + 3 \cdot \left(s \cdot u\right)} \]
11. Add Preprocessing

Alternative 14: 7.4% accurate, 12.6× speedup

Math

\[\left(3 \cdot s\right) \cdot -0.28768208622932434 \]

FPCore

(FPCore (s u)
  :precision binary32
  (* (* 3.0 s) -0.28768208622932434))

C

float code(float s, float u) {
	return (3.0f * s) * -0.28768208622932434f;
}

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(s, u)
use fmin_fmax_functions
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = (3.0e0 * s) * (-0.28768208622932434e0)
end function

Julia

function code(s, u)
	return Float32(Float32(Float32(3.0) * s) * Float32(-0.28768208622932434))
end

MATLAB

function tmp = code(s, u)
	tmp = (single(3.0) * s) * single(-0.28768208622932434);
end

Derivation

1. Initial program 96.0%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]

2. Taylor expanded in u around 0

   \[\leadsto \color{blue}{3 \cdot \left(s \cdot \log \frac{3}{4}\right)} \]
3. Step-by-step derivation

   1. lower-*.f32 N/A

      \[\leadsto 3 \cdot \color{blue}{\left(s \cdot \log \frac{3}{4}\right)} \]

   2. lower-*.f32 N/A

      \[\leadsto 3 \cdot \left(s \cdot \color{blue}{\log \frac{3}{4}}\right) \]

   3. lower-log.f32 7.4%

      \[\leadsto 3 \cdot \left(s \cdot \log 0.75\right) \]

4. Applied rewrites 7.4%

   \[\leadsto \color{blue}{3 \cdot \left(s \cdot \log 0.75\right)} \]

5. Evaluated real constant 7.4%

   \[\leadsto 3 \cdot \left(s \cdot -0.28768208622932434\right) \]
6. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto 3 \cdot \color{blue}{\left(s \cdot \frac{-9653009}{33554432}\right)} \]

   2. lift-*.f32 N/A

      \[\leadsto 3 \cdot \left(s \cdot \color{blue}{\frac{-9653009}{33554432}}\right) \]

   3. associate-*r* N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{\frac{-9653009}{33554432}} \]

   4. lift-*.f32 N/A

      \[\leadsto \left(3 \cdot s\right) \cdot \frac{-9653009}{33554432} \]

   5. lower-*.f32 7.4%

      \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{-0.28768208622932434} \]

7. Applied rewrites 7.4%

   \[\leadsto \left(3 \cdot s\right) \cdot \color{blue}{-0.28768208622932434} \]
8. Add Preprocessing

Alternative 15: 7.4% accurate, 12.6× speedup

Math

\[3 \cdot \left(s \cdot -0.28768208622932434\right) \]

FPCore

(FPCore (s u)
  :precision binary32
  (* 3.0 (* s -0.28768208622932434)))

C

float code(float s, float u) {
	return 3.0f * (s * -0.28768208622932434f);
}

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(s, u)
use fmin_fmax_functions
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = 3.0e0 * (s * (-0.28768208622932434e0))
end function

Julia

function code(s, u)
	return Float32(Float32(3.0) * Float32(s * Float32(-0.28768208622932434)))
end

MATLAB

function tmp = code(s, u)
	tmp = single(3.0) * (s * single(-0.28768208622932434));
end

Derivation

1. Initial program 96.0%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]

2. Taylor expanded in u around 0

   \[\leadsto \color{blue}{3 \cdot \left(s \cdot \log \frac{3}{4}\right)} \]
3. Step-by-step derivation

   1. lower-*.f32 N/A

      \[\leadsto 3 \cdot \color{blue}{\left(s \cdot \log \frac{3}{4}\right)} \]

   2. lower-*.f32 N/A

      \[\leadsto 3 \cdot \left(s \cdot \color{blue}{\log \frac{3}{4}}\right) \]

   3. lower-log.f32 7.4%

      \[\leadsto 3 \cdot \left(s \cdot \log 0.75\right) \]

4. Applied rewrites 7.4%

   \[\leadsto \color{blue}{3 \cdot \left(s \cdot \log 0.75\right)} \]

5. Evaluated real constant 7.4%

   \[\leadsto 3 \cdot \left(s \cdot -0.28768208622932434\right) \]
6. Add Preprocessing

Alternative 16: 7.4% accurate, 23.2× speedup

Math

\[s \cdot -0.863046258687973 \]

FPCore

(FPCore (s u)
  :precision binary32
  (* s -0.863046258687973))

C

float code(float s, float u) {
	return s * -0.863046258687973f;
}

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(s, u)
use fmin_fmax_functions
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = s * (-0.863046258687973e0)
end function

Julia

function code(s, u)
	return Float32(s * Float32(-0.863046258687973))
end

MATLAB

function tmp = code(s, u)
	tmp = s * single(-0.863046258687973);
end

Derivation

1. Initial program 96.0%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]

2. Taylor expanded in u around 0

   \[\leadsto \color{blue}{3 \cdot \left(s \cdot \log \frac{3}{4}\right)} \]
3. Step-by-step derivation

   1. lower-*.f32 N/A

      \[\leadsto 3 \cdot \color{blue}{\left(s \cdot \log \frac{3}{4}\right)} \]

   2. lower-*.f32 N/A

      \[\leadsto 3 \cdot \left(s \cdot \color{blue}{\log \frac{3}{4}}\right) \]

   3. lower-log.f32 7.4%

      \[\leadsto 3 \cdot \left(s \cdot \log 0.75\right) \]

4. Applied rewrites 7.4%

   \[\leadsto \color{blue}{3 \cdot \left(s \cdot \log 0.75\right)} \]

5. Evaluated real constant 7.4%

   \[\leadsto 3 \cdot \left(s \cdot -0.28768208622932434\right) \]
6. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto 3 \cdot \color{blue}{\left(s \cdot \frac{-9653009}{33554432}\right)} \]

   2. *-commutative N/A

      \[\leadsto \left(s \cdot \frac{-9653009}{33554432}\right) \cdot \color{blue}{3} \]

   3. lift-*.f32 N/A

      \[\leadsto \left(s \cdot \frac{-9653009}{33554432}\right) \cdot 3 \]

   4. associate-*l* N/A

      \[\leadsto s \cdot \color{blue}{\left(\frac{-9653009}{33554432} \cdot 3\right)} \]

   5. lower-*.f32 N/A

      \[\leadsto s \cdot \color{blue}{\left(\frac{-9653009}{33554432} \cdot 3\right)} \]

   6. metadata-eval 7.4%

      \[\leadsto s \cdot -0.863046258687973 \]

7. Applied rewrites 7.4%

   \[\leadsto s \cdot \color{blue}{-0.863046258687973} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025258 
(FPCore (s u)
  :name "Disney BSSRDF, sample scattering profile, upper"
  :precision binary32
  :pre (and (and (<= 0.0 s) (<= s 256.0)) (and (<= 0.25 u) (<= u 1.0)))
  (* (* 3.0 s) (log (/ 1.0 (- 1.0 (/ (- u 0.25) 0.75))))))