Disney BSSRDF, sample scattering profile, upper

Percentage Accurate: 95.9% → 98.3%

Time: 4.5s

Alternatives: 8

Speedup: 1.3×

Specification

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

Math

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

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: 95.9% accurate, 1.0× speedup

Math

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

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: 98.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(s \cdot 3\right) \cdot \left(-\mathsf{log1p}\left(\frac{\mathsf{fma}\left(u, -2, 0.5\right)}{1.5}\right)\right) \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (* (* s 3.0) (- (log1p (/ (fma u -2.0 0.5) 1.5)))))

C

float code(float s, float u) {
	return (s * 3.0f) * -log1pf((fmaf(u, -2.0f, 0.5f) / 1.5f));
}

Julia

function code(s, u)
	return Float32(Float32(s * Float32(3.0)) * Float32(-log1p(Float32(fma(u, Float32(-2.0), Float32(0.5)) / Float32(1.5)))))
end

Derivation

1. Initial program 95.9%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 95.9

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

   4. lift-log.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. lift--.f32 N/A

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

   7. lift--.f32 N/A

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

   8. lift-/.f32 N/A

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

   9. log-rec N/A

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

   10. lower-neg.f32 N/A

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

   11. lower-log.f32 N/A

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

   12. lower--.f32 N/A

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

   13. mult-flip N/A

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

   14. metadata-eval N/A

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

   15. *-commutative N/A

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

   16. lower-*.f32 N/A

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

   17. lift--.f32 96.5

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

3. Applied rewrites 96.5%

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

   1. lift-log.f32 N/A

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

   2. lift--.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift--.f32 N/A

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

   5. *-commutative N/A

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

   6. metadata-eval N/A

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

   7. mult-flip N/A

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

   8. metadata-eval N/A

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

   9. frac-sub N/A

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

   10. metadata-eval N/A

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

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

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

   12. metadata-eval N/A

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

   13. div-add N/A

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

   14. metadata-eval N/A

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

   15. lower-log1p.f32 N/A

       \[\leadsto \left(s \cdot 3\right) \cdot \left(-\color{blue}{\mathsf{log1p}\left(\frac{\left(\mathsf{neg}\left(2\right)\right) \cdot \left(u - \frac{1}{4}\right)}{\frac{3}{2}}\right)}\right) \]

   16. lower-/.f32 N/A

       \[\leadsto \left(s \cdot 3\right) \cdot \left(-\mathsf{log1p}\left(\color{blue}{\frac{\left(\mathsf{neg}\left(2\right)\right) \cdot \left(u - \frac{1}{4}\right)}{\frac{3}{2}}}\right)\right) \]

   17. metadata-eval N/A

       \[\leadsto \left(s \cdot 3\right) \cdot \left(-\mathsf{log1p}\left(\frac{\color{blue}{-2} \cdot \left(u - \frac{1}{4}\right)}{\frac{3}{2}}\right)\right) \]

   18. lower-*.f32 N/A

       \[\leadsto \left(s \cdot 3\right) \cdot \left(-\mathsf{log1p}\left(\frac{\color{blue}{-2 \cdot \left(u - \frac{1}{4}\right)}}{\frac{3}{2}}\right)\right) \]

   19. lift--.f32 98.3

       \[\leadsto \left(s \cdot 3\right) \cdot \left(-\mathsf{log1p}\left(\frac{-2 \cdot \color{blue}{\left(u - 0.25\right)}}{1.5}\right)\right) \]

5. Applied rewrites 98.3%

   \[\leadsto \left(s \cdot 3\right) \cdot \left(-\color{blue}{\mathsf{log1p}\left(\frac{-2 \cdot \left(u - 0.25\right)}{1.5}\right)}\right) \]

6. Taylor expanded in u around 0

   \[\leadsto \left(s \cdot 3\right) \cdot \left(-\mathsf{log1p}\left(\frac{\color{blue}{\frac{1}{2} + -2 \cdot u}}{\frac{3}{2}}\right)\right) \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \left(s \cdot 3\right) \cdot \left(-\mathsf{log1p}\left(\frac{-2 \cdot u + \color{blue}{\frac{1}{2}}}{\frac{3}{2}}\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \left(s \cdot 3\right) \cdot \left(-\mathsf{log1p}\left(\frac{u \cdot -2 + \frac{1}{2}}{\frac{3}{2}}\right)\right) \]

   3. lower-fma.f32 98.3

      \[\leadsto \left(s \cdot 3\right) \cdot \left(-\mathsf{log1p}\left(\frac{\mathsf{fma}\left(u, \color{blue}{-2}, 0.5\right)}{1.5}\right)\right) \]

8. Applied rewrites 98.3%

   \[\leadsto \left(s \cdot 3\right) \cdot \left(-\mathsf{log1p}\left(\frac{\color{blue}{\mathsf{fma}\left(u, -2, 0.5\right)}}{1.5}\right)\right) \]
9. Add Preprocessing

Alternative 2: 97.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \left(\left(-\mathsf{log1p}\left(-1.3333333333333333 \cdot \left(u - 0.25\right)\right)\right) \cdot s\right) \cdot 3 \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (* (* (- (log1p (* -1.3333333333333333 (- u 0.25)))) s) 3.0))

C

float code(float s, float u) {
	return (-log1pf((-1.3333333333333333f * (u - 0.25f))) * s) * 3.0f;
}

Julia

function code(s, u)
	return Float32(Float32(Float32(-log1p(Float32(Float32(-1.3333333333333333) * Float32(u - Float32(0.25))))) * s) * Float32(3.0))
end

Derivation

1. Initial program 95.9%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 95.9

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

   4. lift-log.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. lift--.f32 N/A

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

   7. lift--.f32 N/A

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

   8. lift-/.f32 N/A

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

   9. log-rec N/A

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

   10. lower-neg.f32 N/A

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

   11. lower-log.f32 N/A

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

   12. lower--.f32 N/A

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

   13. mult-flip N/A

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

   14. metadata-eval N/A

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

   15. *-commutative N/A

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

   16. lower-*.f32 N/A

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

   17. lift--.f32 96.5

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

3. Applied rewrites 96.5%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-neg.f32 N/A

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

   4. lift-log.f32 N/A

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

   5. lift--.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. lift--.f32 N/A

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

   8. associate-*l* N/A

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

   9. lower-*.f32 N/A

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

   10. lower-*.f32 N/A

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

   11. lift--.f32 N/A

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

   12. lift-*.f32 N/A

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

   13. lift--.f32 N/A

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

   14. lift-log.f32 N/A

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

   15. lift-neg.f32 96.5

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

5. Applied rewrites 96.5%

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

7. Applied rewrites 96.5%

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

   1. lift--.f32 N/A

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

   2. lift-fma.f32 N/A

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

   3. +-commutative N/A

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

   4. lower-log.f32 N/A

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

   5. lower-log1p.f32 N/A

      \[\leadsto \left(\left(-\color{blue}{\mathsf{log1p}\left(\frac{-4}{3} \cdot \left(u - \frac{1}{4}\right)\right)}\right) \cdot s\right) \cdot 3 \]

   6. lift--.f32 N/A

      \[\leadsto \left(\left(-\mathsf{log1p}\left(\frac{-4}{3} \cdot \color{blue}{\left(u - \frac{1}{4}\right)}\right)\right) \cdot s\right) \cdot 3 \]

   7. lift-*.f32 97.9

      \[\leadsto \left(\left(-\mathsf{log1p}\left(\color{blue}{-1.3333333333333333 \cdot \left(u - 0.25\right)}\right)\right) \cdot s\right) \cdot 3 \]

9. Applied rewrites 97.9%

   \[\leadsto \left(\left(-\color{blue}{\mathsf{log1p}\left(-1.3333333333333333 \cdot \left(u - 0.25\right)\right)}\right) \cdot s\right) \cdot 3 \]
10. Add Preprocessing

Alternative 3: 97.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ s \cdot \left(3 \cdot \left(-\mathsf{log1p}\left(-1.3333333333333333 \cdot \left(u - 0.25\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (* s (* 3.0 (- (log1p (* -1.3333333333333333 (- u 0.25)))))))

C

float code(float s, float u) {
	return s * (3.0f * -log1pf((-1.3333333333333333f * (u - 0.25f))));
}

Julia

function code(s, u)
	return Float32(s * Float32(Float32(3.0) * Float32(-log1p(Float32(Float32(-1.3333333333333333) * Float32(u - Float32(0.25)))))))
end

Derivation

1. Initial program 95.9%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 95.9

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

   4. lift-log.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. lift--.f32 N/A

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

   7. lift--.f32 N/A

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

   8. lift-/.f32 N/A

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

   9. log-rec N/A

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

   10. lower-neg.f32 N/A

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

   11. lower-log.f32 N/A

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

   12. lower--.f32 N/A

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

   13. mult-flip N/A

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

   14. metadata-eval N/A

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

   15. *-commutative N/A

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

   16. lower-*.f32 N/A

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

   17. lift--.f32 96.5

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

3. Applied rewrites 96.5%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-neg.f32 N/A

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

   4. lift-log.f32 N/A

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

   5. lift--.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. lift--.f32 N/A

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

   8. associate-*l* N/A

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

   9. lower-*.f32 N/A

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

   10. lower-*.f32 N/A

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

   11. lift--.f32 N/A

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

   12. lift-*.f32 N/A

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

   13. lift--.f32 N/A

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

   14. lift-log.f32 N/A

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

   15. lift-neg.f32 96.5

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

5. Applied rewrites 96.5%

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

   1. lift-log.f32 N/A

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

   2. lift--.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift--.f32 N/A

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

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

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

   6. metadata-eval N/A

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

   7. lower-log1p.f32 N/A

      \[\leadsto s \cdot \left(3 \cdot \left(-\color{blue}{\mathsf{log1p}\left(\frac{-4}{3} \cdot \left(u - \frac{1}{4}\right)\right)}\right)\right) \]

   8. lower-*.f32 N/A

      \[\leadsto s \cdot \left(3 \cdot \left(-\mathsf{log1p}\left(\color{blue}{\frac{-4}{3} \cdot \left(u - \frac{1}{4}\right)}\right)\right)\right) \]

   9. lift--.f32 97.9

      \[\leadsto s \cdot \left(3 \cdot \left(-\mathsf{log1p}\left(-1.3333333333333333 \cdot \color{blue}{\left(u - 0.25\right)}\right)\right)\right) \]

7. Applied rewrites 97.9%

   \[\leadsto s \cdot \left(3 \cdot \left(-\color{blue}{\mathsf{log1p}\left(-1.3333333333333333 \cdot \left(u - 0.25\right)\right)}\right)\right) \]
8. Add Preprocessing

Alternative 4: 97.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \left(s \cdot 3\right) \cdot \left(-\mathsf{log1p}\left(\mathsf{fma}\left(-1.3333333333333333, u, 0.3333333333333333\right)\right)\right) \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (* (* s 3.0) (- (log1p (fma -1.3333333333333333 u 0.3333333333333333)))))

C

float code(float s, float u) {
	return (s * 3.0f) * -log1pf(fmaf(-1.3333333333333333f, u, 0.3333333333333333f));
}

Julia

function code(s, u)
	return Float32(Float32(s * Float32(3.0)) * Float32(-log1p(fma(Float32(-1.3333333333333333), u, Float32(0.3333333333333333)))))
end

Derivation

1. Initial program 95.9%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 95.9

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

   4. lift-log.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. lift--.f32 N/A

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

   7. lift--.f32 N/A

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

   8. lift-/.f32 N/A

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

   9. log-rec N/A

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

   10. lower-neg.f32 N/A

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

   11. lower-log.f32 N/A

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

   12. lower--.f32 N/A

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

   13. mult-flip N/A

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

   14. metadata-eval N/A

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

   15. *-commutative N/A

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

   16. lower-*.f32 N/A

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

   17. lift--.f32 96.5

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

3. Applied rewrites 96.5%

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

   1. lift-log.f32 N/A

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

   2. lift--.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift--.f32 N/A

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

   5. *-commutative N/A

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

   6. metadata-eval N/A

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

   7. mult-flip N/A

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

   8. metadata-eval N/A

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

   9. frac-sub N/A

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

   10. metadata-eval N/A

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

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

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

   12. metadata-eval N/A

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

   13. div-add N/A

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

   14. metadata-eval N/A

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

   15. lower-log1p.f32 N/A

       \[\leadsto \left(s \cdot 3\right) \cdot \left(-\color{blue}{\mathsf{log1p}\left(\frac{\left(\mathsf{neg}\left(2\right)\right) \cdot \left(u - \frac{1}{4}\right)}{\frac{3}{2}}\right)}\right) \]

   16. lower-/.f32 N/A

       \[\leadsto \left(s \cdot 3\right) \cdot \left(-\mathsf{log1p}\left(\color{blue}{\frac{\left(\mathsf{neg}\left(2\right)\right) \cdot \left(u - \frac{1}{4}\right)}{\frac{3}{2}}}\right)\right) \]

   17. metadata-eval N/A

       \[\leadsto \left(s \cdot 3\right) \cdot \left(-\mathsf{log1p}\left(\frac{\color{blue}{-2} \cdot \left(u - \frac{1}{4}\right)}{\frac{3}{2}}\right)\right) \]

   18. lower-*.f32 N/A

       \[\leadsto \left(s \cdot 3\right) \cdot \left(-\mathsf{log1p}\left(\frac{\color{blue}{-2 \cdot \left(u - \frac{1}{4}\right)}}{\frac{3}{2}}\right)\right) \]

   19. lift--.f32 98.3

       \[\leadsto \left(s \cdot 3\right) \cdot \left(-\mathsf{log1p}\left(\frac{-2 \cdot \color{blue}{\left(u - 0.25\right)}}{1.5}\right)\right) \]

5. Applied rewrites 98.3%

   \[\leadsto \left(s \cdot 3\right) \cdot \left(-\color{blue}{\mathsf{log1p}\left(\frac{-2 \cdot \left(u - 0.25\right)}{1.5}\right)}\right) \]

6. Taylor expanded in u around 0

   \[\leadsto \left(s \cdot 3\right) \cdot \left(-\mathsf{log1p}\left(\color{blue}{\frac{1}{3} + \frac{-4}{3} \cdot u}\right)\right) \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \left(s \cdot 3\right) \cdot \left(-\mathsf{log1p}\left(\frac{-4}{3} \cdot u + \color{blue}{\frac{1}{3}}\right)\right) \]

   2. lower-fma.f32 97.9

      \[\leadsto \left(s \cdot 3\right) \cdot \left(-\mathsf{log1p}\left(\mathsf{fma}\left(-1.3333333333333333, \color{blue}{u}, 0.3333333333333333\right)\right)\right) \]

8. Applied rewrites 97.9%

   \[\leadsto \left(s \cdot 3\right) \cdot \left(-\mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(-1.3333333333333333, u, 0.3333333333333333\right)}\right)\right) \]
9. Add Preprocessing

Alternative 5: 96.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ s \cdot \left(3 \cdot \left(-\log \left(\mathsf{fma}\left(-1.3333333333333333, u, 1.3333333333333333\right)\right)\right)\right) \end{array} \]

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 95.9%

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

2. Taylor expanded in u around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f32 96.2

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

4. Applied rewrites 96.2%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f32 96.2

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

   4. lift-log.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. log-rec N/A

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

   7. lower-neg.f32 N/A

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

   8. lower-log.f32 96.8

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

6. Applied rewrites 96.8%

   \[\leadsto \color{blue}{\left(s \cdot 3\right) \cdot \left(-\log \left(\mathsf{fma}\left(-1.3333333333333333, u, 1.3333333333333333\right)\right)\right)} \]
7. Step-by-step derivation

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 96.9

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

8. Applied rewrites 96.9%

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

Alternative 6: 96.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ 3 \cdot \left(s \cdot \left(-\log \left(\mathsf{fma}\left(-1.3333333333333333, u, 1.3333333333333333\right)\right)\right)\right) \end{array} \]

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 95.9%

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

2. Taylor expanded in u around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f32 96.2

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

4. Applied rewrites 96.2%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 96.2

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

   6. lift-log.f32 N/A

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

   7. lift-/.f32 N/A

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

   8. log-rec N/A

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

   9. lower-neg.f32 N/A

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

   10. lower-log.f32 96.8

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

6. Applied rewrites 96.8%

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

Alternative 7: 25.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ s \cdot \mathsf{fma}\left(u, 3, \log 0.421875\right) \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (* s (fma u 3.0 (log 0.421875))))

C

float code(float s, float u) {
	return s * fmaf(u, 3.0f, logf(0.421875f));
}

Julia

function code(s, u)
	return Float32(s * fma(u, Float32(3.0), log(Float32(0.421875))))
end

Derivation

1. Initial program 95.9%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 95.9

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

   4. lift-log.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. lift--.f32 N/A

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

   7. lift--.f32 N/A

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

   8. lift-/.f32 N/A

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

   9. log-rec N/A

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

   10. lower-neg.f32 N/A

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

   11. lower-log.f32 N/A

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

   12. lower--.f32 N/A

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

   13. mult-flip N/A

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

   14. metadata-eval N/A

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

   15. *-commutative N/A

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

   16. lower-*.f32 N/A

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

   17. lift--.f32 96.5

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

3. Applied rewrites 96.5%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-neg.f32 N/A

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

   4. lift-log.f32 N/A

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

   5. lift--.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. lift--.f32 N/A

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

   8. associate-*l* N/A

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

   9. lower-*.f32 N/A

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

   10. lower-*.f32 N/A

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

   11. lift--.f32 N/A

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

   12. lift-*.f32 N/A

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

   13. lift--.f32 N/A

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

   14. lift-log.f32 N/A

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

   15. lift-neg.f32 96.5

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

5. Applied rewrites 96.5%

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

6. Taylor expanded in u around 0

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

   1. metadata-eval N/A

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

   2. log-fabs N/A

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

   3. log-pow-rev N/A

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

   4. lower-log.f32 N/A

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

   5. metadata-eval -0.0

      \[\leadsto s \cdot \log -0.421875 \]

8. Applied rewrites -0.0%

   \[\leadsto s \cdot \color{blue}{\log -0.421875} \]

9. Taylor expanded in u around 0

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

    1. +-commutative N/A

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

    2. *-commutative N/A

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

    3. lower-fma.f32 N/A

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

    4. log-pow-rev N/A

       \[\leadsto s \cdot \mathsf{fma}\left(u, 3, \log \left({\frac{4}{3}}^{-3}\right)\right) \]

    5. lower-log.f32 N/A

       \[\leadsto s \cdot \mathsf{fma}\left(u, 3, \log \left({\frac{4}{3}}^{-3}\right)\right) \]

    6. metadata-eval 25.7

       \[\leadsto s \cdot \mathsf{fma}\left(u, 3, \log 0.421875\right) \]

11. Applied rewrites 25.7%

    \[\leadsto s \cdot \color{blue}{\mathsf{fma}\left(u, 3, \log 0.421875\right)} \]
12. Add Preprocessing

Alternative 8: 7.4% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \log 0.421875 \cdot s \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (* (log 0.421875) s))

C

float code(float s, float u) {
	return logf(0.421875f) * 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(0.421875e0) * s
end function

Julia

function code(s, u)
	return Float32(log(Float32(0.421875)) * s)
end

MATLAB

function tmp = code(s, u)
	tmp = log(single(0.421875)) * s;
end

Derivation

1. Initial program 95.9%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 95.9

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

   4. lift-log.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. lift--.f32 N/A

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

   7. lift--.f32 N/A

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

   8. lift-/.f32 N/A

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

   9. log-rec N/A

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

   10. lower-neg.f32 N/A

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

   11. lower-log.f32 N/A

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

   12. lower--.f32 N/A

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

   13. mult-flip N/A

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

   14. metadata-eval N/A

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

   15. *-commutative N/A

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

   16. lower-*.f32 N/A

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

   17. lift--.f32 96.5

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

3. Applied rewrites 96.5%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-neg.f32 N/A

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

   4. lift-log.f32 N/A

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

   5. lift--.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. lift--.f32 N/A

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

   8. associate-*l* N/A

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

   9. lower-*.f32 N/A

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

   10. lower-*.f32 N/A

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

   11. lift--.f32 N/A

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

   12. lift-*.f32 N/A

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

   13. lift--.f32 N/A

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

   14. lift-log.f32 N/A

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

   15. lift-neg.f32 96.5

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

5. Applied rewrites 96.5%

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

6. Taylor expanded in u around 0

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

   1. metadata-eval N/A

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

   2. log-fabs N/A

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

   3. log-pow-rev N/A

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

   4. lower-log.f32 N/A

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

   5. metadata-eval -0.0

      \[\leadsto s \cdot \log -0.421875 \]

8. Applied rewrites -0.0%

   \[\leadsto s \cdot \color{blue}{\log -0.421875} \]
9. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \color{blue}{s \cdot \log \frac{-27}{64}} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\log \frac{-27}{64} \cdot s} \]

   3. lower-*.f32 -0.0

      \[\leadsto \color{blue}{\log -0.421875 \cdot s} \]

   4. lift-log.f32 N/A

      \[\leadsto \log \frac{-27}{64} \cdot s \]

   5. log-fabs N/A

      \[\leadsto \log \left(\left|\frac{-27}{64}\right|\right) \cdot s \]

   6. metadata-eval N/A

      \[\leadsto \log \frac{27}{64} \cdot s \]

   7. metadata-eval N/A

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

   8. lower-log.f32 N/A

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

   9. metadata-eval 7.4

      \[\leadsto \log 0.421875 \cdot s \]

10. Applied rewrites 7.4%

    \[\leadsto \color{blue}{\log 0.421875 \cdot s} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025134 
(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))))))