Disney BSSRDF, sample scattering profile, upper

Percentage Accurate: 95.9% → 98.1%

Time: 6.4s

Alternatives: 3

Speedup: N/A×

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: 98.1% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -1.3333333333333333 \cdot \left(u - 0.25\right)\\ \left(\left(\mathsf{log1p}\left({\left(u - 0.25\right)}^{3} \cdot -2.3703703703703702\right) - \mathsf{log1p}\left({t\_0}^{2} - t\_0\right)\right) \cdot s\right) \cdot \left(-3\right) \end{array} \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (let* ((t_0 (* -1.3333333333333333 (- u 0.25))))
   (*
    (*
     (-
      (log1p (* (pow (- u 0.25) 3.0) -2.3703703703703702))
      (log1p (- (pow t_0 2.0) t_0)))
     s)
    (- 3.0))))

C

float code(float s, float u) {
	float t_0 = -1.3333333333333333f * (u - 0.25f);
	return ((log1pf((powf((u - 0.25f), 3.0f) * -2.3703703703703702f)) - log1pf((powf(t_0, 2.0f) - t_0))) * s) * -3.0f;
}

Julia

function code(s, u)
	t_0 = Float32(Float32(-1.3333333333333333) * Float32(u - Float32(0.25)))
	return Float32(Float32(Float32(log1p(Float32((Float32(u - Float32(0.25)) ^ Float32(3.0)) * Float32(-2.3703703703703702))) - log1p(Float32((t_0 ^ Float32(2.0)) - t_0))) * s) * Float32(-Float32(3.0)))
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. Add Preprocessing

3. Taylor expanded in s around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. log-rec N/A

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

   6. lower-neg.f32 N/A

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

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

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

   8. lower-log1p.f32 N/A

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

   9. metadata-eval N/A

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

   10. lower-*.f32 N/A

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

   11. lift--.f32 97.8

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

5. Applied rewrites 97.8%

   \[\leadsto \color{blue}{\left(\left(-\mathsf{log1p}\left(-1.3333333333333333 \cdot \left(u - 0.25\right)\right)\right) \cdot s\right) \cdot 3} \]
6. Step-by-step derivation

   1. lift-log1p.f32 N/A

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

   2. lift--.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. flip3-+ N/A

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

   5. log-div N/A

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

   6. lower--.f32 N/A

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

   7. metadata-eval N/A

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

   8. lower-log1p.f32 N/A

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

   9. lower-pow.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. lift--.f32 N/A

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

   12. metadata-eval N/A

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

7. Applied rewrites 97.5%

   \[\leadsto \left(\left(-\left(\mathsf{log1p}\left({\left(-1.3333333333333333 \cdot \left(u - 0.25\right)\right)}^{3}\right) - \mathsf{log1p}\left({\left(-1.3333333333333333 \cdot \left(u - 0.25\right)\right)}^{2} - 1 \cdot \left(-1.3333333333333333 \cdot \left(u - 0.25\right)\right)\right)\right)\right) \cdot s\right) \cdot 3 \]
8. Step-by-step derivation

   1. lift-pow.f32 N/A

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

   2. lift--.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. unpow-prod-down N/A

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

   5. metadata-eval N/A

      \[\leadsto \left(\left(-\left(\mathsf{log1p}\left(\frac{-64}{27} \cdot {\left(u - \frac{1}{4}\right)}^{3}\right) - \mathsf{log1p}\left({\left(\frac{-4}{3} \cdot \left(u - \frac{1}{4}\right)\right)}^{2} - 1 \cdot \left(\frac{-4}{3} \cdot \left(u - \frac{1}{4}\right)\right)\right)\right)\right) \cdot s\right) \cdot 3 \]

   6. *-commutative N/A

      \[\leadsto \left(\left(-\left(\mathsf{log1p}\left({\left(u - \frac{1}{4}\right)}^{3} \cdot \frac{-64}{27}\right) - \mathsf{log1p}\left({\left(\frac{-4}{3} \cdot \left(u - \frac{1}{4}\right)\right)}^{2} - 1 \cdot \left(\frac{-4}{3} \cdot \left(u - \frac{1}{4}\right)\right)\right)\right)\right) \cdot s\right) \cdot 3 \]

   7. lower-*.f32 N/A

      \[\leadsto \left(\left(-\left(\mathsf{log1p}\left({\left(u - \frac{1}{4}\right)}^{3} \cdot \frac{-64}{27}\right) - \mathsf{log1p}\left({\left(\frac{-4}{3} \cdot \left(u - \frac{1}{4}\right)\right)}^{2} - 1 \cdot \left(\frac{-4}{3} \cdot \left(u - \frac{1}{4}\right)\right)\right)\right)\right) \cdot s\right) \cdot 3 \]

   8. lower-pow.f32 N/A

      \[\leadsto \left(\left(-\left(\mathsf{log1p}\left({\left(u - \frac{1}{4}\right)}^{3} \cdot \frac{-64}{27}\right) - \mathsf{log1p}\left({\left(\frac{-4}{3} \cdot \left(u - \frac{1}{4}\right)\right)}^{2} - 1 \cdot \left(\frac{-4}{3} \cdot \left(u - \frac{1}{4}\right)\right)\right)\right)\right) \cdot s\right) \cdot 3 \]

   9. lift--.f32 98.0

      \[\leadsto \left(\left(-\left(\mathsf{log1p}\left({\left(u - 0.25\right)}^{3} \cdot -2.3703703703703702\right) - \mathsf{log1p}\left({\left(-1.3333333333333333 \cdot \left(u - 0.25\right)\right)}^{2} - 1 \cdot \left(-1.3333333333333333 \cdot \left(u - 0.25\right)\right)\right)\right)\right) \cdot s\right) \cdot 3 \]

9. Applied rewrites 98.0%

   \[\leadsto \left(\left(-\left(\mathsf{log1p}\left({\left(u - 0.25\right)}^{3} \cdot -2.3703703703703702\right) - \mathsf{log1p}\left({\left(-1.3333333333333333 \cdot \left(u - 0.25\right)\right)}^{2} - 1 \cdot \left(-1.3333333333333333 \cdot \left(u - 0.25\right)\right)\right)\right)\right) \cdot s\right) \cdot 3 \]

10. Final simplification 98.0%

    \[\leadsto \left(\left(\mathsf{log1p}\left({\left(u - 0.25\right)}^{3} \cdot -2.3703703703703702\right) - \mathsf{log1p}\left({\left(-1.3333333333333333 \cdot \left(u - 0.25\right)\right)}^{2} - -1.3333333333333333 \cdot \left(u - 0.25\right)\right)\right) \cdot s\right) \cdot \left(-3\right) \]
11. Add Preprocessing

Alternative 2: 97.9% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \left(s \cdot 3\right) \cdot \left(\mathsf{log1p}\left(\mathsf{fma}\left({\left(u - 0.25\right)}^{2}, 1.7777777777777777, 1.3333333333333333 \cdot \left(u - 0.25\right)\right)\right) - \mathsf{log1p}\left({\left(-1.3333333333333333 \cdot \left(u - 0.25\right)\right)}^{3}\right)\right) \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (*
  (* s 3.0)
  (-
   (log1p
    (fma
     (pow (- u 0.25) 2.0)
     1.7777777777777777
     (* 1.3333333333333333 (- u 0.25))))
   (log1p (pow (* -1.3333333333333333 (- u 0.25)) 3.0)))))

C

float code(float s, float u) {
	return (s * 3.0f) * (log1pf(fmaf(powf((u - 0.25f), 2.0f), 1.7777777777777777f, (1.3333333333333333f * (u - 0.25f)))) - log1pf(powf((-1.3333333333333333f * (u - 0.25f)), 3.0f)));
}

Julia

function code(s, u)
	return Float32(Float32(s * Float32(3.0)) * Float32(log1p(fma((Float32(u - Float32(0.25)) ^ Float32(2.0)), Float32(1.7777777777777777), Float32(Float32(1.3333333333333333) * Float32(u - Float32(0.25))))) - log1p((Float32(Float32(-1.3333333333333333) * Float32(u - Float32(0.25))) ^ Float32(3.0)))))
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. Add Preprocessing
3. 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. lift-*.f32 N/A

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

   3. lift-log.f32 N/A

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

   4. lift-/.f32 N/A

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

   5. lift--.f32 N/A

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

   6. lift--.f32 N/A

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

   7. lift-/.f32 N/A

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

   8. associate-*l* N/A

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

   9. lower-*.f32 N/A

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

   10. lower-*.f32 N/A

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

   11. log-rec N/A

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

   12. lower-neg.f32 N/A

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

   13. lower-log.f32 N/A

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

   14. lift-/.f32 N/A

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

   15. lift--.f32 N/A

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

   16. lift--.f32 96.8

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

4. Applied rewrites 96.8%

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

   1. lift-log.f32 N/A

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

   2. lift--.f32 N/A

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

   3. lift--.f32 N/A

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

   4. lift-/.f32 N/A

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

   5. flip3-- N/A

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

   6. log-div N/A

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

   7. lower--.f32 N/A

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

   8. lower-log.f32 N/A

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

   9. metadata-eval N/A

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

   10. lower--.f32 N/A

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

   11. lower-pow.f32 N/A

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

   12. lift-/.f32 N/A

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

   13. lift--.f32 N/A

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

6. Applied rewrites 96.9%

   \[\leadsto 3 \cdot \left(s \cdot \left(-\color{blue}{\left(\log \left(1 - {\left(\frac{u - 0.25}{0.75}\right)}^{3}\right) - \mathsf{log1p}\left({\left(\frac{u - 0.25}{0.75}\right)}^{2} + 1 \cdot \frac{u - 0.25}{0.75}\right)\right)}\right)\right) \]

7. Taylor expanded in s around 0

   \[\leadsto \color{blue}{3 \cdot \left(s \cdot \left(\log \left(1 + \left(\frac{4}{3} \cdot \left(u - \frac{1}{4}\right) + \frac{16}{9} \cdot {\left(u - \frac{1}{4}\right)}^{2}\right)\right) - \log \left(1 - \frac{64}{27} \cdot {\left(u - \frac{1}{4}\right)}^{3}\right)\right)\right)} \]

9. Applied rewrites 97.8%

   \[\leadsto \color{blue}{\left(s \cdot 3\right) \cdot \left(\mathsf{log1p}\left(\mathsf{fma}\left({\left(u - 0.25\right)}^{2}, 1.7777777777777777, 1.3333333333333333 \cdot \left(u - 0.25\right)\right)\right) - \mathsf{log1p}\left({\left(-1.3333333333333333 \cdot \left(u - 0.25\right)\right)}^{3}\right)\right)} \]
10. Add Preprocessing

Alternative 3: 97.7% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \left(s \cdot 3\right) \cdot \left(\mathsf{log1p}\left(\mathsf{fma}\left(0.0625 + u \cdot \left(u - 0.5\right), 1.7777777777777777, 1.3333333333333333 \cdot \left(u - 0.25\right)\right)\right) - \mathsf{log1p}\left({\left(-1.3333333333333333 \cdot \left(u - 0.25\right)\right)}^{3}\right)\right) \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (*
  (* s 3.0)
  (-
   (log1p
    (fma
     (+ 0.0625 (* u (- u 0.5)))
     1.7777777777777777
     (* 1.3333333333333333 (- u 0.25))))
   (log1p (pow (* -1.3333333333333333 (- u 0.25)) 3.0)))))

C

float code(float s, float u) {
	return (s * 3.0f) * (log1pf(fmaf((0.0625f + (u * (u - 0.5f))), 1.7777777777777777f, (1.3333333333333333f * (u - 0.25f)))) - log1pf(powf((-1.3333333333333333f * (u - 0.25f)), 3.0f)));
}

Julia

function code(s, u)
	return Float32(Float32(s * Float32(3.0)) * Float32(log1p(fma(Float32(Float32(0.0625) + Float32(u * Float32(u - Float32(0.5)))), Float32(1.7777777777777777), Float32(Float32(1.3333333333333333) * Float32(u - Float32(0.25))))) - log1p((Float32(Float32(-1.3333333333333333) * Float32(u - Float32(0.25))) ^ Float32(3.0)))))
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. Add Preprocessing
3. 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. lift-*.f32 N/A

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

   3. lift-log.f32 N/A

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

   4. lift-/.f32 N/A

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

   5. lift--.f32 N/A

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

   6. lift--.f32 N/A

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

   7. lift-/.f32 N/A

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

   8. associate-*l* N/A

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

   9. lower-*.f32 N/A

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

   10. lower-*.f32 N/A

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

   11. log-rec N/A

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

   12. lower-neg.f32 N/A

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

   13. lower-log.f32 N/A

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

   14. lift-/.f32 N/A

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

   15. lift--.f32 N/A

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

   16. lift--.f32 96.8

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

4. Applied rewrites 96.8%

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

   1. lift-log.f32 N/A

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

   2. lift--.f32 N/A

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

   3. lift--.f32 N/A

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

   4. lift-/.f32 N/A

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

   5. flip3-- N/A

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

   6. log-div N/A

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

   7. lower--.f32 N/A

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

   8. lower-log.f32 N/A

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

   9. metadata-eval N/A

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

   10. lower--.f32 N/A

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

   11. lower-pow.f32 N/A

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

   12. lift-/.f32 N/A

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

   13. lift--.f32 N/A

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

6. Applied rewrites 96.9%

   \[\leadsto 3 \cdot \left(s \cdot \left(-\color{blue}{\left(\log \left(1 - {\left(\frac{u - 0.25}{0.75}\right)}^{3}\right) - \mathsf{log1p}\left({\left(\frac{u - 0.25}{0.75}\right)}^{2} + 1 \cdot \frac{u - 0.25}{0.75}\right)\right)}\right)\right) \]

7. Taylor expanded in s around 0

   \[\leadsto \color{blue}{3 \cdot \left(s \cdot \left(\log \left(1 + \left(\frac{4}{3} \cdot \left(u - \frac{1}{4}\right) + \frac{16}{9} \cdot {\left(u - \frac{1}{4}\right)}^{2}\right)\right) - \log \left(1 - \frac{64}{27} \cdot {\left(u - \frac{1}{4}\right)}^{3}\right)\right)\right)} \]

9. Applied rewrites 97.8%

   \[\leadsto \color{blue}{\left(s \cdot 3\right) \cdot \left(\mathsf{log1p}\left(\mathsf{fma}\left({\left(u - 0.25\right)}^{2}, 1.7777777777777777, 1.3333333333333333 \cdot \left(u - 0.25\right)\right)\right) - \mathsf{log1p}\left({\left(-1.3333333333333333 \cdot \left(u - 0.25\right)\right)}^{3}\right)\right)} \]

10. Taylor expanded in u around 0

    \[\leadsto \left(s \cdot 3\right) \cdot \left(\mathsf{log1p}\left(\mathsf{fma}\left(\frac{1}{16} + u \cdot \left(u - \frac{1}{2}\right), \frac{16}{9}, \frac{4}{3} \cdot \left(u - \frac{1}{4}\right)\right)\right) - \mathsf{log1p}\left({\left(\color{blue}{\frac{-4}{3}} \cdot \left(u - \frac{1}{4}\right)\right)}^{3}\right)\right) \]
11. Step-by-step derivation

    1. lower-+.f32 N/A

       \[\leadsto \left(s \cdot 3\right) \cdot \left(\mathsf{log1p}\left(\mathsf{fma}\left(\frac{1}{16} + u \cdot \left(u - \frac{1}{2}\right), \frac{16}{9}, \frac{4}{3} \cdot \left(u - \frac{1}{4}\right)\right)\right) - \mathsf{log1p}\left({\left(\frac{-4}{3} \cdot \left(u - \frac{1}{4}\right)\right)}^{3}\right)\right) \]

    2. lower-*.f32 N/A

       \[\leadsto \left(s \cdot 3\right) \cdot \left(\mathsf{log1p}\left(\mathsf{fma}\left(\frac{1}{16} + u \cdot \left(u - \frac{1}{2}\right), \frac{16}{9}, \frac{4}{3} \cdot \left(u - \frac{1}{4}\right)\right)\right) - \mathsf{log1p}\left({\left(\frac{-4}{3} \cdot \left(u - \frac{1}{4}\right)\right)}^{3}\right)\right) \]

    3. lower--.f32 97.7

       \[\leadsto \left(s \cdot 3\right) \cdot \left(\mathsf{log1p}\left(\mathsf{fma}\left(0.0625 + u \cdot \left(u - 0.5\right), 1.7777777777777777, 1.3333333333333333 \cdot \left(u - 0.25\right)\right)\right) - \mathsf{log1p}\left({\left(-1.3333333333333333 \cdot \left(u - 0.25\right)\right)}^{3}\right)\right) \]

12. Applied rewrites 97.7%

    \[\leadsto \left(s \cdot 3\right) \cdot \left(\mathsf{log1p}\left(\mathsf{fma}\left(0.0625 + u \cdot \left(u - 0.5\right), 1.7777777777777777, 1.3333333333333333 \cdot \left(u - 0.25\right)\right)\right) - \mathsf{log1p}\left({\left(\color{blue}{-1.3333333333333333} \cdot \left(u - 0.25\right)\right)}^{3}\right)\right) \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2025057 
(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))))))