Disney BSSRDF, sample scattering profile, lower

Percentage Accurate: 61.1% → 99.4%

Time: 12.6s

Alternatives: 6

Speedup: 21.8×

Specification

\[\left(0 \leq s \land s \leq 256\right) \land \left(2.328306437 \cdot 10^{-10} \leq u \land u \leq 0.25\right)\]

Math

\[\begin{array}{l} \\ s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (* s (log (/ 1.0 (- 1.0 (* 4.0 u))))))

C

float code(float s, float u) {
	return s * logf((1.0f / (1.0f - (4.0f * u))));
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = s * log((1.0e0 / (1.0e0 - (4.0e0 * u))))
end function

Julia

function code(s, u)
	return Float32(s * log(Float32(Float32(1.0) / Float32(Float32(1.0) - Float32(Float32(4.0) * u)))))
end

MATLAB

function tmp = code(s, u)
	tmp = s * log((single(1.0) / (single(1.0) - (single(4.0) * u))));
end

Initial Program: 61.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (* s (log (/ 1.0 (- 1.0 (* 4.0 u))))))

C

float code(float s, float u) {
	return s * logf((1.0f / (1.0f - (4.0f * u))));
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = s * log((1.0e0 / (1.0e0 - (4.0e0 * u))))
end function

Julia

function code(s, u)
	return Float32(s * log(Float32(Float32(1.0) / Float32(Float32(1.0) - Float32(Float32(4.0) * u)))))
end

MATLAB

function tmp = code(s, u)
	tmp = s * log((single(1.0) / (single(1.0) - (single(4.0) * u))));
end

Alternative 1: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ s \cdot \left(-\mathsf{log1p}\left(u \cdot -4\right)\right) \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (* s (- (log1p (* u -4.0)))))

C

float code(float s, float u) {
	return s * -log1pf((u * -4.0f));
}

Julia

function code(s, u)
	return Float32(s * Float32(-log1p(Float32(u * Float32(-4.0)))))
end

Derivation

1. Initial program 62.2%

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

   1. log-rec 64.6%

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

   2. distribute-rgt-neg-out 64.6%

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

   3. distribute-lft-neg-out 64.6%

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

   4. cancel-sign-sub-inv 64.6%

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

   5. log1p-define 99.3%

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

   6. *-commutative 99.3%

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

   7. metadata-eval 99.3%

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

3. Simplified 99.3%

   \[\leadsto \color{blue}{\left(-s\right) \cdot \mathsf{log1p}\left(u \cdot -4\right)} \]
4. Add Preprocessing

5. Final simplification 99.3%

   \[\leadsto s \cdot \left(-\mathsf{log1p}\left(u \cdot -4\right)\right) \]
6. Add Preprocessing

Alternative 2: 87.1% accurate, 9.9× speedup

Math

\[\begin{array}{l} \\ u \cdot \left(s \cdot \left(u \cdot 8\right) + s \cdot 4\right) \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (* u (+ (* s (* u 8.0)) (* s 4.0))))

C

float code(float s, float u) {
	return u * ((s * (u * 8.0f)) + (s * 4.0f));
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = u * ((s * (u * 8.0e0)) + (s * 4.0e0))
end function

Julia

function code(s, u)
	return Float32(u * Float32(Float32(s * Float32(u * Float32(8.0))) + Float32(s * Float32(4.0))))
end

MATLAB

function tmp = code(s, u)
	tmp = u * ((s * (u * single(8.0))) + (s * single(4.0)));
end

Derivation

1. Initial program 62.2%

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

   1. log-rec 64.6%

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

   2. cancel-sign-sub-inv 64.6%

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

   3. metadata-eval 64.6%

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

3. Simplified 64.6%

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

5. Taylor expanded in u around 0 86.1%

   \[\leadsto \color{blue}{4 \cdot \left(s \cdot u\right) + 8 \cdot \left(s \cdot {u}^{2}\right)} \]
6. Step-by-step derivation

   1. +-commutative 86.1%

      \[\leadsto \color{blue}{8 \cdot \left(s \cdot {u}^{2}\right) + 4 \cdot \left(s \cdot u\right)} \]

   2. associate-*r* 86.1%

      \[\leadsto \color{blue}{\left(8 \cdot s\right) \cdot {u}^{2}} + 4 \cdot \left(s \cdot u\right) \]

   3. unpow2 86.1%

      \[\leadsto \left(8 \cdot s\right) \cdot \color{blue}{\left(u \cdot u\right)} + 4 \cdot \left(s \cdot u\right) \]

   4. associate-*r* 86.1%

      \[\leadsto \color{blue}{\left(\left(8 \cdot s\right) \cdot u\right) \cdot u} + 4 \cdot \left(s \cdot u\right) \]

   5. fma-define 86.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(8 \cdot s\right) \cdot u, u, 4 \cdot \left(s \cdot u\right)\right)} \]

   6. *-commutative 86.1%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(s \cdot 8\right)} \cdot u, u, 4 \cdot \left(s \cdot u\right)\right) \]

   7. *-commutative 86.1%

      \[\leadsto \mathsf{fma}\left(\left(s \cdot 8\right) \cdot u, u, \color{blue}{\left(s \cdot u\right) \cdot 4}\right) \]

   8. *-commutative 86.1%

      \[\leadsto \mathsf{fma}\left(\left(s \cdot 8\right) \cdot u, u, \color{blue}{\left(u \cdot s\right)} \cdot 4\right) \]

   9. associate-*l* 86.5%

      \[\leadsto \mathsf{fma}\left(\left(s \cdot 8\right) \cdot u, u, \color{blue}{u \cdot \left(s \cdot 4\right)}\right) \]

7. Applied egg-rr 86.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\left(s \cdot 8\right) \cdot u, u, u \cdot \left(s \cdot 4\right)\right)} \]
8. Step-by-step derivation

   1. fma-undefine 86.5%

      \[\leadsto \color{blue}{\left(\left(s \cdot 8\right) \cdot u\right) \cdot u + u \cdot \left(s \cdot 4\right)} \]

   2. *-commutative 86.5%

      \[\leadsto \left(\left(s \cdot 8\right) \cdot u\right) \cdot u + \color{blue}{\left(s \cdot 4\right) \cdot u} \]

   3. distribute-rgt-out 86.6%

      \[\leadsto \color{blue}{u \cdot \left(\left(s \cdot 8\right) \cdot u + s \cdot 4\right)} \]

   4. associate-*l* 86.6%

      \[\leadsto u \cdot \left(\color{blue}{s \cdot \left(8 \cdot u\right)} + s \cdot 4\right) \]

   5. *-commutative 86.6%

      \[\leadsto u \cdot \left(s \cdot \color{blue}{\left(u \cdot 8\right)} + s \cdot 4\right) \]

   6. *-commutative 86.6%

      \[\leadsto u \cdot \left(s \cdot \left(u \cdot 8\right) + \color{blue}{4 \cdot s}\right) \]

9. Simplified 86.6%

   \[\leadsto \color{blue}{u \cdot \left(s \cdot \left(u \cdot 8\right) + 4 \cdot s\right)} \]

10. Final simplification 86.6%

    \[\leadsto u \cdot \left(s \cdot \left(u \cdot 8\right) + s \cdot 4\right) \]
11. Add Preprocessing

Alternative 3: 86.9% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ s \cdot \left(u \cdot \left(u \cdot 8 + 4\right)\right) \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (* s (* u (+ (* u 8.0) 4.0))))

C

float code(float s, float u) {
	return s * (u * ((u * 8.0f) + 4.0f));
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = s * (u * ((u * 8.0e0) + 4.0e0))
end function

Julia

function code(s, u)
	return Float32(s * Float32(u * Float32(Float32(u * Float32(8.0)) + Float32(4.0))))
end

MATLAB

function tmp = code(s, u)
	tmp = s * (u * ((u * single(8.0)) + single(4.0)));
end

Derivation

1. Initial program 62.2%

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

   1. log-rec 64.6%

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

   2. cancel-sign-sub-inv 64.6%

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

   3. metadata-eval 64.6%

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

3. Simplified 64.6%

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

5. Taylor expanded in u around 0 86.1%

   \[\leadsto \color{blue}{4 \cdot \left(s \cdot u\right) + 8 \cdot \left(s \cdot {u}^{2}\right)} \]
6. Step-by-step derivation

   1. +-commutative 86.1%

      \[\leadsto \color{blue}{8 \cdot \left(s \cdot {u}^{2}\right) + 4 \cdot \left(s \cdot u\right)} \]

   2. associate-*r* 86.1%

      \[\leadsto \color{blue}{\left(8 \cdot s\right) \cdot {u}^{2}} + 4 \cdot \left(s \cdot u\right) \]

   3. unpow2 86.1%

      \[\leadsto \left(8 \cdot s\right) \cdot \color{blue}{\left(u \cdot u\right)} + 4 \cdot \left(s \cdot u\right) \]

   4. associate-*r* 86.1%

      \[\leadsto \color{blue}{\left(\left(8 \cdot s\right) \cdot u\right) \cdot u} + 4 \cdot \left(s \cdot u\right) \]

   5. fma-define 86.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(8 \cdot s\right) \cdot u, u, 4 \cdot \left(s \cdot u\right)\right)} \]

   6. *-commutative 86.1%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(s \cdot 8\right)} \cdot u, u, 4 \cdot \left(s \cdot u\right)\right) \]

   7. *-commutative 86.1%

      \[\leadsto \mathsf{fma}\left(\left(s \cdot 8\right) \cdot u, u, \color{blue}{\left(s \cdot u\right) \cdot 4}\right) \]

   8. *-commutative 86.1%

      \[\leadsto \mathsf{fma}\left(\left(s \cdot 8\right) \cdot u, u, \color{blue}{\left(u \cdot s\right)} \cdot 4\right) \]

   9. associate-*l* 86.5%

      \[\leadsto \mathsf{fma}\left(\left(s \cdot 8\right) \cdot u, u, \color{blue}{u \cdot \left(s \cdot 4\right)}\right) \]

7. Applied egg-rr 86.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\left(s \cdot 8\right) \cdot u, u, u \cdot \left(s \cdot 4\right)\right)} \]
8. Step-by-step derivation

   1. fma-undefine 86.5%

      \[\leadsto \color{blue}{\left(\left(s \cdot 8\right) \cdot u\right) \cdot u + u \cdot \left(s \cdot 4\right)} \]

   2. *-commutative 86.5%

      \[\leadsto \left(\left(s \cdot 8\right) \cdot u\right) \cdot u + \color{blue}{\left(s \cdot 4\right) \cdot u} \]

   3. distribute-rgt-out 86.6%

      \[\leadsto \color{blue}{u \cdot \left(\left(s \cdot 8\right) \cdot u + s \cdot 4\right)} \]

   4. associate-*l* 86.6%

      \[\leadsto u \cdot \left(\color{blue}{s \cdot \left(8 \cdot u\right)} + s \cdot 4\right) \]

   5. *-commutative 86.6%

      \[\leadsto u \cdot \left(s \cdot \color{blue}{\left(u \cdot 8\right)} + s \cdot 4\right) \]

   6. *-commutative 86.6%

      \[\leadsto u \cdot \left(s \cdot \left(u \cdot 8\right) + \color{blue}{4 \cdot s}\right) \]

9. Simplified 86.6%

   \[\leadsto \color{blue}{u \cdot \left(s \cdot \left(u \cdot 8\right) + 4 \cdot s\right)} \]

10. Taylor expanded in s around 0 86.4%

    \[\leadsto \color{blue}{s \cdot \left(u \cdot \left(4 + 8 \cdot u\right)\right)} \]

11. Final simplification 86.4%

    \[\leadsto s \cdot \left(u \cdot \left(u \cdot 8 + 4\right)\right) \]
12. Add Preprocessing

Alternative 4: 86.9% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ u \cdot \left(s \cdot \left(u \cdot 8 + 4\right)\right) \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (* u (* s (+ (* u 8.0) 4.0))))

C

float code(float s, float u) {
	return u * (s * ((u * 8.0f) + 4.0f));
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = u * (s * ((u * 8.0e0) + 4.0e0))
end function

Julia

function code(s, u)
	return Float32(u * Float32(s * Float32(Float32(u * Float32(8.0)) + Float32(4.0))))
end

MATLAB

function tmp = code(s, u)
	tmp = u * (s * ((u * single(8.0)) + single(4.0)));
end

Derivation

1. Initial program 62.2%

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

   1. log-rec 64.6%

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

   2. cancel-sign-sub-inv 64.6%

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

   3. metadata-eval 64.6%

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

3. Simplified 64.6%

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

5. Taylor expanded in u around 0 86.1%

   \[\leadsto \color{blue}{4 \cdot \left(s \cdot u\right) + 8 \cdot \left(s \cdot {u}^{2}\right)} \]
6. Step-by-step derivation

   1. +-commutative 86.1%

      \[\leadsto \color{blue}{8 \cdot \left(s \cdot {u}^{2}\right) + 4 \cdot \left(s \cdot u\right)} \]

   2. associate-*r* 86.1%

      \[\leadsto \color{blue}{\left(8 \cdot s\right) \cdot {u}^{2}} + 4 \cdot \left(s \cdot u\right) \]

   3. unpow2 86.1%

      \[\leadsto \left(8 \cdot s\right) \cdot \color{blue}{\left(u \cdot u\right)} + 4 \cdot \left(s \cdot u\right) \]

   4. associate-*r* 86.1%

      \[\leadsto \color{blue}{\left(\left(8 \cdot s\right) \cdot u\right) \cdot u} + 4 \cdot \left(s \cdot u\right) \]

   5. fma-define 86.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(8 \cdot s\right) \cdot u, u, 4 \cdot \left(s \cdot u\right)\right)} \]

   6. *-commutative 86.1%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(s \cdot 8\right)} \cdot u, u, 4 \cdot \left(s \cdot u\right)\right) \]

   7. *-commutative 86.1%

      \[\leadsto \mathsf{fma}\left(\left(s \cdot 8\right) \cdot u, u, \color{blue}{\left(s \cdot u\right) \cdot 4}\right) \]

   8. *-commutative 86.1%

      \[\leadsto \mathsf{fma}\left(\left(s \cdot 8\right) \cdot u, u, \color{blue}{\left(u \cdot s\right)} \cdot 4\right) \]

   9. associate-*l* 86.5%

      \[\leadsto \mathsf{fma}\left(\left(s \cdot 8\right) \cdot u, u, \color{blue}{u \cdot \left(s \cdot 4\right)}\right) \]

7. Applied egg-rr 86.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\left(s \cdot 8\right) \cdot u, u, u \cdot \left(s \cdot 4\right)\right)} \]
8. Step-by-step derivation

   1. fma-undefine 86.5%

      \[\leadsto \color{blue}{\left(\left(s \cdot 8\right) \cdot u\right) \cdot u + u \cdot \left(s \cdot 4\right)} \]

   2. *-commutative 86.5%

      \[\leadsto \left(\left(s \cdot 8\right) \cdot u\right) \cdot u + \color{blue}{\left(s \cdot 4\right) \cdot u} \]

   3. distribute-rgt-out 86.6%

      \[\leadsto \color{blue}{u \cdot \left(\left(s \cdot 8\right) \cdot u + s \cdot 4\right)} \]

   4. associate-*l* 86.6%

      \[\leadsto u \cdot \left(\color{blue}{s \cdot \left(8 \cdot u\right)} + s \cdot 4\right) \]

   5. *-commutative 86.6%

      \[\leadsto u \cdot \left(s \cdot \color{blue}{\left(u \cdot 8\right)} + s \cdot 4\right) \]

   6. *-commutative 86.6%

      \[\leadsto u \cdot \left(s \cdot \left(u \cdot 8\right) + \color{blue}{4 \cdot s}\right) \]

9. Simplified 86.6%

   \[\leadsto \color{blue}{u \cdot \left(s \cdot \left(u \cdot 8\right) + 4 \cdot s\right)} \]

10. Taylor expanded in s around 0 86.4%

    \[\leadsto u \cdot \color{blue}{\left(s \cdot \left(4 + 8 \cdot u\right)\right)} \]
11. Step-by-step derivation

    1. *-commutative 86.4%

       \[\leadsto u \cdot \left(s \cdot \left(4 + \color{blue}{u \cdot 8}\right)\right) \]

12. Simplified 86.4%

    \[\leadsto u \cdot \color{blue}{\left(s \cdot \left(4 + u \cdot 8\right)\right)} \]

13. Final simplification 86.4%

    \[\leadsto u \cdot \left(s \cdot \left(u \cdot 8 + 4\right)\right) \]
14. Add Preprocessing

Alternative 5: 73.9% accurate, 21.8× speedup

Math

\[\begin{array}{l} \\ 4 \cdot \left(s \cdot u\right) \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (* 4.0 (* s u)))

C

float code(float s, float u) {
	return 4.0f * (s * u);
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = 4.0e0 * (s * u)
end function

Julia

function code(s, u)
	return Float32(Float32(4.0) * Float32(s * u))
end

MATLAB

function tmp = code(s, u)
	tmp = single(4.0) * (s * u);
end

Derivation

1. Initial program 62.2%

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

   1. log-rec 64.6%

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

   2. cancel-sign-sub-inv 64.6%

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

   3. metadata-eval 64.6%

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

3. Simplified 64.6%

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

5. Taylor expanded in u around 0 73.4%

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

   1. *-commutative 73.4%

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

7. Simplified 73.4%

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

8. Final simplification 73.4%

   \[\leadsto 4 \cdot \left(s \cdot u\right) \]
9. Add Preprocessing

Alternative 6: 74.1% accurate, 21.8× speedup

Math

\[\begin{array}{l} \\ u \cdot \left(s \cdot 4\right) \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (* u (* s 4.0)))

C

float code(float s, float u) {
	return u * (s * 4.0f);
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = u * (s * 4.0e0)
end function

Julia

function code(s, u)
	return Float32(u * Float32(s * Float32(4.0)))
end

MATLAB

function tmp = code(s, u)
	tmp = u * (s * single(4.0));
end

Derivation

1. Initial program 62.2%

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

   1. log-rec 64.6%

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

   2. cancel-sign-sub-inv 64.6%

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

   3. metadata-eval 64.6%

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

3. Simplified 64.6%

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

5. Taylor expanded in u around 0 73.4%

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

   1. associate-*r* 73.5%

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

   2. *-commutative 73.5%

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

7. Simplified 73.5%

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

8. Final simplification 73.5%

   \[\leadsto u \cdot \left(s \cdot 4\right) \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024041 
(FPCore (s u)
  :name "Disney BSSRDF, sample scattering profile, lower"
  :precision binary32
  :pre (and (and (<= 0.0 s) (<= s 256.0)) (and (<= 2.328306437e-10 u) (<= u 0.25)))
  (* s (log (/ 1.0 (- 1.0 (* 4.0 u))))))