Disney BSSRDF, sample scattering profile, lower

Percentage Accurate: 61.5% → 99.4%

Time: 10.0s

Alternatives: 9

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: 61.5% 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

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

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(Float32(-s) * log1p(Float32(u * Float32(-4.0))))
end

Derivation

1. Initial program 63.6%

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

   1. log-rec 66.2%

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

   2. distribute-rgt-neg-out 66.2%

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

   3. distribute-lft-neg-in 66.2%

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

   4. cancel-sign-sub-inv 66.2%

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

   5. log1p-def 99.4%

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

   6. *-commutative 99.4%

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

   7. metadata-eval 99.4%

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

3. Simplified 99.4%

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

4. Final simplification 99.4%

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

Alternative 2: 91.4% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ s \cdot \left(u \cdot 4 + \left(\left(u \cdot u\right) \cdot 8 + \left(u \cdot u\right) \cdot \left(u \cdot 21.333333333333332\right)\right)\right) \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (* s (+ (* u 4.0) (+ (* (* u u) 8.0) (* (* u u) (* u 21.333333333333332))))))

C

float code(float s, float u) {
	return s * ((u * 4.0f) + (((u * u) * 8.0f) + ((u * u) * (u * 21.333333333333332f))));
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = s * ((u * 4.0e0) + (((u * u) * 8.0e0) + ((u * u) * (u * 21.333333333333332e0))))
end function

Julia

function code(s, u)
	return Float32(s * Float32(Float32(u * Float32(4.0)) + Float32(Float32(Float32(u * u) * Float32(8.0)) + Float32(Float32(u * u) * Float32(u * Float32(21.333333333333332))))))
end

MATLAB

function tmp = code(s, u)
	tmp = s * ((u * single(4.0)) + (((u * u) * single(8.0)) + ((u * u) * (u * single(21.333333333333332)))));
end

Derivation

1. Initial program 63.6%

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

   1. log-rec 66.2%

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

   2. distribute-rgt-neg-out 66.2%

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

   3. distribute-lft-neg-in 66.2%

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

   4. cancel-sign-sub-inv 66.2%

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

   5. log1p-def 99.4%

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

   6. *-commutative 99.4%

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

   7. metadata-eval 99.4%

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

3. Simplified 99.4%

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

4. Taylor expanded in u around 0 90.6%

   \[\leadsto \color{blue}{4 \cdot \left(s \cdot u\right) + \left(8 \cdot \left(s \cdot {u}^{2}\right) + 21.333333333333332 \cdot \left(s \cdot {u}^{3}\right)\right)} \]
5. Step-by-step derivation

   1. *-commutative 90.6%

      \[\leadsto \color{blue}{\left(s \cdot u\right) \cdot 4} + \left(8 \cdot \left(s \cdot {u}^{2}\right) + 21.333333333333332 \cdot \left(s \cdot {u}^{3}\right)\right) \]

   2. associate-*l* 90.8%

      \[\leadsto \color{blue}{s \cdot \left(u \cdot 4\right)} + \left(8 \cdot \left(s \cdot {u}^{2}\right) + 21.333333333333332 \cdot \left(s \cdot {u}^{3}\right)\right) \]

   3. *-commutative 90.8%

      \[\leadsto s \cdot \left(u \cdot 4\right) + \left(8 \cdot \color{blue}{\left({u}^{2} \cdot s\right)} + 21.333333333333332 \cdot \left(s \cdot {u}^{3}\right)\right) \]

   4. associate-*r* 91.0%

      \[\leadsto s \cdot \left(u \cdot 4\right) + \left(\color{blue}{\left(8 \cdot {u}^{2}\right) \cdot s} + 21.333333333333332 \cdot \left(s \cdot {u}^{3}\right)\right) \]

   5. *-commutative 91.0%

      \[\leadsto s \cdot \left(u \cdot 4\right) + \left(\left(8 \cdot {u}^{2}\right) \cdot s + 21.333333333333332 \cdot \color{blue}{\left({u}^{3} \cdot s\right)}\right) \]

   6. associate-*r* 91.0%

      \[\leadsto s \cdot \left(u \cdot 4\right) + \left(\left(8 \cdot {u}^{2}\right) \cdot s + \color{blue}{\left(21.333333333333332 \cdot {u}^{3}\right) \cdot s}\right) \]

   7. distribute-rgt-out 91.0%

      \[\leadsto s \cdot \left(u \cdot 4\right) + \color{blue}{s \cdot \left(8 \cdot {u}^{2} + 21.333333333333332 \cdot {u}^{3}\right)} \]

   8. distribute-lft-out 90.9%

      \[\leadsto \color{blue}{s \cdot \left(u \cdot 4 + \left(8 \cdot {u}^{2} + 21.333333333333332 \cdot {u}^{3}\right)\right)} \]

   9. *-commutative 90.9%

      \[\leadsto s \cdot \left(u \cdot 4 + \left(\color{blue}{{u}^{2} \cdot 8} + 21.333333333333332 \cdot {u}^{3}\right)\right) \]

   10. *-commutative 90.9%

       \[\leadsto s \cdot \left(u \cdot 4 + \left({u}^{2} \cdot 8 + \color{blue}{{u}^{3} \cdot 21.333333333333332}\right)\right) \]

   11. unpow3 90.9%

       \[\leadsto s \cdot \left(u \cdot 4 + \left({u}^{2} \cdot 8 + \color{blue}{\left(\left(u \cdot u\right) \cdot u\right)} \cdot 21.333333333333332\right)\right) \]

   12. unpow2 90.9%

       \[\leadsto s \cdot \left(u \cdot 4 + \left({u}^{2} \cdot 8 + \left(\color{blue}{{u}^{2}} \cdot u\right) \cdot 21.333333333333332\right)\right) \]

   13. associate-*l* 90.9%

       \[\leadsto s \cdot \left(u \cdot 4 + \left({u}^{2} \cdot 8 + \color{blue}{{u}^{2} \cdot \left(u \cdot 21.333333333333332\right)}\right)\right) \]

   14. distribute-lft-out 90.9%

       \[\leadsto s \cdot \left(u \cdot 4 + \color{blue}{{u}^{2} \cdot \left(8 + u \cdot 21.333333333333332\right)}\right) \]

   15. unpow2 90.9%

       \[\leadsto s \cdot \left(u \cdot 4 + \color{blue}{\left(u \cdot u\right)} \cdot \left(8 + u \cdot 21.333333333333332\right)\right) \]

6. Simplified 90.9%

   \[\leadsto \color{blue}{s \cdot \left(u \cdot 4 + \left(u \cdot u\right) \cdot \left(8 + u \cdot 21.333333333333332\right)\right)} \]
7. Step-by-step derivation

   1. distribute-lft-in 90.9%

      \[\leadsto s \cdot \left(u \cdot 4 + \color{blue}{\left(\left(u \cdot u\right) \cdot 8 + \left(u \cdot u\right) \cdot \left(u \cdot 21.333333333333332\right)\right)}\right) \]

   2. *-commutative 90.9%

      \[\leadsto s \cdot \left(u \cdot 4 + \left(\left(u \cdot u\right) \cdot 8 + \left(u \cdot u\right) \cdot \color{blue}{\left(21.333333333333332 \cdot u\right)}\right)\right) \]

8. Applied egg-rr 90.9%

   \[\leadsto s \cdot \left(u \cdot 4 + \color{blue}{\left(\left(u \cdot u\right) \cdot 8 + \left(u \cdot u\right) \cdot \left(21.333333333333332 \cdot u\right)\right)}\right) \]

9. Final simplification 90.9%

   \[\leadsto s \cdot \left(u \cdot 4 + \left(\left(u \cdot u\right) \cdot 8 + \left(u \cdot u\right) \cdot \left(u \cdot 21.333333333333332\right)\right)\right) \]

Alternative 3: 91.4% accurate, 7.3× speedup

Math

\[\begin{array}{l} \\ s \cdot \left(u \cdot 4 + \left(u \cdot u\right) \cdot \left(8 + u \cdot 21.333333333333332\right)\right) \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (* s (+ (* u 4.0) (* (* u u) (+ 8.0 (* u 21.333333333333332))))))

C

float code(float s, float u) {
	return s * ((u * 4.0f) + ((u * u) * (8.0f + (u * 21.333333333333332f))));
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = s * ((u * 4.0e0) + ((u * u) * (8.0e0 + (u * 21.333333333333332e0))))
end function

Julia

function code(s, u)
	return Float32(s * Float32(Float32(u * Float32(4.0)) + Float32(Float32(u * u) * Float32(Float32(8.0) + Float32(u * Float32(21.333333333333332))))))
end

MATLAB

function tmp = code(s, u)
	tmp = s * ((u * single(4.0)) + ((u * u) * (single(8.0) + (u * single(21.333333333333332)))));
end

Derivation

1. Initial program 63.6%

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

   1. log-rec 66.2%

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

   2. distribute-rgt-neg-out 66.2%

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

   3. distribute-lft-neg-in 66.2%

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

   4. cancel-sign-sub-inv 66.2%

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

   5. log1p-def 99.4%

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

   6. *-commutative 99.4%

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

   7. metadata-eval 99.4%

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

3. Simplified 99.4%

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

4. Taylor expanded in u around 0 90.6%

   \[\leadsto \color{blue}{4 \cdot \left(s \cdot u\right) + \left(8 \cdot \left(s \cdot {u}^{2}\right) + 21.333333333333332 \cdot \left(s \cdot {u}^{3}\right)\right)} \]
5. Step-by-step derivation

   1. *-commutative 90.6%

      \[\leadsto \color{blue}{\left(s \cdot u\right) \cdot 4} + \left(8 \cdot \left(s \cdot {u}^{2}\right) + 21.333333333333332 \cdot \left(s \cdot {u}^{3}\right)\right) \]

   2. associate-*l* 90.8%

      \[\leadsto \color{blue}{s \cdot \left(u \cdot 4\right)} + \left(8 \cdot \left(s \cdot {u}^{2}\right) + 21.333333333333332 \cdot \left(s \cdot {u}^{3}\right)\right) \]

   3. *-commutative 90.8%

      \[\leadsto s \cdot \left(u \cdot 4\right) + \left(8 \cdot \color{blue}{\left({u}^{2} \cdot s\right)} + 21.333333333333332 \cdot \left(s \cdot {u}^{3}\right)\right) \]

   4. associate-*r* 91.0%

      \[\leadsto s \cdot \left(u \cdot 4\right) + \left(\color{blue}{\left(8 \cdot {u}^{2}\right) \cdot s} + 21.333333333333332 \cdot \left(s \cdot {u}^{3}\right)\right) \]

   5. *-commutative 91.0%

      \[\leadsto s \cdot \left(u \cdot 4\right) + \left(\left(8 \cdot {u}^{2}\right) \cdot s + 21.333333333333332 \cdot \color{blue}{\left({u}^{3} \cdot s\right)}\right) \]

   6. associate-*r* 91.0%

      \[\leadsto s \cdot \left(u \cdot 4\right) + \left(\left(8 \cdot {u}^{2}\right) \cdot s + \color{blue}{\left(21.333333333333332 \cdot {u}^{3}\right) \cdot s}\right) \]

   7. distribute-rgt-out 91.0%

      \[\leadsto s \cdot \left(u \cdot 4\right) + \color{blue}{s \cdot \left(8 \cdot {u}^{2} + 21.333333333333332 \cdot {u}^{3}\right)} \]

   8. distribute-lft-out 90.9%

      \[\leadsto \color{blue}{s \cdot \left(u \cdot 4 + \left(8 \cdot {u}^{2} + 21.333333333333332 \cdot {u}^{3}\right)\right)} \]

   9. *-commutative 90.9%

      \[\leadsto s \cdot \left(u \cdot 4 + \left(\color{blue}{{u}^{2} \cdot 8} + 21.333333333333332 \cdot {u}^{3}\right)\right) \]

   10. *-commutative 90.9%

       \[\leadsto s \cdot \left(u \cdot 4 + \left({u}^{2} \cdot 8 + \color{blue}{{u}^{3} \cdot 21.333333333333332}\right)\right) \]

   11. unpow3 90.9%

       \[\leadsto s \cdot \left(u \cdot 4 + \left({u}^{2} \cdot 8 + \color{blue}{\left(\left(u \cdot u\right) \cdot u\right)} \cdot 21.333333333333332\right)\right) \]

   12. unpow2 90.9%

       \[\leadsto s \cdot \left(u \cdot 4 + \left({u}^{2} \cdot 8 + \left(\color{blue}{{u}^{2}} \cdot u\right) \cdot 21.333333333333332\right)\right) \]

   13. associate-*l* 90.9%

       \[\leadsto s \cdot \left(u \cdot 4 + \left({u}^{2} \cdot 8 + \color{blue}{{u}^{2} \cdot \left(u \cdot 21.333333333333332\right)}\right)\right) \]

   14. distribute-lft-out 90.9%

       \[\leadsto s \cdot \left(u \cdot 4 + \color{blue}{{u}^{2} \cdot \left(8 + u \cdot 21.333333333333332\right)}\right) \]

   15. unpow2 90.9%

       \[\leadsto s \cdot \left(u \cdot 4 + \color{blue}{\left(u \cdot u\right)} \cdot \left(8 + u \cdot 21.333333333333332\right)\right) \]

6. Simplified 90.9%

   \[\leadsto \color{blue}{s \cdot \left(u \cdot 4 + \left(u \cdot u\right) \cdot \left(8 + u \cdot 21.333333333333332\right)\right)} \]

7. Final simplification 90.9%

   \[\leadsto s \cdot \left(u \cdot 4 + \left(u \cdot u\right) \cdot \left(8 + u \cdot 21.333333333333332\right)\right) \]

Alternative 4: 91.1% accurate, 8.4× speedup

Math

\[\begin{array}{l} \\ s \cdot \left(u \cdot \left(4 + u \cdot \left(8 + u \cdot 21.333333333333332\right)\right)\right) \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (* s (* u (+ 4.0 (* u (+ 8.0 (* u 21.333333333333332)))))))

C

float code(float s, float u) {
	return s * (u * (4.0f + (u * (8.0f + (u * 21.333333333333332f)))));
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = s * (u * (4.0e0 + (u * (8.0e0 + (u * 21.333333333333332e0)))))
end function

Julia

function code(s, u)
	return Float32(s * Float32(u * Float32(Float32(4.0) + Float32(u * Float32(Float32(8.0) + Float32(u * Float32(21.333333333333332)))))))
end

MATLAB

function tmp = code(s, u)
	tmp = s * (u * (single(4.0) + (u * (single(8.0) + (u * single(21.333333333333332))))));
end

Derivation

1. Initial program 63.6%

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

   1. log-rec 66.2%

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

   2. distribute-rgt-neg-out 66.2%

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

   3. distribute-lft-neg-in 66.2%

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

   4. cancel-sign-sub-inv 66.2%

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

   5. log1p-def 99.4%

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

   6. *-commutative 99.4%

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

   7. metadata-eval 99.4%

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

3. Simplified 99.4%

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

4. Taylor expanded in u around 0 90.6%

   \[\leadsto \color{blue}{4 \cdot \left(s \cdot u\right) + \left(8 \cdot \left(s \cdot {u}^{2}\right) + 21.333333333333332 \cdot \left(s \cdot {u}^{3}\right)\right)} \]
5. Step-by-step derivation

   1. *-commutative 90.6%

      \[\leadsto \color{blue}{\left(s \cdot u\right) \cdot 4} + \left(8 \cdot \left(s \cdot {u}^{2}\right) + 21.333333333333332 \cdot \left(s \cdot {u}^{3}\right)\right) \]

   2. associate-*l* 90.8%

      \[\leadsto \color{blue}{s \cdot \left(u \cdot 4\right)} + \left(8 \cdot \left(s \cdot {u}^{2}\right) + 21.333333333333332 \cdot \left(s \cdot {u}^{3}\right)\right) \]

   3. *-commutative 90.8%

      \[\leadsto s \cdot \left(u \cdot 4\right) + \left(8 \cdot \color{blue}{\left({u}^{2} \cdot s\right)} + 21.333333333333332 \cdot \left(s \cdot {u}^{3}\right)\right) \]

   4. associate-*r* 91.0%

      \[\leadsto s \cdot \left(u \cdot 4\right) + \left(\color{blue}{\left(8 \cdot {u}^{2}\right) \cdot s} + 21.333333333333332 \cdot \left(s \cdot {u}^{3}\right)\right) \]

   5. *-commutative 91.0%

      \[\leadsto s \cdot \left(u \cdot 4\right) + \left(\left(8 \cdot {u}^{2}\right) \cdot s + 21.333333333333332 \cdot \color{blue}{\left({u}^{3} \cdot s\right)}\right) \]

   6. associate-*r* 91.0%

      \[\leadsto s \cdot \left(u \cdot 4\right) + \left(\left(8 \cdot {u}^{2}\right) \cdot s + \color{blue}{\left(21.333333333333332 \cdot {u}^{3}\right) \cdot s}\right) \]

   7. distribute-rgt-out 91.0%

      \[\leadsto s \cdot \left(u \cdot 4\right) + \color{blue}{s \cdot \left(8 \cdot {u}^{2} + 21.333333333333332 \cdot {u}^{3}\right)} \]

   8. distribute-lft-out 90.9%

      \[\leadsto \color{blue}{s \cdot \left(u \cdot 4 + \left(8 \cdot {u}^{2} + 21.333333333333332 \cdot {u}^{3}\right)\right)} \]

   9. *-commutative 90.9%

      \[\leadsto s \cdot \left(u \cdot 4 + \left(\color{blue}{{u}^{2} \cdot 8} + 21.333333333333332 \cdot {u}^{3}\right)\right) \]

   10. *-commutative 90.9%

       \[\leadsto s \cdot \left(u \cdot 4 + \left({u}^{2} \cdot 8 + \color{blue}{{u}^{3} \cdot 21.333333333333332}\right)\right) \]

   11. unpow3 90.9%

       \[\leadsto s \cdot \left(u \cdot 4 + \left({u}^{2} \cdot 8 + \color{blue}{\left(\left(u \cdot u\right) \cdot u\right)} \cdot 21.333333333333332\right)\right) \]

   12. unpow2 90.9%

       \[\leadsto s \cdot \left(u \cdot 4 + \left({u}^{2} \cdot 8 + \left(\color{blue}{{u}^{2}} \cdot u\right) \cdot 21.333333333333332\right)\right) \]

   13. associate-*l* 90.9%

       \[\leadsto s \cdot \left(u \cdot 4 + \left({u}^{2} \cdot 8 + \color{blue}{{u}^{2} \cdot \left(u \cdot 21.333333333333332\right)}\right)\right) \]

   14. distribute-lft-out 90.9%

       \[\leadsto s \cdot \left(u \cdot 4 + \color{blue}{{u}^{2} \cdot \left(8 + u \cdot 21.333333333333332\right)}\right) \]

   15. unpow2 90.9%

       \[\leadsto s \cdot \left(u \cdot 4 + \color{blue}{\left(u \cdot u\right)} \cdot \left(8 + u \cdot 21.333333333333332\right)\right) \]

6. Simplified 90.9%

   \[\leadsto \color{blue}{s \cdot \left(u \cdot 4 + \left(u \cdot u\right) \cdot \left(8 + u \cdot 21.333333333333332\right)\right)} \]

7. Taylor expanded in s around 0 90.9%

   \[\leadsto \color{blue}{s \cdot \left(4 \cdot u + {u}^{2} \cdot \left(8 + 21.333333333333332 \cdot u\right)\right)} \]
8. Step-by-step derivation

   1. unpow2 90.9%

      \[\leadsto s \cdot \left(4 \cdot u + \color{blue}{\left(u \cdot u\right)} \cdot \left(8 + 21.333333333333332 \cdot u\right)\right) \]

   2. +-commutative 90.9%

      \[\leadsto s \cdot \left(4 \cdot u + \left(u \cdot u\right) \cdot \color{blue}{\left(21.333333333333332 \cdot u + 8\right)}\right) \]

   3. fma-udef 90.9%

      \[\leadsto s \cdot \left(4 \cdot u + \left(u \cdot u\right) \cdot \color{blue}{\mathsf{fma}\left(21.333333333333332, u, 8\right)}\right) \]

   4. associate-*r* 90.9%

      \[\leadsto s \cdot \left(4 \cdot u + \color{blue}{u \cdot \left(u \cdot \mathsf{fma}\left(21.333333333333332, u, 8\right)\right)}\right) \]

   5. *-commutative 90.9%

      \[\leadsto s \cdot \left(\color{blue}{u \cdot 4} + u \cdot \left(u \cdot \mathsf{fma}\left(21.333333333333332, u, 8\right)\right)\right) \]

   6. distribute-lft-out 91.0%

      \[\leadsto \color{blue}{s \cdot \left(u \cdot 4\right) + s \cdot \left(u \cdot \left(u \cdot \mathsf{fma}\left(21.333333333333332, u, 8\right)\right)\right)} \]

   7. +-commutative 91.0%

      \[\leadsto \color{blue}{s \cdot \left(u \cdot \left(u \cdot \mathsf{fma}\left(21.333333333333332, u, 8\right)\right)\right) + s \cdot \left(u \cdot 4\right)} \]

   8. distribute-lft-in 90.9%

      \[\leadsto \color{blue}{s \cdot \left(u \cdot \left(u \cdot \mathsf{fma}\left(21.333333333333332, u, 8\right)\right) + u \cdot 4\right)} \]

   9. distribute-lft-out 90.7%

      \[\leadsto s \cdot \color{blue}{\left(u \cdot \left(u \cdot \mathsf{fma}\left(21.333333333333332, u, 8\right) + 4\right)\right)} \]

   10. fma-udef 90.7%

       \[\leadsto s \cdot \left(u \cdot \left(u \cdot \color{blue}{\left(21.333333333333332 \cdot u + 8\right)} + 4\right)\right) \]

   11. *-commutative 90.7%

       \[\leadsto s \cdot \left(u \cdot \left(u \cdot \left(\color{blue}{u \cdot 21.333333333333332} + 8\right) + 4\right)\right) \]

   12. fma-def 90.7%

       \[\leadsto s \cdot \left(u \cdot \left(u \cdot \color{blue}{\mathsf{fma}\left(u, 21.333333333333332, 8\right)} + 4\right)\right) \]

9. Simplified 90.7%

   \[\leadsto \color{blue}{s \cdot \left(u \cdot \left(u \cdot \mathsf{fma}\left(u, 21.333333333333332, 8\right) + 4\right)\right)} \]

10. Taylor expanded in s around 0 90.7%

    \[\leadsto \color{blue}{s \cdot \left(u \cdot \left(4 + u \cdot \left(8 + 21.333333333333332 \cdot u\right)\right)\right)} \]

11. Final simplification 90.7%

    \[\leadsto s \cdot \left(u \cdot \left(4 + u \cdot \left(8 + u \cdot 21.333333333333332\right)\right)\right) \]

Alternative 5: 87.1% accurate, 9.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 63.6%

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

   1. log-rec 66.2%

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

   2. distribute-rgt-neg-out 66.2%

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

   3. distribute-lft-neg-in 66.2%

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

   4. cancel-sign-sub-inv 66.2%

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

   5. log1p-def 99.4%

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

   6. *-commutative 99.4%

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

   7. metadata-eval 99.4%

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

3. Simplified 99.4%

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

4. Taylor expanded in u around 0 85.3%

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

   1. *-commutative 85.3%

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

   2. associate-*l* 85.5%

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

   3. *-commutative 85.5%

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

   4. associate-*l* 85.7%

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

   5. distribute-lft-out 85.7%

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

   6. unpow2 85.7%

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

   7. associate-*l* 85.7%

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

6. Simplified 85.7%

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

7. Final simplification 85.7%

   \[\leadsto s \cdot \left(u \cdot 4 + u \cdot \left(u \cdot 8\right)\right) \]

Alternative 6: 86.9% accurate, 12.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 63.6%

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

   1. log-rec 66.2%

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

   2. distribute-rgt-neg-out 66.2%

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

   3. distribute-lft-neg-in 66.2%

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

   4. cancel-sign-sub-inv 66.2%

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

   5. log1p-def 99.4%

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

   6. *-commutative 99.4%

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

   7. metadata-eval 99.4%

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

3. Simplified 99.4%

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

4. Taylor expanded in u around 0 85.3%

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

   1. *-commutative 85.3%

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

   2. associate-*l* 85.5%

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

   3. *-commutative 85.5%

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

   4. associate-*l* 85.7%

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

   5. distribute-lft-out 85.7%

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

   6. unpow2 85.7%

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

   7. associate-*l* 85.7%

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

   8. distribute-lft-out 85.6%

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

6. Simplified 85.6%

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

7. Final simplification 85.6%

   \[\leadsto s \cdot \left(u \cdot \left(4 + u \cdot 8\right)\right) \]

Alternative 7: 86.9% accurate, 12.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 63.6%

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

   1. log-rec 66.2%

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

   2. distribute-rgt-neg-out 66.2%

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

   3. distribute-lft-neg-in 66.2%

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

   4. cancel-sign-sub-inv 66.2%

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

   5. log1p-def 99.4%

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

   6. *-commutative 99.4%

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

   7. metadata-eval 99.4%

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

3. Simplified 99.4%

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

4. Taylor expanded in u around 0 85.3%

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

   1. *-commutative 85.3%

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

   2. associate-*l* 85.5%

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

   3. *-commutative 85.5%

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

   4. associate-*l* 85.7%

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

   5. distribute-lft-out 85.7%

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

   6. unpow2 85.7%

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

   7. associate-*l* 85.7%

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

   8. distribute-lft-out 85.6%

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

6. Simplified 85.6%

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

7. Taylor expanded in s around 0 85.6%

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

   1. associate-*r* 85.3%

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

   2. *-commutative 85.3%

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

   3. +-commutative 85.3%

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

   4. *-commutative 85.3%

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

   5. fma-udef 85.3%

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

   6. associate-*l* 85.6%

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

   7. fma-udef 85.6%

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

   8. *-commutative 85.6%

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

   9. fma-def 85.6%

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

9. Simplified 85.6%

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

10. Taylor expanded in s around 0 85.6%

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

11. Final simplification 85.6%

    \[\leadsto u \cdot \left(s \cdot \left(4 + u \cdot 8\right)\right) \]

Alternative 8: 73.8% 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 63.6%

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

   1. log-rec 66.2%

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

   2. distribute-rgt-neg-out 66.2%

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

   3. distribute-lft-neg-in 66.2%

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

   4. cancel-sign-sub-inv 66.2%

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

   5. log1p-def 99.4%

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

   6. *-commutative 99.4%

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

   7. metadata-eval 99.4%

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

3. Simplified 99.4%

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

4. Taylor expanded in u around 0 71.9%

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

5. Final simplification 71.9%

   \[\leadsto 4 \cdot \left(s \cdot u\right) \]

Alternative 9: 74.0% accurate, 21.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 63.6%

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

   1. log-rec 66.2%

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

   2. distribute-rgt-neg-out 66.2%

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

   3. distribute-lft-neg-in 66.2%

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

   4. cancel-sign-sub-inv 66.2%

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

   5. log1p-def 99.4%

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

   6. *-commutative 99.4%

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

   7. metadata-eval 99.4%

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

3. Simplified 99.4%

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

4. Taylor expanded in u around 0 71.9%

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

   1. *-commutative 71.9%

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

   2. associate-*l* 72.2%

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

6. Simplified 72.2%

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

7. Final simplification 72.2%

   \[\leadsto s \cdot \left(u \cdot 4\right) \]

Reproduce

herbie shell --seed 2023275 
(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))))))