Disney BSSRDF, sample scattering profile, lower

Percentage Accurate: 62.0% → 99.4%

Time: 9.0s

Alternatives: 7

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: 62.0% 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

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 64.8%

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

   1. *-commutative 64.8%

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

   2. log-rec 67.8%

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

   3. distribute-lft-neg-out 67.8%

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

   4. distribute-rgt-neg-in 67.8%

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

   5. sub-neg 67.8%

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

   6. log1p-def 99.4%

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

   7. *-commutative 99.4%

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

   8. distribute-rgt-neg-in 99.4%

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

   9. metadata-eval 99.4%

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

3. Simplified 99.4%

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

4. Final simplification 99.4%

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

Alternative 2: 86.2% accurate, 8.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 64.8%

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

   1. flip3-- 64.4%

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

   2. associate-/r/ 64.5%

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

   3. log-prod 64.7%

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

   4. metadata-eval 64.7%

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

   5. *-commutative 64.7%

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

   6. unpow-prod-down 64.7%

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

   7. metadata-eval 64.7%

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

   8. metadata-eval 64.7%

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

   9. log1p-udef 95.4%

      \[\leadsto s \cdot \left(\log \left(\frac{1}{1 - {u}^{3} \cdot 64}\right) + \color{blue}{\mathsf{log1p}\left(\left(4 \cdot u\right) \cdot \left(4 \cdot u\right) + 1 \cdot \left(4 \cdot u\right)\right)}\right) \]

   10. *-un-lft-identity 95.4%

       \[\leadsto s \cdot \left(\log \left(\frac{1}{1 - {u}^{3} \cdot 64}\right) + \mathsf{log1p}\left(\left(4 \cdot u\right) \cdot \left(4 \cdot u\right) + \color{blue}{4 \cdot u}\right)\right) \]

   11. distribute-lft1-in 95.3%

       \[\leadsto s \cdot \left(\log \left(\frac{1}{1 - {u}^{3} \cdot 64}\right) + \mathsf{log1p}\left(\color{blue}{\left(4 \cdot u + 1\right) \cdot \left(4 \cdot u\right)}\right)\right) \]

   12. fma-def 95.3%

       \[\leadsto s \cdot \left(\log \left(\frac{1}{1 - {u}^{3} \cdot 64}\right) + \mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(4, u, 1\right)} \cdot \left(4 \cdot u\right)\right)\right) \]

3. Applied egg-rr 95.3%

   \[\leadsto s \cdot \color{blue}{\left(\log \left(\frac{1}{1 - {u}^{3} \cdot 64}\right) + \mathsf{log1p}\left(\mathsf{fma}\left(4, u, 1\right) \cdot \left(4 \cdot u\right)\right)\right)} \]
4. Step-by-step derivation

   1. +-commutative 95.3%

      \[\leadsto s \cdot \color{blue}{\left(\mathsf{log1p}\left(\mathsf{fma}\left(4, u, 1\right) \cdot \left(4 \cdot u\right)\right) + \log \left(\frac{1}{1 - {u}^{3} \cdot 64}\right)\right)} \]

   2. log-rec 96.0%

      \[\leadsto s \cdot \left(\mathsf{log1p}\left(\mathsf{fma}\left(4, u, 1\right) \cdot \left(4 \cdot u\right)\right) + \color{blue}{\left(-\log \left(1 - {u}^{3} \cdot 64\right)\right)}\right) \]

   3. sub-neg 96.0%

      \[\leadsto s \cdot \left(\mathsf{log1p}\left(\mathsf{fma}\left(4, u, 1\right) \cdot \left(4 \cdot u\right)\right) + \left(-\log \color{blue}{\left(1 + \left(-{u}^{3} \cdot 64\right)\right)}\right)\right) \]

   4. log1p-def 98.6%

      \[\leadsto s \cdot \left(\mathsf{log1p}\left(\mathsf{fma}\left(4, u, 1\right) \cdot \left(4 \cdot u\right)\right) + \left(-\color{blue}{\mathsf{log1p}\left(-{u}^{3} \cdot 64\right)}\right)\right) \]

   5. unsub-neg 98.6%

      \[\leadsto s \cdot \color{blue}{\left(\mathsf{log1p}\left(\mathsf{fma}\left(4, u, 1\right) \cdot \left(4 \cdot u\right)\right) - \mathsf{log1p}\left(-{u}^{3} \cdot 64\right)\right)} \]

   6. *-commutative 98.6%

      \[\leadsto s \cdot \left(\mathsf{log1p}\left(\color{blue}{\left(4 \cdot u\right) \cdot \mathsf{fma}\left(4, u, 1\right)}\right) - \mathsf{log1p}\left(-{u}^{3} \cdot 64\right)\right) \]

   7. associate-*l* 98.6%

      \[\leadsto s \cdot \left(\mathsf{log1p}\left(\color{blue}{4 \cdot \left(u \cdot \mathsf{fma}\left(4, u, 1\right)\right)}\right) - \mathsf{log1p}\left(-{u}^{3} \cdot 64\right)\right) \]

   8. distribute-rgt-neg-in 98.6%

      \[\leadsto s \cdot \left(\mathsf{log1p}\left(4 \cdot \left(u \cdot \mathsf{fma}\left(4, u, 1\right)\right)\right) - \mathsf{log1p}\left(\color{blue}{{u}^{3} \cdot \left(-64\right)}\right)\right) \]

   9. metadata-eval 98.6%

      \[\leadsto s \cdot \left(\mathsf{log1p}\left(4 \cdot \left(u \cdot \mathsf{fma}\left(4, u, 1\right)\right)\right) - \mathsf{log1p}\left({u}^{3} \cdot \color{blue}{-64}\right)\right) \]

5. Simplified 98.6%

   \[\leadsto s \cdot \color{blue}{\left(\mathsf{log1p}\left(4 \cdot \left(u \cdot \mathsf{fma}\left(4, u, 1\right)\right)\right) - \mathsf{log1p}\left({u}^{3} \cdot -64\right)\right)} \]

6. Taylor expanded in u around 0 84.8%

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

   1. associate-*r* 85.0%

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

   2. *-commutative 85.0%

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

   3. *-commutative 85.0%

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

   4. unpow2 85.0%

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

   5. fma-udef 85.3%

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

   6. *-commutative 85.3%

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

   7. associate-*r* 85.3%

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

   8. *-commutative 85.3%

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

8. Simplified 85.3%

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

   1. fma-udef 85.0%

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

   2. *-commutative 85.0%

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

10. Applied egg-rr 85.0%

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

11. Final simplification 85.0%

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

Alternative 3: 86.4% accurate, 8.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 64.8%

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

   1. flip3-- 64.4%

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

   2. associate-/r/ 64.5%

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

   3. log-prod 64.7%

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

   4. metadata-eval 64.7%

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

   5. *-commutative 64.7%

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

   6. unpow-prod-down 64.7%

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

   7. metadata-eval 64.7%

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

   8. metadata-eval 64.7%

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

   9. log1p-udef 95.4%

      \[\leadsto s \cdot \left(\log \left(\frac{1}{1 - {u}^{3} \cdot 64}\right) + \color{blue}{\mathsf{log1p}\left(\left(4 \cdot u\right) \cdot \left(4 \cdot u\right) + 1 \cdot \left(4 \cdot u\right)\right)}\right) \]

   10. *-un-lft-identity 95.4%

       \[\leadsto s \cdot \left(\log \left(\frac{1}{1 - {u}^{3} \cdot 64}\right) + \mathsf{log1p}\left(\left(4 \cdot u\right) \cdot \left(4 \cdot u\right) + \color{blue}{4 \cdot u}\right)\right) \]

   11. distribute-lft1-in 95.3%

       \[\leadsto s \cdot \left(\log \left(\frac{1}{1 - {u}^{3} \cdot 64}\right) + \mathsf{log1p}\left(\color{blue}{\left(4 \cdot u + 1\right) \cdot \left(4 \cdot u\right)}\right)\right) \]

   12. fma-def 95.3%

       \[\leadsto s \cdot \left(\log \left(\frac{1}{1 - {u}^{3} \cdot 64}\right) + \mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(4, u, 1\right)} \cdot \left(4 \cdot u\right)\right)\right) \]

3. Applied egg-rr 95.3%

   \[\leadsto s \cdot \color{blue}{\left(\log \left(\frac{1}{1 - {u}^{3} \cdot 64}\right) + \mathsf{log1p}\left(\mathsf{fma}\left(4, u, 1\right) \cdot \left(4 \cdot u\right)\right)\right)} \]
4. Step-by-step derivation

   1. +-commutative 95.3%

      \[\leadsto s \cdot \color{blue}{\left(\mathsf{log1p}\left(\mathsf{fma}\left(4, u, 1\right) \cdot \left(4 \cdot u\right)\right) + \log \left(\frac{1}{1 - {u}^{3} \cdot 64}\right)\right)} \]

   2. log-rec 96.0%

      \[\leadsto s \cdot \left(\mathsf{log1p}\left(\mathsf{fma}\left(4, u, 1\right) \cdot \left(4 \cdot u\right)\right) + \color{blue}{\left(-\log \left(1 - {u}^{3} \cdot 64\right)\right)}\right) \]

   3. sub-neg 96.0%

      \[\leadsto s \cdot \left(\mathsf{log1p}\left(\mathsf{fma}\left(4, u, 1\right) \cdot \left(4 \cdot u\right)\right) + \left(-\log \color{blue}{\left(1 + \left(-{u}^{3} \cdot 64\right)\right)}\right)\right) \]

   4. log1p-def 98.6%

      \[\leadsto s \cdot \left(\mathsf{log1p}\left(\mathsf{fma}\left(4, u, 1\right) \cdot \left(4 \cdot u\right)\right) + \left(-\color{blue}{\mathsf{log1p}\left(-{u}^{3} \cdot 64\right)}\right)\right) \]

   5. unsub-neg 98.6%

      \[\leadsto s \cdot \color{blue}{\left(\mathsf{log1p}\left(\mathsf{fma}\left(4, u, 1\right) \cdot \left(4 \cdot u\right)\right) - \mathsf{log1p}\left(-{u}^{3} \cdot 64\right)\right)} \]

   6. *-commutative 98.6%

      \[\leadsto s \cdot \left(\mathsf{log1p}\left(\color{blue}{\left(4 \cdot u\right) \cdot \mathsf{fma}\left(4, u, 1\right)}\right) - \mathsf{log1p}\left(-{u}^{3} \cdot 64\right)\right) \]

   7. associate-*l* 98.6%

      \[\leadsto s \cdot \left(\mathsf{log1p}\left(\color{blue}{4 \cdot \left(u \cdot \mathsf{fma}\left(4, u, 1\right)\right)}\right) - \mathsf{log1p}\left(-{u}^{3} \cdot 64\right)\right) \]

   8. distribute-rgt-neg-in 98.6%

      \[\leadsto s \cdot \left(\mathsf{log1p}\left(4 \cdot \left(u \cdot \mathsf{fma}\left(4, u, 1\right)\right)\right) - \mathsf{log1p}\left(\color{blue}{{u}^{3} \cdot \left(-64\right)}\right)\right) \]

   9. metadata-eval 98.6%

      \[\leadsto s \cdot \left(\mathsf{log1p}\left(4 \cdot \left(u \cdot \mathsf{fma}\left(4, u, 1\right)\right)\right) - \mathsf{log1p}\left({u}^{3} \cdot \color{blue}{-64}\right)\right) \]

5. Simplified 98.6%

   \[\leadsto s \cdot \color{blue}{\left(\mathsf{log1p}\left(4 \cdot \left(u \cdot \mathsf{fma}\left(4, u, 1\right)\right)\right) - \mathsf{log1p}\left({u}^{3} \cdot -64\right)\right)} \]

6. Taylor expanded in u around 0 84.8%

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

   1. associate-*r* 85.0%

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

   2. *-commutative 85.0%

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

   3. *-commutative 85.0%

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

   4. unpow2 85.0%

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

   5. fma-udef 85.3%

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

   6. *-commutative 85.3%

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

   7. associate-*r* 85.3%

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

   8. *-commutative 85.3%

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

8. Simplified 85.3%

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

9. Taylor expanded in u around 0 85.3%

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

    1. *-commutative 85.3%

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

    2. unpow2 85.3%

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

    3. associate-*r* 85.3%

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

    4. *-commutative 85.3%

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

    5. associate-*l* 85.4%

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

11. Simplified 85.4%

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

    1. fma-udef 85.2%

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

    2. *-commutative 85.2%

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

    3. *-commutative 85.2%

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

    4. *-commutative 85.2%

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

13. Applied egg-rr 85.2%

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

14. Final simplification 85.2%

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

Alternative 4: 85.9% accurate, 12.1× speedup

Math

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

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(Float32(u * 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 64.8%

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

   1. flip3-- 64.4%

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

   2. associate-/r/ 64.5%

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

   3. log-prod 64.7%

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

   4. metadata-eval 64.7%

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

   5. *-commutative 64.7%

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

   6. unpow-prod-down 64.7%

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

   7. metadata-eval 64.7%

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

   8. metadata-eval 64.7%

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

   9. log1p-udef 95.4%

      \[\leadsto s \cdot \left(\log \left(\frac{1}{1 - {u}^{3} \cdot 64}\right) + \color{blue}{\mathsf{log1p}\left(\left(4 \cdot u\right) \cdot \left(4 \cdot u\right) + 1 \cdot \left(4 \cdot u\right)\right)}\right) \]

   10. *-un-lft-identity 95.4%

       \[\leadsto s \cdot \left(\log \left(\frac{1}{1 - {u}^{3} \cdot 64}\right) + \mathsf{log1p}\left(\left(4 \cdot u\right) \cdot \left(4 \cdot u\right) + \color{blue}{4 \cdot u}\right)\right) \]

   11. distribute-lft1-in 95.3%

       \[\leadsto s \cdot \left(\log \left(\frac{1}{1 - {u}^{3} \cdot 64}\right) + \mathsf{log1p}\left(\color{blue}{\left(4 \cdot u + 1\right) \cdot \left(4 \cdot u\right)}\right)\right) \]

   12. fma-def 95.3%

       \[\leadsto s \cdot \left(\log \left(\frac{1}{1 - {u}^{3} \cdot 64}\right) + \mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(4, u, 1\right)} \cdot \left(4 \cdot u\right)\right)\right) \]

3. Applied egg-rr 95.3%

   \[\leadsto s \cdot \color{blue}{\left(\log \left(\frac{1}{1 - {u}^{3} \cdot 64}\right) + \mathsf{log1p}\left(\mathsf{fma}\left(4, u, 1\right) \cdot \left(4 \cdot u\right)\right)\right)} \]
4. Step-by-step derivation

   1. +-commutative 95.3%

      \[\leadsto s \cdot \color{blue}{\left(\mathsf{log1p}\left(\mathsf{fma}\left(4, u, 1\right) \cdot \left(4 \cdot u\right)\right) + \log \left(\frac{1}{1 - {u}^{3} \cdot 64}\right)\right)} \]

   2. log-rec 96.0%

      \[\leadsto s \cdot \left(\mathsf{log1p}\left(\mathsf{fma}\left(4, u, 1\right) \cdot \left(4 \cdot u\right)\right) + \color{blue}{\left(-\log \left(1 - {u}^{3} \cdot 64\right)\right)}\right) \]

   3. sub-neg 96.0%

      \[\leadsto s \cdot \left(\mathsf{log1p}\left(\mathsf{fma}\left(4, u, 1\right) \cdot \left(4 \cdot u\right)\right) + \left(-\log \color{blue}{\left(1 + \left(-{u}^{3} \cdot 64\right)\right)}\right)\right) \]

   4. log1p-def 98.6%

      \[\leadsto s \cdot \left(\mathsf{log1p}\left(\mathsf{fma}\left(4, u, 1\right) \cdot \left(4 \cdot u\right)\right) + \left(-\color{blue}{\mathsf{log1p}\left(-{u}^{3} \cdot 64\right)}\right)\right) \]

   5. unsub-neg 98.6%

      \[\leadsto s \cdot \color{blue}{\left(\mathsf{log1p}\left(\mathsf{fma}\left(4, u, 1\right) \cdot \left(4 \cdot u\right)\right) - \mathsf{log1p}\left(-{u}^{3} \cdot 64\right)\right)} \]

   6. *-commutative 98.6%

      \[\leadsto s \cdot \left(\mathsf{log1p}\left(\color{blue}{\left(4 \cdot u\right) \cdot \mathsf{fma}\left(4, u, 1\right)}\right) - \mathsf{log1p}\left(-{u}^{3} \cdot 64\right)\right) \]

   7. associate-*l* 98.6%

      \[\leadsto s \cdot \left(\mathsf{log1p}\left(\color{blue}{4 \cdot \left(u \cdot \mathsf{fma}\left(4, u, 1\right)\right)}\right) - \mathsf{log1p}\left(-{u}^{3} \cdot 64\right)\right) \]

   8. distribute-rgt-neg-in 98.6%

      \[\leadsto s \cdot \left(\mathsf{log1p}\left(4 \cdot \left(u \cdot \mathsf{fma}\left(4, u, 1\right)\right)\right) - \mathsf{log1p}\left(\color{blue}{{u}^{3} \cdot \left(-64\right)}\right)\right) \]

   9. metadata-eval 98.6%

      \[\leadsto s \cdot \left(\mathsf{log1p}\left(4 \cdot \left(u \cdot \mathsf{fma}\left(4, u, 1\right)\right)\right) - \mathsf{log1p}\left({u}^{3} \cdot \color{blue}{-64}\right)\right) \]

5. Simplified 98.6%

   \[\leadsto s \cdot \color{blue}{\left(\mathsf{log1p}\left(4 \cdot \left(u \cdot \mathsf{fma}\left(4, u, 1\right)\right)\right) - \mathsf{log1p}\left({u}^{3} \cdot -64\right)\right)} \]

6. Taylor expanded in u around 0 84.8%

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

   1. associate-*r* 85.0%

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

   2. *-commutative 85.0%

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

   3. *-commutative 85.0%

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

   4. unpow2 85.0%

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

   5. fma-udef 85.3%

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

   6. *-commutative 85.3%

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

   7. associate-*r* 85.3%

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

   8. *-commutative 85.3%

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

8. Simplified 85.3%

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

   1. fma-udef 85.0%

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

   2. *-commutative 85.0%

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

10. Applied egg-rr 85.0%

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

11. Taylor expanded in u around 0 84.8%

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

    1. +-commutative 84.8%

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

    2. unpow2 84.8%

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

    3. associate-*r* 84.8%

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

    4. *-commutative 84.8%

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

    5. associate-*r* 84.9%

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

    6. distribute-rgt-out 84.6%

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

    7. *-commutative 84.6%

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

13. Simplified 84.6%

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

14. Final simplification 84.6%

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

Alternative 5: 73.0% accurate, 21.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 64.8%

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

2. Taylor expanded in u around 0 70.4%

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

3. Final simplification 70.4%

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

Alternative 6: 73.2% 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 64.8%

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

2. Taylor expanded in u around 0 70.4%

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

   1. associate-*r* 70.6%

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

4. Simplified 70.6%

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

5. Final simplification 70.6%

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

Alternative 7: 16.6% accurate, 36.3× speedup

Math

\[\begin{array}{l} \\ s \cdot 0 \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (* s 0.0))

C

float code(float s, float u) {
	return s * 0.0f;
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = s * 0.0e0
end function

Julia

function code(s, u)
	return Float32(s * Float32(0.0))
end

MATLAB

function tmp = code(s, u)
	tmp = s * single(0.0);
end

Derivation

1. Initial program 64.8%

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

3. Applied egg-rr 16.6%

   \[\leadsto s \cdot \color{blue}{\left(0.5 \cdot \mathsf{log1p}\left(4 \cdot u\right) - 0.5 \cdot \mathsf{log1p}\left(4 \cdot u\right)\right)} \]
4. Step-by-step derivation

   1. +-inverses 16.6%

      \[\leadsto s \cdot \color{blue}{0} \]

5. Simplified 16.6%

   \[\leadsto s \cdot \color{blue}{0} \]

6. Final simplification 16.6%

   \[\leadsto s \cdot 0 \]

Reproduce

herbie shell --seed 2023230 
(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))))))