Disney BSSRDF, sample scattering profile, lower

Percentage Accurate: 61.5% → 99.4%

Time: 17.6s

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: 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 62.7%

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

   1. log-rec 65.3%

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

   2. distribute-rgt-neg-out 65.3%

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

   3. distribute-lft-neg-in 65.3%

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

   4. cancel-sign-sub-inv 65.3%

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

   5. log1p-def 99.2%

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

   6. *-commutative 99.2%

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

   7. metadata-eval 99.2%

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

3. Simplified 99.2%

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

4. Final simplification 99.2%

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

Alternative 2: 86.7% accurate, 8.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 62.7%

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

2. Taylor expanded in u around 0 85.9%

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

   1. fma-def 85.9%

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

   2. *-commutative 85.9%

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

   3. *-commutative 85.9%

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

   4. unpow2 85.9%

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

4. Simplified 85.9%

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

   1. fma-udef 85.9%

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

   2. *-commutative 85.9%

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

   3. *-commutative 85.9%

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

   4. pow2 85.9%

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

   5. +-commutative 85.9%

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

   6. pow2 85.9%

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

   7. *-commutative 85.9%

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

   8. associate-*r* 86.0%

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

   9. *-commutative 86.0%

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

6. Applied egg-rr 86.0%

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

   1. +-commutative 86.0%

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

   2. associate-*r* 86.3%

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

   3. fma-def 86.5%

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

   4. *-commutative 86.5%

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

8. Applied egg-rr 86.5%

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

   1. fma-udef 86.3%

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

   2. associate-*r* 86.3%

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

10. Applied egg-rr 86.3%

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

11. Final simplification 86.3%

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

Alternative 3: 86.5% 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 62.7%

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

   1. flip3-- 62.5%

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

   2. associate-/r/ 62.4%

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

   3. log-prod 62.5%

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

   4. metadata-eval 62.5%

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

   5. *-commutative 62.5%

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

   6. unpow-prod-down 62.5%

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

   7. metadata-eval 62.5%

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

   8. metadata-eval 62.5%

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

   9. log1p-udef 96.2%

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

   10. *-un-lft-identity 96.2%

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

   11. distribute-lft1-in 96.1%

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

   12. fma-def 96.1%

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

3. Applied egg-rr 96.1%

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

   1. +-commutative 96.1%

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

   2. log-rec 96.7%

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

   3. sub-neg 96.7%

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

   4. log1p-def 98.8%

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

   5. unsub-neg 98.8%

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

   6. *-commutative 98.8%

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

   7. associate-*l* 98.8%

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

   8. distribute-rgt-neg-in 98.8%

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

   9. metadata-eval 98.8%

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

5. Simplified 98.8%

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

6. Taylor expanded in u around 0 85.9%

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

   1. +-commutative 85.9%

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

   2. *-commutative 85.9%

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

   3. associate-*l* 86.0%

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

   4. *-commutative 86.0%

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

   5. associate-*r* 86.3%

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

   6. distribute-rgt-in 86.1%

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

   7. *-commutative 86.1%

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

   8. +-commutative 86.1%

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

   9. unpow2 86.1%

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

   10. associate-*r* 86.1%

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

   11. distribute-rgt-out 85.9%

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

   12. *-commutative 85.9%

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

8. Simplified 85.9%

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

9. Final simplification 85.9%

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

Alternative 4: 73.5% accurate, 15.6× speedup

Math

\[\begin{array}{l} \\ s \cdot \left(0.3333333333333333 \cdot \left(u \cdot 12\right)\right) \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (* s (* 0.3333333333333333 (* u 12.0))))

C

float code(float s, float u) {
	return s * (0.3333333333333333f * (u * 12.0f));
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = s * (0.3333333333333333e0 * (u * 12.0e0))
end function

Julia

function code(s, u)
	return Float32(s * Float32(Float32(0.3333333333333333) * Float32(u * Float32(12.0))))
end

MATLAB

function tmp = code(s, u)
	tmp = s * (single(0.3333333333333333) * (u * single(12.0)));
end

Derivation

1. Initial program 62.7%

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

   1. add-sqr-sqrt 62.6%

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

   2. sqrt-unprod 62.7%

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

   3. log-rec 64.3%

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

   4. cancel-sign-sub-inv 64.3%

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

   5. metadata-eval 64.3%

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

   6. *-commutative 64.3%

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

   7. log1p-udef 65.0%

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

   8. log-rec 67.5%

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

   9. cancel-sign-sub-inv 67.5%

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

   10. metadata-eval 67.5%

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

   11. *-commutative 67.5%

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

   12. log1p-udef 99.2%

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

   13. sqr-neg 99.2%

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

   14. sqrt-unprod -0.0%

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

   15. add-sqr-sqrt 13.1%

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

   16. log1p-udef 13.5%

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

   17. *-commutative 13.5%

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

3. Applied egg-rr 70.1%

   \[\leadsto s \cdot \color{blue}{\left(0.3333333333333333 \cdot \left(3 \cdot \mathsf{log1p}\left(4 \cdot u\right)\right)\right)} \]

4. Taylor expanded in u around 0 72.4%

   \[\leadsto s \cdot \left(0.3333333333333333 \cdot \color{blue}{\left(12 \cdot u\right)}\right) \]
5. Step-by-step derivation

   1. *-commutative 72.4%

      \[\leadsto s \cdot \left(0.3333333333333333 \cdot \color{blue}{\left(u \cdot 12\right)}\right) \]

6. Simplified 72.4%

   \[\leadsto s \cdot \left(0.3333333333333333 \cdot \color{blue}{\left(u \cdot 12\right)}\right) \]

7. Final simplification 72.4%

   \[\leadsto s \cdot \left(0.3333333333333333 \cdot \left(u \cdot 12\right)\right) \]

Alternative 5: 73.4% 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 62.7%

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

2. Taylor expanded in u around 0 72.2%

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

   1. *-commutative 72.2%

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

4. Simplified 72.2%

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

5. Final simplification 72.2%

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

Alternative 6: 73.6% 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 62.7%

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

2. Taylor expanded in u around 0 72.4%

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

3. Final simplification 72.4%

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

Alternative 7: 16.5% 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 62.7%

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

3. Applied egg-rr 16.2%

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

   1. +-inverses 16.2%

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

5. Simplified 16.2%

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

6. Final simplification 16.2%

   \[\leadsto s \cdot 0 \]

Reproduce

herbie shell --seed 2023274 
(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))))))