Disney BSSRDF, sample scattering profile, lower

Percentage Accurate: 62.0% → 99.4%

Time: 13.7s

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: 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

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

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 62.2%

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

   1. log-rec 65.2%

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

   2. distribute-rgt-neg-out 65.2%

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

   3. distribute-lft-neg-out 65.2%

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

   4. cancel-sign-sub-inv 65.2%

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

   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. Add Preprocessing

5. Final simplification 99.4%

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

Alternative 2: 90.8% 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 62.2%

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

3. Taylor expanded in u around 0 90.5%

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

   1. +-commutative 90.5%

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

   2. unpow2 90.5%

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

   3. associate-*r* 90.5%

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

   4. unpow3 90.5%

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

   5. unpow2 90.5%

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

   6. associate-*r* 90.5%

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

   7. distribute-rgt-out 90.5%

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

   8. *-commutative 90.5%

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

   9. distribute-lft-out 90.2%

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

   10. unpow2 90.2%

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

   11. associate-*r* 90.2%

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

   12. *-commutative 90.2%

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

   13. distribute-rgt-out 90.2%

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

5. Simplified 90.2%

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

6. Final simplification 90.2%

   \[\leadsto s \cdot \left(u \cdot \left(4 + u \cdot \left(8 + u \cdot 21.333333333333332\right)\right)\right) \]
7. Add Preprocessing

Alternative 3: 90.8% accurate, 8.4× speedup

Math

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

FPCore

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

C

float code(float s, float u) {
	return u * (s * (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 = u * (s * (4.0e0 + (u * (8.0e0 + (u * 21.333333333333332e0)))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 62.2%

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

3. Taylor expanded in u around 0 90.1%

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

   1. fma-def 90.1%

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

   2. *-commutative 90.1%

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

   3. *-commutative 90.1%

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

   4. associate-*r* 90.1%

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

   5. *-commutative 90.1%

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

   6. associate-*r* 90.1%

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

   7. distribute-rgt-out 90.2%

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

   8. cube-mult 90.2%

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

   9. unpow2 90.2%

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

   10. associate-*r* 90.2%

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

   11. distribute-rgt-out 90.2%

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

   12. *-commutative 90.2%

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

5. Simplified 90.2%

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

   1. fma-udef 90.2%

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

   2. associate-*r* 90.5%

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

   3. *-commutative 90.5%

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

   4. distribute-rgt-out 90.5%

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

   5. *-commutative 90.5%

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

   6. unpow2 90.5%

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

   7. associate-*r* 90.5%

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

   8. distribute-lft-in 90.2%

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

   9. +-commutative 90.2%

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

   10. *-commutative 90.2%

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

   11. associate-*r* 90.3%

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

   12. fma-def 90.3%

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

   13. +-commutative 90.3%

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

   14. fma-def 90.3%

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

7. Applied egg-rr 90.3%

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

8. Taylor expanded in s around 0 90.3%

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

9. Final simplification 90.3%

   \[\leadsto u \cdot \left(s \cdot \left(4 + u \cdot \left(8 + u \cdot 21.333333333333332\right)\right)\right) \]
10. Add Preprocessing

Alternative 4: 86.6% accurate, 9.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 62.2%

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

3. Taylor expanded in u around 0 92.5%

   \[\leadsto s \cdot \color{blue}{\left(4 \cdot u + \left(8 \cdot {u}^{2} + \left(21.333333333333332 \cdot {u}^{3} + 64 \cdot {u}^{4}\right)\right)\right)} \]
4. Step-by-step derivation

   1. fma-def 92.5%

      \[\leadsto s \cdot \color{blue}{\mathsf{fma}\left(4, u, 8 \cdot {u}^{2} + \left(21.333333333333332 \cdot {u}^{3} + 64 \cdot {u}^{4}\right)\right)} \]

   2. +-commutative 92.5%

      \[\leadsto s \cdot \mathsf{fma}\left(4, u, \color{blue}{\left(21.333333333333332 \cdot {u}^{3} + 64 \cdot {u}^{4}\right) + 8 \cdot {u}^{2}}\right) \]

   3. +-commutative 92.5%

      \[\leadsto s \cdot \mathsf{fma}\left(4, u, \color{blue}{\left(64 \cdot {u}^{4} + 21.333333333333332 \cdot {u}^{3}\right)} + 8 \cdot {u}^{2}\right) \]

   4. *-commutative 92.5%

      \[\leadsto s \cdot \mathsf{fma}\left(4, u, \left(\color{blue}{{u}^{4} \cdot 64} + 21.333333333333332 \cdot {u}^{3}\right) + 8 \cdot {u}^{2}\right) \]

   5. metadata-eval 92.5%

      \[\leadsto s \cdot \mathsf{fma}\left(4, u, \left({u}^{\color{blue}{\left(2 \cdot 2\right)}} \cdot 64 + 21.333333333333332 \cdot {u}^{3}\right) + 8 \cdot {u}^{2}\right) \]

   6. pow-sqr 92.5%

      \[\leadsto s \cdot \mathsf{fma}\left(4, u, \left(\color{blue}{\left({u}^{2} \cdot {u}^{2}\right)} \cdot 64 + 21.333333333333332 \cdot {u}^{3}\right) + 8 \cdot {u}^{2}\right) \]

   7. associate-*l* 92.5%

      \[\leadsto s \cdot \mathsf{fma}\left(4, u, \left(\color{blue}{{u}^{2} \cdot \left({u}^{2} \cdot 64\right)} + 21.333333333333332 \cdot {u}^{3}\right) + 8 \cdot {u}^{2}\right) \]

   8. *-commutative 92.5%

      \[\leadsto s \cdot \mathsf{fma}\left(4, u, \left({u}^{2} \cdot \left({u}^{2} \cdot 64\right) + \color{blue}{{u}^{3} \cdot 21.333333333333332}\right) + 8 \cdot {u}^{2}\right) \]

   9. unpow3 92.5%

      \[\leadsto s \cdot \mathsf{fma}\left(4, u, \left({u}^{2} \cdot \left({u}^{2} \cdot 64\right) + \color{blue}{\left(\left(u \cdot u\right) \cdot u\right)} \cdot 21.333333333333332\right) + 8 \cdot {u}^{2}\right) \]

   10. unpow2 92.5%

       \[\leadsto s \cdot \mathsf{fma}\left(4, u, \left({u}^{2} \cdot \left({u}^{2} \cdot 64\right) + \left(\color{blue}{{u}^{2}} \cdot u\right) \cdot 21.333333333333332\right) + 8 \cdot {u}^{2}\right) \]

   11. associate-*l* 92.5%

       \[\leadsto s \cdot \mathsf{fma}\left(4, u, \left({u}^{2} \cdot \left({u}^{2} \cdot 64\right) + \color{blue}{{u}^{2} \cdot \left(u \cdot 21.333333333333332\right)}\right) + 8 \cdot {u}^{2}\right) \]

   12. distribute-lft-out 92.5%

       \[\leadsto s \cdot \mathsf{fma}\left(4, u, \color{blue}{{u}^{2} \cdot \left({u}^{2} \cdot 64 + u \cdot 21.333333333333332\right)} + 8 \cdot {u}^{2}\right) \]

   13. *-commutative 92.5%

       \[\leadsto s \cdot \mathsf{fma}\left(4, u, {u}^{2} \cdot \left({u}^{2} \cdot 64 + u \cdot 21.333333333333332\right) + \color{blue}{{u}^{2} \cdot 8}\right) \]

   14. distribute-lft-out 92.5%

       \[\leadsto s \cdot \mathsf{fma}\left(4, u, \color{blue}{{u}^{2} \cdot \left(\left({u}^{2} \cdot 64 + u \cdot 21.333333333333332\right) + 8\right)}\right) \]

   15. unpow2 92.5%

       \[\leadsto s \cdot \mathsf{fma}\left(4, u, {u}^{2} \cdot \left(\left(\color{blue}{\left(u \cdot u\right)} \cdot 64 + u \cdot 21.333333333333332\right) + 8\right)\right) \]

   16. associate-*l* 92.5%

       \[\leadsto s \cdot \mathsf{fma}\left(4, u, {u}^{2} \cdot \left(\left(\color{blue}{u \cdot \left(u \cdot 64\right)} + u \cdot 21.333333333333332\right) + 8\right)\right) \]

   17. distribute-lft-out 92.5%

       \[\leadsto s \cdot \mathsf{fma}\left(4, u, {u}^{2} \cdot \left(\color{blue}{u \cdot \left(u \cdot 64 + 21.333333333333332\right)} + 8\right)\right) \]

5. Simplified 92.5%

   \[\leadsto s \cdot \color{blue}{\mathsf{fma}\left(4, u, {u}^{2} \cdot \left(u \cdot \left(u \cdot 64 + 21.333333333333332\right) + 8\right)\right)} \]

6. Taylor expanded in s around 0 92.5%

   \[\leadsto \color{blue}{s \cdot \left(4 \cdot u + {u}^{2} \cdot \left(8 + u \cdot \left(21.333333333333332 + 64 \cdot u\right)\right)\right)} \]

7. Taylor expanded in u around 0 86.1%

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

   1. *-commutative 86.1%

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

   2. associate-*r* 86.3%

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

   3. *-commutative 86.3%

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

   4. *-commutative 86.3%

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

   5. associate-*l* 86.4%

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

   6. distribute-rgt-out 86.3%

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

   7. +-commutative 86.3%

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

   8. *-commutative 86.3%

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

   9. unpow2 86.3%

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

   10. associate-*r* 86.3%

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

   11. distribute-lft-out 86.2%

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

   12. *-commutative 86.2%

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

9. Simplified 86.2%

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

    1. distribute-rgt-in 86.3%

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

    2. *-commutative 86.3%

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

    3. *-commutative 86.3%

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

11. Applied egg-rr 86.3%

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

12. Final simplification 86.3%

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

Alternative 5: 86.4% 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 62.2%

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

3. Taylor expanded in u around 0 86.3%

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

   1. unpow2 86.3%

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

   2. associate-*r* 86.3%

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

   3. distribute-rgt-out 86.2%

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

   4. *-commutative 86.2%

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

5. Simplified 86.2%

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

6. Final simplification 86.2%

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

Alternative 6: 86.4% 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 62.2%

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

3. Taylor expanded in u around 0 86.1%

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

   1. +-commutative 86.1%

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

   2. associate-*r* 86.1%

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

   3. unpow2 86.1%

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

   4. associate-*r* 86.1%

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

   5. associate-*r* 86.4%

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

   6. distribute-rgt-out 86.3%

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

   7. *-commutative 86.3%

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

   8. associate-*l* 86.3%

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

   9. *-commutative 86.3%

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

   10. distribute-lft-out 86.2%

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

   11. *-commutative 86.2%

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

5. Simplified 86.2%

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

6. Final simplification 86.2%

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

Alternative 7: 73.3% accurate, 21.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 62.2%

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

3. Taylor expanded in u around 0 72.6%

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

   1. *-commutative 72.6%

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

5. Simplified 72.6%

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

6. Final simplification 72.6%

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

Alternative 8: 73.5% accurate, 21.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 62.2%

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

3. Taylor expanded in u around 0 72.6%

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

   1. associate-*r* 72.8%

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

5. Simplified 72.8%

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

6. Final simplification 72.8%

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

Alternative 9: 16.7% 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 62.2%

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

   1. add-sqr-sqrt 59.3%

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

4. Applied egg-rr 16.7%

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

5. Final simplification 16.7%

   \[\leadsto s \cdot 0 \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024024 
(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))))))