Disney BSSRDF, sample scattering profile, lower

Percentage Accurate: 60.7% → 99.4%

Time: 7.4s

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: 60.7% 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 62.2%

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

   1. *-commutative 62.2%

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

   2. log-rec 64.9%

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

   3. distribute-lft-neg-out 64.9%

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

   4. distribute-rgt-neg-in 64.9%

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

   5. sub-neg 64.9%

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

   6. log1p-def 99.3%

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

   7. *-commutative 99.3%

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

   8. distribute-rgt-neg-in 99.3%

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

   9. metadata-eval 99.3%

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

3. Simplified 99.3%

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

4. Final simplification 99.3%

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

Alternative 2: 91.2% 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. Taylor expanded in u around 0 89.8%

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

   1. associate-+r+ 89.8%

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

   2. +-commutative 89.8%

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

   3. *-commutative 89.8%

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

   4. unpow2 89.8%

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

   5. associate-*r* 89.8%

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

   6. unpow3 89.8%

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

   7. unpow2 89.8%

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

   8. associate-*r* 89.8%

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

   9. distribute-rgt-out 89.8%

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

   10. distribute-lft-out 89.7%

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

   11. unpow2 89.7%

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

   12. associate-*r* 89.7%

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

   13. *-commutative 89.7%

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

   14. distribute-rgt-out 89.7%

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

4. Simplified 89.7%

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

5. Final simplification 89.7%

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

Alternative 3: 88.3% accurate, 9.9× speedup

Math

\[\begin{array}{l} \\ 16 \cdot \frac{u \cdot s}{4 + u \cdot -8} \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (* 16.0 (/ (* u s) (+ 4.0 (* u -8.0)))))

C

float code(float s, float u) {
	return 16.0f * ((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 = 16.0e0 * ((u * s) / (4.0e0 + (u * (-8.0e0))))
end function

Julia

function code(s, u)
	return Float32(Float32(16.0) * Float32(Float32(u * s) / Float32(Float32(4.0) + Float32(u * Float32(-8.0)))))
end

MATLAB

function tmp = code(s, u)
	tmp = single(16.0) * ((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. Taylor expanded in u around 0 85.3%

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

   1. associate-*r* 85.6%

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

   2. associate-*r* 85.7%

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

   3. unpow2 85.7%

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

   4. associate-*r* 85.7%

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

   5. distribute-rgt-out 85.7%

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

   6. *-commutative 85.7%

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

   7. *-commutative 85.7%

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

   8. associate-*l* 85.7%

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

   9. distribute-lft-out 85.6%

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

   10. *-commutative 85.6%

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

4. Simplified 85.6%

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

   1. distribute-lft-in 85.7%

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

   2. *-commutative 85.7%

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

   3. flip-+ 59.9%

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

   4. swap-sqr 59.0%

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

   5. metadata-eval 59.0%

      \[\leadsto u \cdot \frac{\color{blue}{16} \cdot \left(s \cdot s\right) - \left(s \cdot \left(u \cdot 8\right)\right) \cdot \left(s \cdot \left(u \cdot 8\right)\right)}{4 \cdot s - s \cdot \left(u \cdot 8\right)} \]

   6. *-commutative 59.0%

      \[\leadsto u \cdot \frac{16 \cdot \left(s \cdot s\right) - \left(s \cdot \left(u \cdot 8\right)\right) \cdot \left(s \cdot \left(u \cdot 8\right)\right)}{\color{blue}{s \cdot 4} - s \cdot \left(u \cdot 8\right)} \]

6. Applied egg-rr 59.0%

   \[\leadsto u \cdot \color{blue}{\frac{16 \cdot \left(s \cdot s\right) - \left(s \cdot \left(u \cdot 8\right)\right) \cdot \left(s \cdot \left(u \cdot 8\right)\right)}{s \cdot 4 - s \cdot \left(u \cdot 8\right)}} \]
7. Step-by-step derivation

   1. swap-sqr 59.0%

      \[\leadsto u \cdot \frac{16 \cdot \left(s \cdot s\right) - \color{blue}{\left(s \cdot s\right) \cdot \left(\left(u \cdot 8\right) \cdot \left(u \cdot 8\right)\right)}}{s \cdot 4 - s \cdot \left(u \cdot 8\right)} \]

   2. *-commutative 59.0%

      \[\leadsto u \cdot \frac{\color{blue}{\left(s \cdot s\right) \cdot 16} - \left(s \cdot s\right) \cdot \left(\left(u \cdot 8\right) \cdot \left(u \cdot 8\right)\right)}{s \cdot 4 - s \cdot \left(u \cdot 8\right)} \]

   3. distribute-lft-out-- 59.0%

      \[\leadsto u \cdot \frac{\color{blue}{\left(s \cdot s\right) \cdot \left(16 - \left(u \cdot 8\right) \cdot \left(u \cdot 8\right)\right)}}{s \cdot 4 - s \cdot \left(u \cdot 8\right)} \]

   4. swap-sqr 59.0%

      \[\leadsto u \cdot \frac{\left(s \cdot s\right) \cdot \left(16 - \color{blue}{\left(u \cdot u\right) \cdot \left(8 \cdot 8\right)}\right)}{s \cdot 4 - s \cdot \left(u \cdot 8\right)} \]

   5. metadata-eval 59.0%

      \[\leadsto u \cdot \frac{\left(s \cdot s\right) \cdot \left(16 - \left(u \cdot u\right) \cdot \color{blue}{64}\right)}{s \cdot 4 - s \cdot \left(u \cdot 8\right)} \]

   6. metadata-eval 59.0%

      \[\leadsto u \cdot \frac{\left(s \cdot s\right) \cdot \left(16 - \left(u \cdot u\right) \cdot \color{blue}{\left(--64\right)}\right)}{s \cdot 4 - s \cdot \left(u \cdot 8\right)} \]

   7. distribute-rgt-neg-in 59.0%

      \[\leadsto u \cdot \frac{\left(s \cdot s\right) \cdot \left(16 - \color{blue}{\left(-\left(u \cdot u\right) \cdot -64\right)}\right)}{s \cdot 4 - s \cdot \left(u \cdot 8\right)} \]

   8. associate-*l* 59.0%

      \[\leadsto u \cdot \frac{\left(s \cdot s\right) \cdot \left(16 - \left(-\color{blue}{u \cdot \left(u \cdot -64\right)}\right)\right)}{s \cdot 4 - s \cdot \left(u \cdot 8\right)} \]

   9. distribute-rgt-neg-in 59.0%

      \[\leadsto u \cdot \frac{\left(s \cdot s\right) \cdot \left(16 - \color{blue}{u \cdot \left(-u \cdot -64\right)}\right)}{s \cdot 4 - s \cdot \left(u \cdot 8\right)} \]

   10. distribute-rgt-neg-in 59.0%

       \[\leadsto u \cdot \frac{\left(s \cdot s\right) \cdot \left(16 - u \cdot \color{blue}{\left(u \cdot \left(--64\right)\right)}\right)}{s \cdot 4 - s \cdot \left(u \cdot 8\right)} \]

   11. metadata-eval 59.0%

       \[\leadsto u \cdot \frac{\left(s \cdot s\right) \cdot \left(16 - u \cdot \left(u \cdot \color{blue}{64}\right)\right)}{s \cdot 4 - s \cdot \left(u \cdot 8\right)} \]

   12. distribute-lft-out-- 58.9%

       \[\leadsto u \cdot \frac{\left(s \cdot s\right) \cdot \left(16 - u \cdot \left(u \cdot 64\right)\right)}{\color{blue}{s \cdot \left(4 - u \cdot 8\right)}} \]

   13. *-commutative 58.9%

       \[\leadsto u \cdot \frac{\left(s \cdot s\right) \cdot \left(16 - u \cdot \left(u \cdot 64\right)\right)}{s \cdot \left(4 - \color{blue}{8 \cdot u}\right)} \]

   14. cancel-sign-sub-inv 58.9%

       \[\leadsto u \cdot \frac{\left(s \cdot s\right) \cdot \left(16 - u \cdot \left(u \cdot 64\right)\right)}{s \cdot \color{blue}{\left(4 + \left(-8\right) \cdot u\right)}} \]

   15. metadata-eval 58.9%

       \[\leadsto u \cdot \frac{\left(s \cdot s\right) \cdot \left(16 - u \cdot \left(u \cdot 64\right)\right)}{s \cdot \left(4 + \color{blue}{-8} \cdot u\right)} \]

8. Simplified 58.9%

   \[\leadsto u \cdot \color{blue}{\frac{\left(s \cdot s\right) \cdot \left(16 - u \cdot \left(u \cdot 64\right)\right)}{s \cdot \left(4 + -8 \cdot u\right)}} \]

9. Taylor expanded in u around 0 60.1%

   \[\leadsto u \cdot \frac{\color{blue}{16 \cdot {s}^{2}}}{s \cdot \left(4 + -8 \cdot u\right)} \]
10. Step-by-step derivation

    1. *-commutative 60.1%

       \[\leadsto u \cdot \frac{\color{blue}{{s}^{2} \cdot 16}}{s \cdot \left(4 + -8 \cdot u\right)} \]

    2. unpow2 60.1%

       \[\leadsto u \cdot \frac{\color{blue}{\left(s \cdot s\right)} \cdot 16}{s \cdot \left(4 + -8 \cdot u\right)} \]

    3. associate-*l* 61.0%

       \[\leadsto u \cdot \frac{\color{blue}{s \cdot \left(s \cdot 16\right)}}{s \cdot \left(4 + -8 \cdot u\right)} \]

11. Simplified 61.0%

    \[\leadsto u \cdot \frac{\color{blue}{s \cdot \left(s \cdot 16\right)}}{s \cdot \left(4 + -8 \cdot u\right)} \]

12. Taylor expanded in s around 0 86.6%

    \[\leadsto \color{blue}{16 \cdot \frac{s \cdot u}{4 + -8 \cdot u}} \]

13. Final simplification 86.6%

    \[\leadsto 16 \cdot \frac{u \cdot s}{4 + u \cdot -8} \]

Alternative 4: 89.1% accurate, 9.9× speedup

Math

\[\begin{array}{l} \\ u \cdot \left(16 \cdot \frac{s}{4 + u \cdot -8}\right) \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (* u (* 16.0 (/ s (+ 4.0 (* u -8.0))))))

C

float code(float s, float u) {
	return u * (16.0f * (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 * (16.0e0 * (s / (4.0e0 + (u * (-8.0e0)))))
end function

Julia

function code(s, u)
	return Float32(u * Float32(Float32(16.0) * Float32(s / Float32(Float32(4.0) + Float32(u * Float32(-8.0))))))
end

MATLAB

function tmp = code(s, u)
	tmp = u * (single(16.0) * (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. Taylor expanded in u around 0 85.3%

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

   1. associate-*r* 85.6%

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

   2. associate-*r* 85.7%

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

   3. unpow2 85.7%

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

   4. associate-*r* 85.7%

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

   5. distribute-rgt-out 85.7%

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

   6. *-commutative 85.7%

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

   7. *-commutative 85.7%

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

   8. associate-*l* 85.7%

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

   9. distribute-lft-out 85.6%

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

   10. *-commutative 85.6%

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

4. Simplified 85.6%

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

   1. distribute-lft-in 85.7%

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

   2. *-commutative 85.7%

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

   3. flip-+ 59.9%

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

   4. swap-sqr 59.0%

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

   5. metadata-eval 59.0%

      \[\leadsto u \cdot \frac{\color{blue}{16} \cdot \left(s \cdot s\right) - \left(s \cdot \left(u \cdot 8\right)\right) \cdot \left(s \cdot \left(u \cdot 8\right)\right)}{4 \cdot s - s \cdot \left(u \cdot 8\right)} \]

   6. *-commutative 59.0%

      \[\leadsto u \cdot \frac{16 \cdot \left(s \cdot s\right) - \left(s \cdot \left(u \cdot 8\right)\right) \cdot \left(s \cdot \left(u \cdot 8\right)\right)}{\color{blue}{s \cdot 4} - s \cdot \left(u \cdot 8\right)} \]

6. Applied egg-rr 59.0%

   \[\leadsto u \cdot \color{blue}{\frac{16 \cdot \left(s \cdot s\right) - \left(s \cdot \left(u \cdot 8\right)\right) \cdot \left(s \cdot \left(u \cdot 8\right)\right)}{s \cdot 4 - s \cdot \left(u \cdot 8\right)}} \]
7. Step-by-step derivation

   1. swap-sqr 59.0%

      \[\leadsto u \cdot \frac{16 \cdot \left(s \cdot s\right) - \color{blue}{\left(s \cdot s\right) \cdot \left(\left(u \cdot 8\right) \cdot \left(u \cdot 8\right)\right)}}{s \cdot 4 - s \cdot \left(u \cdot 8\right)} \]

   2. *-commutative 59.0%

      \[\leadsto u \cdot \frac{\color{blue}{\left(s \cdot s\right) \cdot 16} - \left(s \cdot s\right) \cdot \left(\left(u \cdot 8\right) \cdot \left(u \cdot 8\right)\right)}{s \cdot 4 - s \cdot \left(u \cdot 8\right)} \]

   3. distribute-lft-out-- 59.0%

      \[\leadsto u \cdot \frac{\color{blue}{\left(s \cdot s\right) \cdot \left(16 - \left(u \cdot 8\right) \cdot \left(u \cdot 8\right)\right)}}{s \cdot 4 - s \cdot \left(u \cdot 8\right)} \]

   4. swap-sqr 59.0%

      \[\leadsto u \cdot \frac{\left(s \cdot s\right) \cdot \left(16 - \color{blue}{\left(u \cdot u\right) \cdot \left(8 \cdot 8\right)}\right)}{s \cdot 4 - s \cdot \left(u \cdot 8\right)} \]

   5. metadata-eval 59.0%

      \[\leadsto u \cdot \frac{\left(s \cdot s\right) \cdot \left(16 - \left(u \cdot u\right) \cdot \color{blue}{64}\right)}{s \cdot 4 - s \cdot \left(u \cdot 8\right)} \]

   6. metadata-eval 59.0%

      \[\leadsto u \cdot \frac{\left(s \cdot s\right) \cdot \left(16 - \left(u \cdot u\right) \cdot \color{blue}{\left(--64\right)}\right)}{s \cdot 4 - s \cdot \left(u \cdot 8\right)} \]

   7. distribute-rgt-neg-in 59.0%

      \[\leadsto u \cdot \frac{\left(s \cdot s\right) \cdot \left(16 - \color{blue}{\left(-\left(u \cdot u\right) \cdot -64\right)}\right)}{s \cdot 4 - s \cdot \left(u \cdot 8\right)} \]

   8. associate-*l* 59.0%

      \[\leadsto u \cdot \frac{\left(s \cdot s\right) \cdot \left(16 - \left(-\color{blue}{u \cdot \left(u \cdot -64\right)}\right)\right)}{s \cdot 4 - s \cdot \left(u \cdot 8\right)} \]

   9. distribute-rgt-neg-in 59.0%

      \[\leadsto u \cdot \frac{\left(s \cdot s\right) \cdot \left(16 - \color{blue}{u \cdot \left(-u \cdot -64\right)}\right)}{s \cdot 4 - s \cdot \left(u \cdot 8\right)} \]

   10. distribute-rgt-neg-in 59.0%

       \[\leadsto u \cdot \frac{\left(s \cdot s\right) \cdot \left(16 - u \cdot \color{blue}{\left(u \cdot \left(--64\right)\right)}\right)}{s \cdot 4 - s \cdot \left(u \cdot 8\right)} \]

   11. metadata-eval 59.0%

       \[\leadsto u \cdot \frac{\left(s \cdot s\right) \cdot \left(16 - u \cdot \left(u \cdot \color{blue}{64}\right)\right)}{s \cdot 4 - s \cdot \left(u \cdot 8\right)} \]

   12. distribute-lft-out-- 58.9%

       \[\leadsto u \cdot \frac{\left(s \cdot s\right) \cdot \left(16 - u \cdot \left(u \cdot 64\right)\right)}{\color{blue}{s \cdot \left(4 - u \cdot 8\right)}} \]

   13. *-commutative 58.9%

       \[\leadsto u \cdot \frac{\left(s \cdot s\right) \cdot \left(16 - u \cdot \left(u \cdot 64\right)\right)}{s \cdot \left(4 - \color{blue}{8 \cdot u}\right)} \]

   14. cancel-sign-sub-inv 58.9%

       \[\leadsto u \cdot \frac{\left(s \cdot s\right) \cdot \left(16 - u \cdot \left(u \cdot 64\right)\right)}{s \cdot \color{blue}{\left(4 + \left(-8\right) \cdot u\right)}} \]

   15. metadata-eval 58.9%

       \[\leadsto u \cdot \frac{\left(s \cdot s\right) \cdot \left(16 - u \cdot \left(u \cdot 64\right)\right)}{s \cdot \left(4 + \color{blue}{-8} \cdot u\right)} \]

8. Simplified 58.9%

   \[\leadsto u \cdot \color{blue}{\frac{\left(s \cdot s\right) \cdot \left(16 - u \cdot \left(u \cdot 64\right)\right)}{s \cdot \left(4 + -8 \cdot u\right)}} \]

9. Taylor expanded in u around 0 60.1%

   \[\leadsto u \cdot \frac{\color{blue}{16 \cdot {s}^{2}}}{s \cdot \left(4 + -8 \cdot u\right)} \]
10. Step-by-step derivation

    1. *-commutative 60.1%

       \[\leadsto u \cdot \frac{\color{blue}{{s}^{2} \cdot 16}}{s \cdot \left(4 + -8 \cdot u\right)} \]

    2. unpow2 60.1%

       \[\leadsto u \cdot \frac{\color{blue}{\left(s \cdot s\right)} \cdot 16}{s \cdot \left(4 + -8 \cdot u\right)} \]

    3. associate-*l* 61.0%

       \[\leadsto u \cdot \frac{\color{blue}{s \cdot \left(s \cdot 16\right)}}{s \cdot \left(4 + -8 \cdot u\right)} \]

11. Simplified 61.0%

    \[\leadsto u \cdot \frac{\color{blue}{s \cdot \left(s \cdot 16\right)}}{s \cdot \left(4 + -8 \cdot u\right)} \]

12. Taylor expanded in s around 0 87.6%

    \[\leadsto u \cdot \color{blue}{\left(16 \cdot \frac{s}{4 + -8 \cdot u}\right)} \]

13. Final simplification 87.6%

    \[\leadsto u \cdot \left(16 \cdot \frac{s}{4 + u \cdot -8}\right) \]

Alternative 5: 87.0% 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. Taylor expanded in u around 0 85.7%

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

   1. +-commutative 85.7%

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

   2. unpow2 85.7%

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

   3. associate-*r* 85.7%

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

   4. distribute-rgt-out 85.5%

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

   5. *-commutative 85.5%

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

4. Simplified 85.5%

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

5. Final simplification 85.5%

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

Alternative 6: 87.0% 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. Taylor expanded in u around 0 85.3%

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

   1. associate-*r* 85.6%

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

   2. associate-*r* 85.7%

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

   3. unpow2 85.7%

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

   4. associate-*r* 85.7%

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

   5. distribute-rgt-out 85.7%

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

   6. *-commutative 85.7%

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

   7. *-commutative 85.7%

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

   8. associate-*l* 85.7%

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

   9. distribute-lft-out 85.6%

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

   10. *-commutative 85.6%

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

4. Simplified 85.6%

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

5. Final simplification 85.6%

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

Alternative 7: 74.0% 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 62.2%

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

2. Taylor expanded in u around 0 72.7%

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

   1. *-commutative 72.7%

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

4. Simplified 72.7%

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

5. Final simplification 72.7%

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

Alternative 8: 74.2% 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 62.2%

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

2. Taylor expanded in u around 0 72.7%

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

   1. associate-*r* 73.1%

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

   2. *-commutative 73.1%

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

4. Simplified 73.1%

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

5. Final simplification 73.1%

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

Alternative 9: 16.4% 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%

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

3. Applied egg-rr 18.0%

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

4. Final simplification 18.0%

   \[\leadsto s \cdot 0 \]

Reproduce

herbie shell --seed 2023200 
(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))))))