Disney BSSRDF, sample scattering profile, lower

Percentage Accurate: 61.3% → 99.4%

Time: 6.7s

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: 61.3% 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 61.6%

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

   1. *-commutative 61.6%

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

   2. log-rec 64.2%

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

   3. distribute-lft-neg-out 64.2%

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

   4. distribute-rgt-neg-in 64.2%

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

   5. sub-neg 64.2%

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

   6. log1p-def 99.2%

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

   7. *-commutative 99.2%

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

   8. distribute-rgt-neg-in 99.2%

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

   9. metadata-eval 99.2%

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

3. Simplified 99.2%

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

4. Final simplification 99.2%

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

Alternative 2: 91.0% 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 61.6%

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

2. Taylor expanded in u around 0 90.1%

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

   1. associate-+r+ 90.1%

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

   2. +-commutative 90.1%

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

   3. *-commutative 90.1%

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

   4. unpow2 90.1%

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

   5. associate-*r* 90.1%

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

   6. unpow3 90.1%

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

   7. unpow2 90.1%

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

   8. associate-*r* 90.1%

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

   9. distribute-rgt-out 90.1%

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

   10. distribute-lft-out 90.0%

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

   11. unpow2 90.0%

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

   12. associate-*r* 90.0%

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

   13. *-commutative 90.0%

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

   14. distribute-rgt-out 90.0%

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

4. Simplified 90.0%

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

5. Final simplification 90.0%

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

Alternative 3: 88.1% 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 61.6%

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

2. Taylor expanded in u around 0 85.6%

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

   1. associate-*r* 85.9%

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

   2. associate-*r* 85.9%

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

   3. unpow2 85.9%

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

   4. associate-*r* 85.9%

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

   5. distribute-rgt-out 85.8%

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

   6. *-commutative 85.8%

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

   7. *-commutative 85.8%

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

   8. associate-*l* 85.8%

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

   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. associate-*r* 85.4%

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

   2. flip-+ 85.3%

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

   3. associate-*r/ 85.6%

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

   4. metadata-eval 85.6%

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

   5. swap-sqr 85.6%

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

   6. metadata-eval 85.6%

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

   7. *-commutative 85.6%

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

   8. cancel-sign-sub-inv 85.6%

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

   9. metadata-eval 85.6%

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

6. Applied egg-rr 85.6%

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

7. Taylor expanded in u around 0 87.6%

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

8. Taylor expanded in s around 0 87.2%

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

9. Final simplification 87.2%

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

Alternative 4: 88.8% accurate, 9.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, u)
	tmp = (s * (u * single(16.0))) / (single(4.0) + (u * single(-8.0)));
end

Derivation

1. Initial program 61.6%

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

2. Taylor expanded in u around 0 85.6%

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

   1. associate-*r* 85.9%

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

   2. associate-*r* 85.9%

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

   3. unpow2 85.9%

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

   4. associate-*r* 85.9%

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

   5. distribute-rgt-out 85.8%

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

   6. *-commutative 85.8%

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

   7. *-commutative 85.8%

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

   8. associate-*l* 85.8%

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

   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. associate-*r* 85.4%

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

   2. flip-+ 85.3%

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

   3. associate-*r/ 85.6%

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

   4. metadata-eval 85.6%

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

   5. swap-sqr 85.6%

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

   6. metadata-eval 85.6%

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

   7. *-commutative 85.6%

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

   8. cancel-sign-sub-inv 85.6%

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

   9. metadata-eval 85.6%

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

6. Applied egg-rr 85.6%

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

7. Taylor expanded in u around 0 87.6%

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

   1. *-commutative 87.6%

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

   2. associate-*l* 87.9%

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

9. Simplified 87.9%

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

10. Final simplification 87.9%

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

Alternative 5: 86.8% 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 61.6%

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

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 85.6%

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

   5. *-commutative 85.6%

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

4. Simplified 85.6%

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

5. Final simplification 85.6%

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

Alternative 6: 86.8% 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 61.6%

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

2. Taylor expanded in u around 0 85.6%

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

   1. associate-*r* 85.9%

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

   2. associate-*r* 85.9%

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

   3. unpow2 85.9%

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

   4. associate-*r* 85.9%

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

   5. distribute-rgt-out 85.8%

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

   6. *-commutative 85.8%

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

   7. *-commutative 85.8%

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

   8. associate-*l* 85.8%

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

   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: 73.8% 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 61.6%

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

2. Taylor expanded in u around 0 73.1%

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

   1. *-commutative 73.1%

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

4. Simplified 73.1%

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

5. Final simplification 73.1%

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

Alternative 8: 74.1% 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 61.6%

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

2. Taylor expanded in u around 0 73.1%

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

   1. associate-*r* 73.4%

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

   2. *-commutative 73.4%

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

4. Simplified 73.4%

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

5. Final simplification 73.4%

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

Alternative 9: 17.0% 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 61.6%

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

3. Applied egg-rr 16.5%

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

4. Final simplification 16.5%

   \[\leadsto s \cdot 0 \]

Reproduce

herbie shell --seed 2023195 
(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))))))