Disney BSSRDF, sample scattering profile, lower

Percentage Accurate: 61.2% → 99.4%

Time: 8.7s

Alternatives: 11

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: 61.2% 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 63.1%

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

   1. *-commutative 63.1%

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

   2. log-rec 65.2%

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

   3. distribute-lft-neg-out 65.2%

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

   4. distribute-rgt-neg-in 65.2%

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

   5. sub-neg 65.2%

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

   6. log1p-def 99.4%

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

   7. *-commutative 99.4%

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

   8. distribute-rgt-neg-in 99.4%

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

   9. metadata-eval 99.4%

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

3. Simplified 99.4%

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

4. Final simplification 99.4%

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

Alternative 2: 91.0% accurate, 6.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 63.1%

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

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 89.9%

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

   11. unpow2 89.9%

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

   12. associate-*r* 89.9%

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

   13. *-commutative 89.9%

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

   14. distribute-rgt-out 89.9%

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

4. Simplified 89.9%

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

   1. +-commutative 89.9%

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

   2. distribute-lft-in 89.9%

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

6. Applied egg-rr 89.9%

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

   1. associate-+r+ 89.9%

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

   2. *-commutative 89.9%

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

   3. distribute-rgt-in 90.0%

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

8. Applied egg-rr 90.0%

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

9. Final simplification 90.0%

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

Alternative 3: 90.9% accurate, 7.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 63.1%

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

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 89.9%

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

   11. unpow2 89.9%

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

   12. associate-*r* 89.9%

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

   13. *-commutative 89.9%

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

   14. distribute-rgt-out 89.9%

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

4. Simplified 89.9%

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

   1. +-commutative 89.9%

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

   2. distribute-lft-in 89.9%

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

6. Applied egg-rr 89.9%

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

7. Final simplification 89.9%

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

Alternative 4: 90.9% accurate, 8.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 63.1%

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

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 89.9%

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

   11. unpow2 89.9%

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

   12. associate-*r* 89.9%

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

   13. *-commutative 89.9%

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

   14. distribute-rgt-out 89.9%

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

4. Simplified 89.9%

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

5. Final simplification 89.9%

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

Alternative 5: 86.8% 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 63.1%

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

2. Taylor expanded in u around 0 85.5%

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

   1. associate-*r* 85.7%

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

   2. associate-*r* 85.8%

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

   3. unpow2 85.8%

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

   4. associate-*r* 85.8%

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

   5. distribute-rgt-out 85.9%

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

   6. *-commutative 85.9%

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

   7. *-commutative 85.9%

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

   8. associate-*l* 85.9%

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

   9. distribute-lft-out 85.7%

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

   10. *-commutative 85.7%

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

4. Simplified 85.7%

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

5. Taylor expanded in s around 0 85.6%

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

   1. *-commutative 85.6%

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

   2. +-commutative 85.6%

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

   3. distribute-rgt-in 85.8%

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

7. Applied egg-rr 85.8%

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

8. Final simplification 85.8%

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

Alternative 6: 86.8% accurate, 9.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 63.1%

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

2. Taylor expanded in u around 0 85.5%

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

   1. associate-*r* 85.7%

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

   2. associate-*r* 85.8%

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

   3. unpow2 85.8%

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

   4. associate-*r* 85.8%

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

   5. distribute-rgt-out 85.9%

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

   6. *-commutative 85.9%

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

   7. *-commutative 85.9%

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

   8. associate-*l* 85.9%

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

   9. distribute-lft-out 85.7%

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

   10. *-commutative 85.7%

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

4. Simplified 85.7%

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

   1. +-commutative 85.7%

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

   2. distribute-rgt-in 85.9%

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

6. Applied egg-rr 85.9%

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

7. Final simplification 85.9%

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

Alternative 7: 86.6% 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 63.1%

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

2. Taylor expanded in u around 0 85.8%

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

   1. +-commutative 85.8%

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

   2. unpow2 85.8%

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

   3. associate-*r* 85.8%

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

   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 8: 86.6% 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 63.1%

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

2. Taylor expanded in u around 0 85.5%

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

   1. associate-*r* 85.7%

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

   2. associate-*r* 85.8%

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

   3. unpow2 85.8%

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

   4. associate-*r* 85.8%

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

   5. distribute-rgt-out 85.9%

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

   6. *-commutative 85.9%

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

   7. *-commutative 85.9%

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

   8. associate-*l* 85.9%

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

   9. distribute-lft-out 85.7%

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

   10. *-commutative 85.7%

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

4. Simplified 85.7%

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

5. Final simplification 85.7%

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

Alternative 9: 73.6% 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 63.1%

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

2. Taylor expanded in u around 0 72.8%

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

   1. *-commutative 72.8%

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

4. Simplified 72.8%

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

5. Final simplification 72.8%

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

Alternative 10: 73.8% 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 63.1%

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

2. Taylor expanded in u around 0 72.8%

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

   1. associate-*r* 73.0%

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

   2. *-commutative 73.0%

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

4. Simplified 73.0%

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

5. Final simplification 73.0%

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

Alternative 11: 16.3% 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 63.1%

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

   1. add-sqr-sqrt 63.1%

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

   2. sqrt-unprod 63.1%

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

   3. log-rec 64.7%

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

   4. cancel-sign-sub-inv 64.7%

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

   5. metadata-eval 64.7%

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

   6. *-commutative 64.7%

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

   7. log1p-udef 65.4%

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

   8. log-rec 67.4%

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

   9. cancel-sign-sub-inv 67.4%

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

   10. metadata-eval 67.4%

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

   11. *-commutative 67.4%

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

   12. log1p-udef 99.4%

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

   13. sqr-neg 99.4%

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

   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. *-commutative 13.1%

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

3. Applied egg-rr 16.2%

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

4. Final simplification 16.2%

   \[\leadsto s \cdot 0 \]

Reproduce

herbie shell --seed 2023181 
(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))))))