Disney BSSRDF, sample scattering profile, lower

Percentage Accurate: 61.4% → 99.3%

Time: 14.4s

Alternatives: 10

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: 61.4% 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: 99.3% accurate, 1.0× speedup

Math

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

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(s * Float32(-log1p(Float32(u * Float32(-4.0)))))
end

Derivation

1. Initial program 63.0%

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

   1. log-rec 66.3%

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

   2. distribute-rgt-neg-out 66.3%

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

   3. distribute-lft-neg-in 66.3%

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

   4. cancel-sign-sub-inv 66.3%

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

   5. log1p-def 99.3%

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

   6. *-commutative 99.3%

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

   7. metadata-eval 99.3%

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

3. Simplified 99.3%

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

4. Final simplification 99.3%

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

Alternative 2: 90.7% accurate, 6.4× speedup

Math

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

FPCore

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

C

float code(float s, float u) {
	return (4.0f * (s * u)) + (s * ((u * u) * (8.0f + (u * 21.333333333333332f))));
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = (4.0e0 * (s * u)) + (s * ((u * u) * (8.0e0 + (u * 21.333333333333332e0))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 63.0%

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

2. Taylor expanded in u around 0 90.3%

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

   1. fma-def 90.3%

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

   2. *-commutative 90.3%

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

   3. *-commutative 90.3%

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

   4. associate-*r* 90.4%

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

   5. *-commutative 90.4%

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

   6. associate-*r* 90.5%

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

   7. distribute-rgt-out 90.5%

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

   8. cube-mult 90.5%

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

   9. unpow2 90.5%

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

   10. associate-*r* 90.5%

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

   11. distribute-rgt-out 90.5%

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

   12. *-commutative 90.5%

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

   13. unpow2 90.5%

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

4. Simplified 90.5%

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

   1. fma-udef 90.5%

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

6. Applied egg-rr 90.5%

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

7. Final simplification 90.5%

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

Alternative 3: 90.7% 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 63.0%

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

2. Taylor expanded in u around 0 90.7%

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

   1. *-commutative 90.7%

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

   2. unpow2 90.7%

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

   3. associate-*r* 90.7%

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

   4. unpow3 90.7%

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

   5. unpow2 90.7%

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

   6. associate-*r* 90.7%

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

   7. distribute-rgt-out 90.7%

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

   8. distribute-lft-out 90.5%

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

   9. unpow2 90.5%

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

   10. associate-*r* 90.5%

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

   11. distribute-rgt-out 90.5%

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

   12. *-commutative 90.5%

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

4. Simplified 90.5%

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

5. Final simplification 90.5%

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

Alternative 4: 86.7% accurate, 9.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 63.0%

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

2. Taylor expanded in u around 0 86.2%

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

   1. unpow2 86.2%

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

   2. associate-*r* 86.2%

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

   3. distribute-rgt-out 86.1%

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

   4. *-commutative 86.1%

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

4. Simplified 86.1%

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

   1. distribute-rgt-in 86.2%

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

   2. *-commutative 86.2%

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

6. Applied egg-rr 86.2%

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

7. Final simplification 86.2%

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

Alternative 5: 86.7% accurate, 9.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 63.0%

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

2. Taylor expanded in u around 0 92.6%

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

   1. +-commutative 92.6%

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

   2. associate-+r+ 92.6%

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

   3. associate-+l+ 92.6%

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

   4. *-commutative 92.6%

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

   5. associate-*r* 92.7%

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

   6. *-commutative 92.7%

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

   7. associate-*r* 92.8%

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

   8. distribute-rgt-out 92.8%

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

   9. *-commutative 92.8%

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

   10. associate-*r* 92.8%

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

   11. *-commutative 92.8%

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

   12. associate-*r* 93.1%

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

   13. distribute-rgt-out 93.1%

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

4. Simplified 93.1%

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

5. Taylor expanded in u around 0 85.9%

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

   1. +-commutative 85.9%

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

   2. unpow2 85.9%

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

   3. *-commutative 85.9%

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

   4. associate-*l* 85.9%

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

   5. associate-*r* 86.0%

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

   6. *-commutative 86.0%

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

   7. distribute-rgt-in 85.8%

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

   8. *-commutative 85.8%

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

   9. fma-udef 85.8%

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

   10. associate-*r* 86.1%

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

7. Simplified 86.1%

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

8. Taylor expanded in u around 0 86.3%

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

9. Final simplification 86.3%

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

Alternative 6: 86.5% accurate, 12.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 63.0%

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

2. Taylor expanded in u around 0 86.2%

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

   1. unpow2 86.2%

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

   2. associate-*r* 86.2%

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

   3. distribute-rgt-out 86.1%

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

   4. *-commutative 86.1%

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

4. Simplified 86.1%

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

5. Final simplification 86.1%

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

Alternative 7: 86.5% 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 63.0%

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

2. Taylor expanded in u around 0 92.6%

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

   1. +-commutative 92.6%

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

   2. associate-+r+ 92.6%

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

   3. associate-+l+ 92.6%

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

   4. *-commutative 92.6%

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

   5. associate-*r* 92.7%

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

   6. *-commutative 92.7%

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

   7. associate-*r* 92.8%

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

   8. distribute-rgt-out 92.8%

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

   9. *-commutative 92.8%

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

   10. associate-*r* 92.8%

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

   11. *-commutative 92.8%

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

   12. associate-*r* 93.1%

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

   13. distribute-rgt-out 93.1%

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

4. Simplified 93.1%

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

5. Taylor expanded in u around 0 85.9%

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

   1. +-commutative 85.9%

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

   2. unpow2 85.9%

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

   3. *-commutative 85.9%

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

   4. associate-*l* 85.9%

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

   5. associate-*r* 86.0%

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

   6. *-commutative 86.0%

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

   7. distribute-rgt-in 85.8%

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

   8. *-commutative 85.8%

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

   9. fma-udef 85.8%

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

   10. associate-*r* 86.1%

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

7. Simplified 86.1%

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

8. Taylor expanded in s around 0 86.1%

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

9. Final simplification 86.1%

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

Alternative 8: 73.5% 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 63.0%

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

2. Taylor expanded in u around 0 72.3%

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

   1. *-commutative 72.3%

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

4. Simplified 72.3%

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

5. Final simplification 72.3%

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

Alternative 9: 73.7% 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 63.0%

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

2. Taylor expanded in u around 0 72.3%

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

   1. associate-*r* 72.5%

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

   2. *-commutative 72.5%

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

4. Simplified 72.5%

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

5. Final simplification 72.5%

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

Alternative 10: 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 63.0%

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

3. Applied egg-rr 16.1%

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

4. Final simplification 16.1%

   \[\leadsto s \cdot 0 \]

Reproduce

herbie shell --seed 2023287 
(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))))))