Disney BSSRDF, sample scattering profile, lower

Percentage Accurate: 61.1% → 99.4%

Time: 13.8s

Alternatives: 8

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: 61.1% 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

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

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 64.3%

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

   1. log-rec 66.6%

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

   2. distribute-rgt-neg-out 66.6%

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

   3. distribute-lft-neg-out 66.6%

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

   4. cancel-sign-sub-inv 66.6%

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

   5. log1p-def 99.4%

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

   6. *-commutative 99.4%

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

   7. metadata-eval 99.4%

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

3. Simplified 99.4%

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

5. Final simplification 99.4%

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

Alternative 2: 91.2% accurate, 7.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 64.3%

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

3. Taylor expanded in u around 0 91.1%

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

   1. *-commutative 91.1%

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

   2. *-commutative 91.1%

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

   3. unpow2 91.1%

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

   4. associate-*l* 91.1%

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

   5. *-commutative 91.1%

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

   6. cube-mult 91.1%

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

   7. unpow2 91.1%

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

   8. associate-*l* 91.1%

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

   9. distribute-lft-out 91.1%

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

   10. distribute-lft-out 90.8%

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

   11. unpow2 90.8%

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

   12. associate-*l* 90.8%

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

   13. distribute-lft-out 90.8%

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

5. Simplified 90.8%

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

   1. associate-*r* 90.4%

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

   2. +-commutative 90.4%

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

   3. distribute-rgt-in 90.7%

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

   4. +-commutative 90.7%

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

   5. fma-def 90.7%

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

7. Applied egg-rr 90.7%

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

   1. associate-*l* 90.7%

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

   2. associate-*r* 91.0%

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

   3. *-commutative 91.0%

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

   4. distribute-lft-out 91.0%

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

   5. *-commutative 91.0%

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

   6. associate-*r* 91.0%

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

   7. *-commutative 91.0%

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

   8. associate-*l* 91.0%

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

   9. *-commutative 91.0%

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

9. Applied egg-rr 91.0%

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

10. Taylor expanded in s around 0 91.0%

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

11. Final simplification 91.0%

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

Alternative 3: 90.9% 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 64.3%

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

3. Taylor expanded in u around 0 91.1%

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

   1. *-commutative 91.1%

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

   2. *-commutative 91.1%

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

   3. unpow2 91.1%

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

   4. associate-*l* 91.1%

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

   5. *-commutative 91.1%

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

   6. cube-mult 91.1%

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

   7. unpow2 91.1%

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

   8. associate-*l* 91.1%

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

   9. distribute-lft-out 91.1%

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

   10. distribute-lft-out 90.8%

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

   11. unpow2 90.8%

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

   12. associate-*l* 90.8%

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

   13. distribute-lft-out 90.8%

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

5. Simplified 90.8%

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

6. Final simplification 90.8%

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

Alternative 4: 86.8% 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 64.3%

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

3. Taylor expanded in u around 0 91.1%

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

   1. *-commutative 91.1%

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

   2. *-commutative 91.1%

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

   3. unpow2 91.1%

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

   4. associate-*l* 91.1%

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

   5. *-commutative 91.1%

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

   6. cube-mult 91.1%

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

   7. unpow2 91.1%

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

   8. associate-*l* 91.1%

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

   9. distribute-lft-out 91.1%

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

   10. distribute-lft-out 90.8%

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

   11. unpow2 90.8%

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

   12. associate-*l* 90.8%

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

   13. distribute-lft-out 90.8%

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

5. Simplified 90.8%

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

   1. associate-*r* 90.4%

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

   2. +-commutative 90.4%

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

   3. distribute-rgt-in 90.7%

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

   4. +-commutative 90.7%

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

   5. fma-def 90.7%

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

7. Applied egg-rr 90.7%

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

   1. associate-*l* 90.7%

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

   2. associate-*r* 91.0%

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

   3. *-commutative 91.0%

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

   4. distribute-lft-out 91.0%

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

   5. *-commutative 91.0%

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

   6. associate-*r* 91.0%

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

   7. *-commutative 91.0%

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

   8. associate-*l* 91.0%

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

   9. *-commutative 91.0%

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

9. Applied egg-rr 91.0%

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

10. Taylor expanded in u around 0 86.7%

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

    1. *-commutative 86.7%

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

12. Simplified 86.7%

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

13. Final simplification 86.7%

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

Alternative 5: 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 64.3%

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

3. Taylor expanded in u around 0 86.8%

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

   1. unpow2 86.8%

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

   2. associate-*r* 86.8%

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

   3. distribute-rgt-out 86.6%

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

   4. *-commutative 86.6%

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

5. Simplified 86.6%

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

6. Final simplification 86.6%

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

Alternative 6: 73.6% 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 64.3%

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

3. Taylor expanded in u around 0 73.0%

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

   1. *-commutative 73.0%

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

5. Simplified 73.0%

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

6. Final simplification 73.0%

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

Alternative 7: 73.9% accurate, 21.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 64.3%

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

3. Taylor expanded in u around 0 73.3%

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

4. Final simplification 73.3%

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

Alternative 8: 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 64.3%

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

   1. add-sqr-sqrt 59.9%

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

4. Applied egg-rr 18.1%

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

5. Final simplification 18.1%

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

Reproduce

herbie shell --seed 2024026 
(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))))))