Disney BSSRDF, sample scattering profile, lower

Percentage Accurate: 61.3% → 99.4%

Time: 12.6s

Alternatives: 15

Speedup: 11.4×

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, 1.1× 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 63.8%

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

3. Taylor expanded in s around 0

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

   1. *-commutative N/A

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

   2. log-rec N/A

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

   3. distribute-lft-neg-out N/A

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

   4. distribute-rgt-neg-in N/A

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

   5. lower-*.f32 N/A

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

   6. cancel-sign-sub-inv N/A

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

   7. metadata-eval N/A

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

   8. lower-log1p.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f32 N/A

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

   11. lower-neg.f32 99.5

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

5. Applied rewrites 99.5%

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

Alternative 2: 94.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ u \cdot \left(s \cdot \mathsf{fma}\left(u, \frac{64 - u \cdot \left(\mathsf{fma}\left(u, 64, 21.333333333333332\right) \cdot \left(u \cdot 21.333333333333332\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(u, 64, 21.333333333333332\right), -u, 8\right)}, 4\right)\right) \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (*
  u
  (*
   s
   (fma
    u
    (/
     (-
      64.0
      (* u (* (fma u 64.0 21.333333333333332) (* u 21.333333333333332))))
     (fma (fma u 64.0 21.333333333333332) (- u) 8.0))
    4.0))))

C

float code(float s, float u) {
	return u * (s * fmaf(u, ((64.0f - (u * (fmaf(u, 64.0f, 21.333333333333332f) * (u * 21.333333333333332f)))) / fmaf(fmaf(u, 64.0f, 21.333333333333332f), -u, 8.0f)), 4.0f));
}

Julia

function code(s, u)
	return Float32(u * Float32(s * fma(u, Float32(Float32(Float32(64.0) - Float32(u * Float32(fma(u, Float32(64.0), Float32(21.333333333333332)) * Float32(u * Float32(21.333333333333332))))) / fma(fma(u, Float32(64.0), Float32(21.333333333333332)), Float32(-u), Float32(8.0))), Float32(4.0))))
end

Derivation

1. Initial program 61.3%

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

3. Taylor expanded in u around 0

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

5. Applied rewrites 92.6%

   \[\leadsto \color{blue}{\left(u \cdot s\right) \cdot \mathsf{fma}\left(u, \mathsf{fma}\left(u, \mathsf{fma}\left(u, 64, 21.333333333333332\right), 8\right), 4\right)} \]
6. Step-by-step derivation

7. Applied rewrites 92.6%

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

8. Taylor expanded in u around 0

   \[\leadsto \left(u \cdot s\right) \cdot \mathsf{fma}\left(u, \frac{64}{8 - u \cdot \mathsf{fma}\left(u, 64, \frac{64}{3}\right)} - \frac{u \cdot \left(\mathsf{fma}\left(u, 64, \frac{64}{3}\right) \cdot \left(u \cdot \frac{64}{3}\right)\right)}{8 - u \cdot \mathsf{fma}\left(u, 64, \frac{64}{3}\right)}, 4\right) \]
9. Step-by-step derivation

10. Applied rewrites 94.0%

    \[\leadsto \left(u \cdot s\right) \cdot \mathsf{fma}\left(u, \frac{64}{8 - u \cdot \mathsf{fma}\left(u, 64, 21.333333333333332\right)} - \frac{u \cdot \left(\mathsf{fma}\left(u, 64, 21.333333333333332\right) \cdot \left(u \cdot 21.333333333333332\right)\right)}{8 - u \cdot \mathsf{fma}\left(u, 64, 21.333333333333332\right)}, 4\right) \]
11. Step-by-step derivation

12. Applied rewrites 94.3%

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

13. Final simplification 94.3%

    \[\leadsto u \cdot \left(s \cdot \mathsf{fma}\left(u, \frac{64 - u \cdot \left(\mathsf{fma}\left(u, 64, 21.333333333333332\right) \cdot \left(u \cdot 21.333333333333332\right)\right)}{\mathsf{fma}\left(\mathsf{fma}\left(u, 64, 21.333333333333332\right), -u, 8\right)}, 4\right)\right) \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024222 
(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))))))