Disney BSSRDF, sample scattering profile, lower

Percentage Accurate: 61.4% → 99.4%

Time: 11.6s

Alternatives: 13

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: 61.4% 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: 99.4% 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 61.4%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
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 99.2

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

5. Applied rewrites 99.2%

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

Alternative 2: 93.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(u, \mathsf{fma}\left(u, 64, 21.333333333333332\right), 8\right)\\ \frac{\mathsf{fma}\left(u \cdot u, t\_0 \cdot \mathsf{fma}\left(u, 21.333333333333332, 8\right), -16\right) \cdot \left(s \cdot u\right)}{\mathsf{fma}\left(u, t\_0, -4\right)} \end{array} \end{array} \]

FPCore

(FPCore (s u)
 :precision binary32
 (let* ((t_0 (fma u (fma u 64.0 21.333333333333332) 8.0)))
   (/
    (* (fma (* u u) (* t_0 (fma u 21.333333333333332 8.0)) -16.0) (* s u))
    (fma u t_0 -4.0))))

C

float code(float s, float u) {
	float t_0 = fmaf(u, fmaf(u, 64.0f, 21.333333333333332f), 8.0f);
	return (fmaf((u * u), (t_0 * fmaf(u, 21.333333333333332f, 8.0f)), -16.0f) * (s * u)) / fmaf(u, t_0, -4.0f);
}

Julia

function code(s, u)
	t_0 = fma(u, fma(u, Float32(64.0), Float32(21.333333333333332)), Float32(8.0))
	return Float32(Float32(fma(Float32(u * u), Float32(t_0 * fma(u, Float32(21.333333333333332), Float32(8.0))), Float32(-16.0)) * Float32(s * u)) / fma(u, t_0, Float32(-4.0)))
end

Derivation

1. Initial program 61.4%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
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 92.7%

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

7. Applied rewrites 92.9%

   \[\leadsto \frac{\mathsf{fma}\left(u \cdot u, \mathsf{fma}\left(u, \mathsf{fma}\left(u, 64, 21.333333333333332\right), 8\right) \cdot \mathsf{fma}\left(u, \mathsf{fma}\left(u, 64, 21.333333333333332\right), 8\right), -16\right) \cdot \left(s \cdot u\right)}{\color{blue}{\mathsf{fma}\left(u, \mathsf{fma}\left(u, \mathsf{fma}\left(u, 64, 21.333333333333332\right), 8\right), -4\right)}} \]

8. Taylor expanded in u around 0

   \[\leadsto \frac{\mathsf{fma}\left(u \cdot u, \mathsf{fma}\left(u, \mathsf{fma}\left(u, 64, \frac{64}{3}\right), 8\right) \cdot \mathsf{fma}\left(u, \frac{64}{3}, 8\right), -16\right) \cdot \left(s \cdot u\right)}{\mathsf{fma}\left(u, \mathsf{fma}\left(u, \mathsf{fma}\left(u, 64, \frac{64}{3}\right), 8\right), -4\right)} \]
9. Step-by-step derivation

10. Applied rewrites 93.7%

    \[\leadsto \frac{\mathsf{fma}\left(u \cdot u, \mathsf{fma}\left(u, \mathsf{fma}\left(u, 64, 21.333333333333332\right), 8\right) \cdot \mathsf{fma}\left(u, 21.333333333333332, 8\right), -16\right) \cdot \left(s \cdot u\right)}{\mathsf{fma}\left(u, \mathsf{fma}\left(u, \mathsf{fma}\left(u, 64, 21.333333333333332\right), 8\right), -4\right)} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024230 
(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))))))