Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.6%

Time: 16.7s

Alternatives: 26

Speedup: N/A×

Specification

\[\left(0 \leq s \land s \leq 256\right) \land \left(10^{-6} < r \land r < 1000000\right)\]

Math

\[\begin{array}{l} \\ \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/ (* 0.25 (exp (/ (- r) s))) (* (* (* 2.0 PI) s) r))
  (/ (* 0.75 (exp (/ (- r) (* 3.0 s)))) (* (* (* 6.0 PI) s) r))))

C

float code(float s, float r) {
	return ((0.25f * expf((-r / s))) / (((2.0f * ((float) M_PI)) * s) * r)) + ((0.75f * expf((-r / (3.0f * s)))) / (((6.0f * ((float) M_PI)) * s) * r));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.25) * exp(Float32(Float32(-r) / s))) / Float32(Float32(Float32(Float32(2.0) * Float32(pi)) * s) * r)) + Float32(Float32(Float32(0.75) * exp(Float32(Float32(-r) / Float32(Float32(3.0) * s)))) / Float32(Float32(Float32(Float32(6.0) * Float32(pi)) * s) * r)))
end

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.25) * exp((-r / s))) / (((single(2.0) * single(pi)) * s) * r)) + ((single(0.75) * exp((-r / (single(3.0) * s)))) / (((single(6.0) * single(pi)) * s) * r));
end

Initial Program: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/ (* 0.25 (exp (/ (- r) s))) (* (* (* 2.0 PI) s) r))
  (/ (* 0.75 (exp (/ (- r) (* 3.0 s)))) (* (* (* 6.0 PI) s) r))))

C

float code(float s, float r) {
	return ((0.25f * expf((-r / s))) / (((2.0f * ((float) M_PI)) * s) * r)) + ((0.75f * expf((-r / (3.0f * s)))) / (((6.0f * ((float) M_PI)) * s) * r));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.25) * exp(Float32(Float32(-r) / s))) / Float32(Float32(Float32(Float32(2.0) * Float32(pi)) * s) * r)) + Float32(Float32(Float32(0.75) * exp(Float32(Float32(-r) / Float32(Float32(3.0) * s)))) / Float32(Float32(Float32(Float32(6.0) * Float32(pi)) * s) * r)))
end

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.25) * exp((-r / s))) / (((single(2.0) * single(pi)) * s) * r)) + ((single(0.75) * exp((-r / (single(3.0) * s)))) / (((single(6.0) * single(pi)) * s) * r));
end

Alternative 1: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{0.25 \cdot e^{\frac{r}{-s}}}{r \cdot \left(s \cdot \left(2 \cdot \pi\right)\right)} + \frac{\frac{e^{\frac{r}{s \cdot -3}}}{\pi \cdot 6}}{\left(r \cdot s\right) \cdot 1.3333333333333333} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/ (* 0.25 (exp (/ r (- s)))) (* r (* s (* 2.0 PI))))
  (/ (/ (exp (/ r (* s -3.0))) (* PI 6.0)) (* (* r s) 1.3333333333333333))))

C

float code(float s, float r) {
	return ((0.25f * expf((r / -s))) / (r * (s * (2.0f * ((float) M_PI))))) + ((expf((r / (s * -3.0f))) / (((float) M_PI) * 6.0f)) / ((r * s) * 1.3333333333333333f));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.25) * exp(Float32(r / Float32(-s)))) / Float32(r * Float32(s * Float32(Float32(2.0) * Float32(pi))))) + Float32(Float32(exp(Float32(r / Float32(s * Float32(-3.0)))) / Float32(Float32(pi) * Float32(6.0))) / Float32(Float32(r * s) * Float32(1.3333333333333333))))
end

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.25) * exp((r / -s))) / (r * (s * (single(2.0) * single(pi))))) + ((exp((r / (s * single(-3.0)))) / (single(pi) * single(6.0))) / ((r * s) * single(1.3333333333333333)));
end

Derivation

1. Initial program 99.7%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \color{blue}{\frac{\frac{3}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r}} \]

   2. clear-num N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \color{blue}{\frac{1}{\frac{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r}{\frac{3}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}}} \]

   3. lift-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{1}{\frac{\color{blue}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r}}{\frac{3}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}} \]

   4. lift-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{1}{\frac{\color{blue}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right)} \cdot r}{\frac{3}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}} \]

   5. associate-*l* N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{1}{\frac{\color{blue}{\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(s \cdot r\right)}}{\frac{3}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}} \]

   6. lift-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{1}{\frac{\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(s \cdot r\right)}{\color{blue}{\frac{3}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}}} \]

   7. *-commutative N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{1}{\frac{\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(s \cdot r\right)}{\color{blue}{e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}} \cdot \frac{3}{4}}}} \]

   8. times-frac N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{1}{\color{blue}{\frac{6 \cdot \mathsf{PI}\left(\right)}{e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}} \cdot \frac{s \cdot r}{\frac{3}{4}}}} \]

   9. associate-/r* N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \color{blue}{\frac{\frac{1}{\frac{6 \cdot \mathsf{PI}\left(\right)}{e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}}}{\frac{s \cdot r}{\frac{3}{4}}}} \]

   10. clear-num N/A

       \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\color{blue}{\frac{e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{6 \cdot \mathsf{PI}\left(\right)}}}{\frac{s \cdot r}{\frac{3}{4}}} \]

   11. lower-/.f32 N/A

       \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \color{blue}{\frac{\frac{e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{6 \cdot \mathsf{PI}\left(\right)}}{\frac{s \cdot r}{\frac{3}{4}}}} \]

4. Applied rewrites 99.7%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \color{blue}{\frac{\frac{e^{\frac{r}{s \cdot -3}}}{\pi \cdot 6}}{\left(r \cdot s\right) \cdot 1.3333333333333333}} \]

5. Final simplification 99.7%

   \[\leadsto \frac{0.25 \cdot e^{\frac{r}{-s}}}{r \cdot \left(s \cdot \left(2 \cdot \pi\right)\right)} + \frac{\frac{e^{\frac{r}{s \cdot -3}}}{\pi \cdot 6}}{\left(r \cdot s\right) \cdot 1.3333333333333333} \]
6. Add Preprocessing

Alternative 2: 99.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{0.125 \cdot \left(e^{\frac{r}{s \cdot -3}} + e^{\frac{r}{-s}}\right)}{r \cdot \left(s \cdot \pi\right)} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (/ (* 0.125 (+ (exp (/ r (* s -3.0))) (exp (/ r (- s))))) (* r (* s PI))))

C

float code(float s, float r) {
	return (0.125f * (expf((r / (s * -3.0f))) + expf((r / -s)))) / (r * (s * ((float) M_PI)));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(0.125) * Float32(exp(Float32(r / Float32(s * Float32(-3.0)))) + exp(Float32(r / Float32(-s))))) / Float32(r * Float32(s * Float32(pi))))
end

MATLAB

function tmp = code(s, r)
	tmp = (single(0.125) * (exp((r / (s * single(-3.0)))) + exp((r / -s)))) / (r * (s * single(pi)));
end

Derivation

1. Initial program 99.6%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \color{blue}{\frac{\frac{3}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r}} \]

   2. clear-num N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \color{blue}{\frac{1}{\frac{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r}{\frac{3}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}}} \]

   3. lift-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{1}{\frac{\color{blue}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r}}{\frac{3}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}} \]

   4. lift-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{1}{\frac{\color{blue}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right)} \cdot r}{\frac{3}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}} \]

   5. associate-*l* N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{1}{\frac{\color{blue}{\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(s \cdot r\right)}}{\frac{3}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}} \]

   6. lift-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{1}{\frac{\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(s \cdot r\right)}{\color{blue}{\frac{3}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}}} \]

   7. *-commutative N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{1}{\frac{\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(s \cdot r\right)}{\color{blue}{e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}} \cdot \frac{3}{4}}}} \]

   8. times-frac N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{1}{\color{blue}{\frac{6 \cdot \mathsf{PI}\left(\right)}{e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}} \cdot \frac{s \cdot r}{\frac{3}{4}}}} \]

   9. associate-/r* N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \color{blue}{\frac{\frac{1}{\frac{6 \cdot \mathsf{PI}\left(\right)}{e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}}}{\frac{s \cdot r}{\frac{3}{4}}}} \]

   10. clear-num N/A

       \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\color{blue}{\frac{e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{6 \cdot \mathsf{PI}\left(\right)}}}{\frac{s \cdot r}{\frac{3}{4}}} \]

   11. lower-/.f32 N/A

       \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \color{blue}{\frac{\frac{e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{6 \cdot \mathsf{PI}\left(\right)}}{\frac{s \cdot r}{\frac{3}{4}}}} \]

4. Applied rewrites 99.6%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \color{blue}{\frac{\frac{e^{\frac{r}{s \cdot -3}}}{\pi \cdot 6}}{\left(r \cdot s\right) \cdot 1.3333333333333333}} \]

6. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\frac{0.125 \cdot \left(e^{\frac{r}{-s}} + e^{\frac{r}{s \cdot -3}}\right)}{r \cdot \left(s \cdot \pi\right)}} \]

7. Final simplification 99.5%

   \[\leadsto \frac{0.125 \cdot \left(e^{\frac{r}{s \cdot -3}} + e^{\frac{r}{-s}}\right)}{r \cdot \left(s \cdot \pi\right)} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024227 
(FPCore (s r)
  :name "Disney BSSRDF, PDF of scattering profile"
  :precision binary32
  :pre (and (and (<= 0.0 s) (<= s 256.0)) (and (< 1e-6 r) (< r 1000000.0)))
  (+ (/ (* 0.25 (exp (/ (- r) s))) (* (* (* 2.0 PI) s) r)) (/ (* 0.75 (exp (/ (- r) (* 3.0 s)))) (* (* (* 6.0 PI) s) r))))