Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.6%

Time: 15.1s

Alternatives: 19

Speedup: 1.0×

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

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{e^{\frac{r}{s \cdot -3}}}{r \cdot \left(s \cdot \left(\pi \cdot 6\right)\right)}, 0.75, \frac{0.125}{\left(r \cdot \left(s \cdot \pi\right)\right) \cdot e^{\frac{r}{s}}}\right) \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (fma
  (/ (exp (/ r (* s -3.0))) (* r (* s (* PI 6.0))))
  0.75
  (/ 0.125 (* (* r (* s PI)) (exp (/ r s))))))

C

float code(float s, float r) {
	return fmaf((expf((r / (s * -3.0f))) / (r * (s * (((float) M_PI) * 6.0f)))), 0.75f, (0.125f / ((r * (s * ((float) M_PI))) * expf((r / s)))));
}

Julia

function code(s, r)
	return fma(Float32(exp(Float32(r / Float32(s * Float32(-3.0)))) / Float32(r * Float32(s * Float32(Float32(pi) * Float32(6.0))))), Float32(0.75), Float32(Float32(0.125) / Float32(Float32(r * Float32(s * Float32(pi))) * exp(Float32(r / s)))))
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

4. Applied rewrites 98.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{e^{\frac{r}{s \cdot -3}}}{r \cdot \left(s \cdot \left(\pi \cdot 6\right)\right)}, 0.75, e^{\frac{r}{-s}} \cdot \frac{0.125}{\pi \cdot \left(r \cdot s\right)}\right)} \]
5. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(\frac{e^{\frac{r}{s \cdot -3}}}{r \cdot \left(s \cdot \left(\mathsf{PI}\left(\right) \cdot 6\right)\right)}, \frac{3}{4}, \color{blue}{e^{\frac{r}{\mathsf{neg}\left(s\right)}} \cdot \frac{\frac{1}{8}}{\mathsf{PI}\left(\right) \cdot \left(r \cdot s\right)}}\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\frac{e^{\frac{r}{s \cdot -3}}}{r \cdot \left(s \cdot \left(\mathsf{PI}\left(\right) \cdot 6\right)\right)}, \frac{3}{4}, \color{blue}{\frac{\frac{1}{8}}{\mathsf{PI}\left(\right) \cdot \left(r \cdot s\right)} \cdot e^{\frac{r}{\mathsf{neg}\left(s\right)}}}\right) \]

   3. lift-/.f32 N/A

      \[\leadsto \mathsf{fma}\left(\frac{e^{\frac{r}{s \cdot -3}}}{r \cdot \left(s \cdot \left(\mathsf{PI}\left(\right) \cdot 6\right)\right)}, \frac{3}{4}, \color{blue}{\frac{\frac{1}{8}}{\mathsf{PI}\left(\right) \cdot \left(r \cdot s\right)}} \cdot e^{\frac{r}{\mathsf{neg}\left(s\right)}}\right) \]

   4. lift-exp.f32 N/A

      \[\leadsto \mathsf{fma}\left(\frac{e^{\frac{r}{s \cdot -3}}}{r \cdot \left(s \cdot \left(\mathsf{PI}\left(\right) \cdot 6\right)\right)}, \frac{3}{4}, \frac{\frac{1}{8}}{\mathsf{PI}\left(\right) \cdot \left(r \cdot s\right)} \cdot \color{blue}{e^{\frac{r}{\mathsf{neg}\left(s\right)}}}\right) \]

   5. lift-/.f32 N/A

      \[\leadsto \mathsf{fma}\left(\frac{e^{\frac{r}{s \cdot -3}}}{r \cdot \left(s \cdot \left(\mathsf{PI}\left(\right) \cdot 6\right)\right)}, \frac{3}{4}, \frac{\frac{1}{8}}{\mathsf{PI}\left(\right) \cdot \left(r \cdot s\right)} \cdot e^{\color{blue}{\frac{r}{\mathsf{neg}\left(s\right)}}}\right) \]

   6. lift-neg.f32 N/A

      \[\leadsto \mathsf{fma}\left(\frac{e^{\frac{r}{s \cdot -3}}}{r \cdot \left(s \cdot \left(\mathsf{PI}\left(\right) \cdot 6\right)\right)}, \frac{3}{4}, \frac{\frac{1}{8}}{\mathsf{PI}\left(\right) \cdot \left(r \cdot s\right)} \cdot e^{\frac{r}{\color{blue}{\mathsf{neg}\left(s\right)}}}\right) \]

   7. distribute-frac-neg2 N/A

      \[\leadsto \mathsf{fma}\left(\frac{e^{\frac{r}{s \cdot -3}}}{r \cdot \left(s \cdot \left(\mathsf{PI}\left(\right) \cdot 6\right)\right)}, \frac{3}{4}, \frac{\frac{1}{8}}{\mathsf{PI}\left(\right) \cdot \left(r \cdot s\right)} \cdot e^{\color{blue}{\mathsf{neg}\left(\frac{r}{s}\right)}}\right) \]

   8. lift-/.f32 N/A

      \[\leadsto \mathsf{fma}\left(\frac{e^{\frac{r}{s \cdot -3}}}{r \cdot \left(s \cdot \left(\mathsf{PI}\left(\right) \cdot 6\right)\right)}, \frac{3}{4}, \frac{\frac{1}{8}}{\mathsf{PI}\left(\right) \cdot \left(r \cdot s\right)} \cdot e^{\mathsf{neg}\left(\color{blue}{\frac{r}{s}}\right)}\right) \]

   9. exp-neg N/A

      \[\leadsto \mathsf{fma}\left(\frac{e^{\frac{r}{s \cdot -3}}}{r \cdot \left(s \cdot \left(\mathsf{PI}\left(\right) \cdot 6\right)\right)}, \frac{3}{4}, \frac{\frac{1}{8}}{\mathsf{PI}\left(\right) \cdot \left(r \cdot s\right)} \cdot \color{blue}{\frac{1}{e^{\frac{r}{s}}}}\right) \]

   10. frac-times N/A

       \[\leadsto \mathsf{fma}\left(\frac{e^{\frac{r}{s \cdot -3}}}{r \cdot \left(s \cdot \left(\mathsf{PI}\left(\right) \cdot 6\right)\right)}, \frac{3}{4}, \color{blue}{\frac{\frac{1}{8} \cdot 1}{\left(\mathsf{PI}\left(\right) \cdot \left(r \cdot s\right)\right) \cdot e^{\frac{r}{s}}}}\right) \]

   11. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(\frac{e^{\frac{r}{s \cdot -3}}}{r \cdot \left(s \cdot \left(\mathsf{PI}\left(\right) \cdot 6\right)\right)}, \frac{3}{4}, \frac{\color{blue}{\frac{1}{8}}}{\left(\mathsf{PI}\left(\right) \cdot \left(r \cdot s\right)\right) \cdot e^{\frac{r}{s}}}\right) \]

   12. lower-/.f32 N/A

       \[\leadsto \mathsf{fma}\left(\frac{e^{\frac{r}{s \cdot -3}}}{r \cdot \left(s \cdot \left(\mathsf{PI}\left(\right) \cdot 6\right)\right)}, \frac{3}{4}, \color{blue}{\frac{\frac{1}{8}}{\left(\mathsf{PI}\left(\right) \cdot \left(r \cdot s\right)\right) \cdot e^{\frac{r}{s}}}}\right) \]

   13. lower-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(\frac{e^{\frac{r}{s \cdot -3}}}{r \cdot \left(s \cdot \left(\mathsf{PI}\left(\right) \cdot 6\right)\right)}, \frac{3}{4}, \frac{\frac{1}{8}}{\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(r \cdot s\right)\right) \cdot e^{\frac{r}{s}}}}\right) \]

   14. lift-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(\frac{e^{\frac{r}{s \cdot -3}}}{r \cdot \left(s \cdot \left(\mathsf{PI}\left(\right) \cdot 6\right)\right)}, \frac{3}{4}, \frac{\frac{1}{8}}{\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(r \cdot s\right)\right)} \cdot e^{\frac{r}{s}}}\right) \]

   15. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\frac{e^{\frac{r}{s \cdot -3}}}{r \cdot \left(s \cdot \left(\mathsf{PI}\left(\right) \cdot 6\right)\right)}, \frac{3}{4}, \frac{\frac{1}{8}}{\color{blue}{\left(\left(r \cdot s\right) \cdot \mathsf{PI}\left(\right)\right)} \cdot e^{\frac{r}{s}}}\right) \]

   16. lift-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(\frac{e^{\frac{r}{s \cdot -3}}}{r \cdot \left(s \cdot \left(\mathsf{PI}\left(\right) \cdot 6\right)\right)}, \frac{3}{4}, \frac{\frac{1}{8}}{\left(\color{blue}{\left(r \cdot s\right)} \cdot \mathsf{PI}\left(\right)\right) \cdot e^{\frac{r}{s}}}\right) \]

   17. associate-*r* N/A

       \[\leadsto \mathsf{fma}\left(\frac{e^{\frac{r}{s \cdot -3}}}{r \cdot \left(s \cdot \left(\mathsf{PI}\left(\right) \cdot 6\right)\right)}, \frac{3}{4}, \frac{\frac{1}{8}}{\color{blue}{\left(r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot e^{\frac{r}{s}}}\right) \]

   18. lift-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(\frac{e^{\frac{r}{s \cdot -3}}}{r \cdot \left(s \cdot \left(\mathsf{PI}\left(\right) \cdot 6\right)\right)}, \frac{3}{4}, \frac{\frac{1}{8}}{\left(r \cdot \color{blue}{\left(s \cdot \mathsf{PI}\left(\right)\right)}\right) \cdot e^{\frac{r}{s}}}\right) \]

   19. lift-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(\frac{e^{\frac{r}{s \cdot -3}}}{r \cdot \left(s \cdot \left(\mathsf{PI}\left(\right) \cdot 6\right)\right)}, \frac{3}{4}, \frac{\frac{1}{8}}{\color{blue}{\left(r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot e^{\frac{r}{s}}}\right) \]

   20. lower-exp.f32 99.7

       \[\leadsto \mathsf{fma}\left(\frac{e^{\frac{r}{s \cdot -3}}}{r \cdot \left(s \cdot \left(\pi \cdot 6\right)\right)}, 0.75, \frac{0.125}{\left(r \cdot \left(s \cdot \pi\right)\right) \cdot \color{blue}{e^{\frac{r}{s}}}}\right) \]

6. Applied rewrites 99.7%

   \[\leadsto \mathsf{fma}\left(\frac{e^{\frac{r}{s \cdot -3}}}{r \cdot \left(s \cdot \left(\pi \cdot 6\right)\right)}, 0.75, \color{blue}{\frac{0.125}{\left(r \cdot \left(s \cdot \pi\right)\right) \cdot e^{\frac{r}{s}}}}\right) \]
7. Add Preprocessing

Alternative 2: 99.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{0.125 \cdot \left(\frac{e^{\frac{-r}{s}}}{s \cdot \pi} + \frac{{e}^{\left(\frac{r}{s} \cdot -0.3333333333333333\right)}}{s \cdot \pi}\right)}{r} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (/
  (*
   0.125
   (+
    (/ (exp (/ (- r) s)) (* s PI))
    (/ (pow E (* (/ r s) -0.3333333333333333)) (* s PI))))
  r))

C

float code(float s, float r) {
	return (0.125f * ((expf((-r / s)) / (s * ((float) M_PI))) + (powf(((float) M_E), ((r / s) * -0.3333333333333333f)) / (s * ((float) M_PI))))) / r;
}

Julia

function code(s, r)
	return Float32(Float32(Float32(0.125) * Float32(Float32(exp(Float32(Float32(-r) / s)) / Float32(s * Float32(pi))) + Float32((Float32(exp(1)) ^ Float32(Float32(r / s) * Float32(-0.3333333333333333))) / Float32(s * Float32(pi))))) / r)
end

MATLAB

function tmp = code(s, r)
	tmp = (single(0.125) * ((exp((-r / s)) / (s * single(pi))) + ((single(2.71828182845904523536) ^ ((r / s) * single(-0.3333333333333333))) / (s * single(pi))))) / r;
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. Taylor expanded in r around inf

   \[\leadsto \color{blue}{\frac{\frac{1}{8} \cdot \frac{e^{-1 \cdot \frac{r}{s}}}{s \cdot \mathsf{PI}\left(\right)} + \frac{1}{8} \cdot \frac{e^{\frac{-1}{3} \cdot \frac{r}{s}}}{s \cdot \mathsf{PI}\left(\right)}}{r}} \]
4. Step-by-step derivation

   1. distribute-lft-out N/A

      \[\leadsto \frac{\color{blue}{\frac{1}{8} \cdot \left(\frac{e^{-1 \cdot \frac{r}{s}}}{s \cdot \mathsf{PI}\left(\right)} + \frac{e^{\frac{-1}{3} \cdot \frac{r}{s}}}{s \cdot \mathsf{PI}\left(\right)}\right)}}{r} \]

   2. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{\left(-1 \cdot \frac{-1}{8}\right)} \cdot \left(\frac{e^{-1 \cdot \frac{r}{s}}}{s \cdot \mathsf{PI}\left(\right)} + \frac{e^{\frac{-1}{3} \cdot \frac{r}{s}}}{s \cdot \mathsf{PI}\left(\right)}\right)}{r} \]

   3. associate-*r* N/A

      \[\leadsto \frac{\color{blue}{-1 \cdot \left(\frac{-1}{8} \cdot \left(\frac{e^{-1 \cdot \frac{r}{s}}}{s \cdot \mathsf{PI}\left(\right)} + \frac{e^{\frac{-1}{3} \cdot \frac{r}{s}}}{s \cdot \mathsf{PI}\left(\right)}\right)\right)}}{r} \]

   4. distribute-lft-out N/A

      \[\leadsto \frac{-1 \cdot \color{blue}{\left(\frac{-1}{8} \cdot \frac{e^{-1 \cdot \frac{r}{s}}}{s \cdot \mathsf{PI}\left(\right)} + \frac{-1}{8} \cdot \frac{e^{\frac{-1}{3} \cdot \frac{r}{s}}}{s \cdot \mathsf{PI}\left(\right)}\right)}}{r} \]

   5. lower-/.f32 N/A

      \[\leadsto \color{blue}{\frac{-1 \cdot \left(\frac{-1}{8} \cdot \frac{e^{-1 \cdot \frac{r}{s}}}{s \cdot \mathsf{PI}\left(\right)} + \frac{-1}{8} \cdot \frac{e^{\frac{-1}{3} \cdot \frac{r}{s}}}{s \cdot \mathsf{PI}\left(\right)}\right)}{r}} \]

5. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\frac{0.125 \cdot \left(\frac{e^{-\frac{r}{s}}}{s \cdot \pi} + \frac{e^{\frac{r}{s} \cdot -0.3333333333333333}}{s \cdot \pi}\right)}{r}} \]
6. Step-by-step derivation

7. Applied rewrites 99.5%

   \[\leadsto \frac{0.125 \cdot \left(\frac{e^{-\frac{r}{s}}}{s \cdot \pi} + \frac{{e}^{\left(\frac{r}{s} \cdot -0.3333333333333333\right)}}{s \cdot \pi}\right)}{r} \]

8. Final simplification 99.5%

   \[\leadsto \frac{0.125 \cdot \left(\frac{e^{\frac{-r}{s}}}{s \cdot \pi} + \frac{{e}^{\left(\frac{r}{s} \cdot -0.3333333333333333\right)}}{s \cdot \pi}\right)}{r} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024230 
(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))))