Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.6%

Time: 3.8s

Alternatives: 1

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, 0.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(0.75, \frac{e^{\frac{\frac{-r}{3}}{s}}}{\left(\pi \cdot 6\right) \cdot \left(s \cdot r\right)}, 0.25 \cdot \frac{{\left(e^{-1}\right)}^{\left(\frac{r}{s}\right)}}{\left(\pi \cdot 2\right) \cdot \left(s \cdot r\right)}\right) \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (fma
  0.75
  (/ (exp (/ (/ (- r) 3.0) s)) (* (* PI 6.0) (* s r)))
  (* 0.25 (/ (pow (exp -1.0) (/ r s)) (* (* PI 2.0) (* s r))))))

C

float code(float s, float r) {
	return fmaf(0.75f, (expf(((-r / 3.0f) / s)) / ((((float) M_PI) * 6.0f) * (s * r))), (0.25f * (powf(expf(-1.0f), (r / s)) / ((((float) M_PI) * 2.0f) * (s * r)))));
}

Julia

function code(s, r)
	return fma(Float32(0.75), Float32(exp(Float32(Float32(Float32(-r) / Float32(3.0)) / s)) / Float32(Float32(Float32(pi) * Float32(6.0)) * Float32(s * r))), Float32(Float32(0.25) * Float32((exp(Float32(-1.0)) ^ Float32(r / s)) / Float32(Float32(Float32(pi) * Float32(2.0)) * Float32(s * r)))))
end

Derivation

1. Initial program 99.8%

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

4. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.75, \frac{e^{\frac{\frac{-r}{3}}{s}}}{\left(\left(\pi \cdot 6\right) \cdot s\right) \cdot r}, 0.25 \cdot \frac{e^{\frac{-r}{s}}}{\left(\left(\pi \cdot 2\right) \cdot s\right) \cdot r}\right)} \]
5. Step-by-step derivation

   1. lift-exp.f32 N/A

      \[\leadsto \mathsf{fma}\left(\frac{3}{4}, \frac{e^{\frac{\frac{-r}{3}}{s}}}{\left(\left(\pi \cdot 6\right) \cdot s\right) \cdot r}, \frac{1}{4} \cdot \frac{\color{blue}{e^{\frac{-r}{s}}}}{\left(\left(\pi \cdot 2\right) \cdot s\right) \cdot r}\right) \]

   2. lift-neg.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. distribute-frac-neg N/A

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

   5. mul-1-neg N/A

      \[\leadsto \mathsf{fma}\left(\frac{3}{4}, \frac{e^{\frac{\frac{-r}{3}}{s}}}{\left(\left(\pi \cdot 6\right) \cdot s\right) \cdot r}, \frac{1}{4} \cdot \frac{e^{\color{blue}{-1 \cdot \frac{r}{s}}}}{\left(\left(\pi \cdot 2\right) \cdot s\right) \cdot r}\right) \]

   6. exp-prod N/A

      \[\leadsto \mathsf{fma}\left(\frac{3}{4}, \frac{e^{\frac{\frac{-r}{3}}{s}}}{\left(\left(\pi \cdot 6\right) \cdot s\right) \cdot r}, \frac{1}{4} \cdot \frac{\color{blue}{{\left(e^{-1}\right)}^{\left(\frac{r}{s}\right)}}}{\left(\left(\pi \cdot 2\right) \cdot s\right) \cdot r}\right) \]

   7. lower-pow.f32 N/A

      \[\leadsto \mathsf{fma}\left(\frac{3}{4}, \frac{e^{\frac{\frac{-r}{3}}{s}}}{\left(\left(\pi \cdot 6\right) \cdot s\right) \cdot r}, \frac{1}{4} \cdot \frac{\color{blue}{{\left(e^{-1}\right)}^{\left(\frac{r}{s}\right)}}}{\left(\left(\pi \cdot 2\right) \cdot s\right) \cdot r}\right) \]

   8. lower-exp.f32 N/A

      \[\leadsto \mathsf{fma}\left(\frac{3}{4}, \frac{e^{\frac{\frac{-r}{3}}{s}}}{\left(\left(\pi \cdot 6\right) \cdot s\right) \cdot r}, \frac{1}{4} \cdot \frac{{\color{blue}{\left(e^{-1}\right)}}^{\left(\frac{r}{s}\right)}}{\left(\left(\pi \cdot 2\right) \cdot s\right) \cdot r}\right) \]

   9. lower-/.f32 99.8

      \[\leadsto \mathsf{fma}\left(0.75, \frac{e^{\frac{\frac{-r}{3}}{s}}}{\left(\left(\pi \cdot 6\right) \cdot s\right) \cdot r}, 0.25 \cdot \frac{{\left(e^{-1}\right)}^{\color{blue}{\left(\frac{r}{s}\right)}}}{\left(\left(\pi \cdot 2\right) \cdot s\right) \cdot r}\right) \]

6. Applied rewrites 99.8%

   \[\leadsto \mathsf{fma}\left(0.75, \frac{e^{\frac{\frac{-r}{3}}{s}}}{\left(\left(\pi \cdot 6\right) \cdot s\right) \cdot r}, 0.25 \cdot \frac{\color{blue}{{\left(e^{-1}\right)}^{\left(\frac{r}{s}\right)}}}{\left(\left(\pi \cdot 2\right) \cdot s\right) \cdot r}\right) \]
7. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(\frac{3}{4}, \frac{e^{\frac{\frac{-r}{3}}{s}}}{\color{blue}{\left(\left(\pi \cdot 6\right) \cdot s\right) \cdot r}}, \frac{1}{4} \cdot \frac{{\left(e^{-1}\right)}^{\left(\frac{r}{s}\right)}}{\left(\left(\pi \cdot 2\right) \cdot s\right) \cdot r}\right) \]

   2. lift-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(\frac{3}{4}, \frac{e^{\frac{\frac{-r}{3}}{s}}}{\color{blue}{\left(\left(\pi \cdot 6\right) \cdot s\right)} \cdot r}, \frac{1}{4} \cdot \frac{{\left(e^{-1}\right)}^{\left(\frac{r}{s}\right)}}{\left(\left(\pi \cdot 2\right) \cdot s\right) \cdot r}\right) \]

   3. associate-*l* N/A

      \[\leadsto \mathsf{fma}\left(\frac{3}{4}, \frac{e^{\frac{\frac{-r}{3}}{s}}}{\color{blue}{\left(\pi \cdot 6\right) \cdot \left(s \cdot r\right)}}, \frac{1}{4} \cdot \frac{{\left(e^{-1}\right)}^{\left(\frac{r}{s}\right)}}{\left(\left(\pi \cdot 2\right) \cdot s\right) \cdot r}\right) \]

   4. lower-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(\frac{3}{4}, \frac{e^{\frac{\frac{-r}{3}}{s}}}{\color{blue}{\left(\pi \cdot 6\right) \cdot \left(s \cdot r\right)}}, \frac{1}{4} \cdot \frac{{\left(e^{-1}\right)}^{\left(\frac{r}{s}\right)}}{\left(\left(\pi \cdot 2\right) \cdot s\right) \cdot r}\right) \]

   5. lower-*.f32 99.8

      \[\leadsto \mathsf{fma}\left(0.75, \frac{e^{\frac{\frac{-r}{3}}{s}}}{\left(\pi \cdot 6\right) \cdot \color{blue}{\left(s \cdot r\right)}}, 0.25 \cdot \frac{{\left(e^{-1}\right)}^{\left(\frac{r}{s}\right)}}{\left(\left(\pi \cdot 2\right) \cdot s\right) \cdot r}\right) \]

8. Applied rewrites 99.8%

   \[\leadsto \mathsf{fma}\left(0.75, \frac{e^{\frac{\frac{-r}{3}}{s}}}{\color{blue}{\left(\pi \cdot 6\right) \cdot \left(s \cdot r\right)}}, 0.25 \cdot \frac{{\left(e^{-1}\right)}^{\left(\frac{r}{s}\right)}}{\left(\left(\pi \cdot 2\right) \cdot s\right) \cdot r}\right) \]
9. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(\frac{3}{4}, \frac{e^{\frac{\frac{-r}{3}}{s}}}{\left(\pi \cdot 6\right) \cdot \left(s \cdot r\right)}, \frac{1}{4} \cdot \frac{{\left(e^{-1}\right)}^{\left(\frac{r}{s}\right)}}{\color{blue}{\left(\left(\pi \cdot 2\right) \cdot s\right) \cdot r}}\right) \]

   2. lift-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(\frac{3}{4}, \frac{e^{\frac{\frac{-r}{3}}{s}}}{\left(\pi \cdot 6\right) \cdot \left(s \cdot r\right)}, \frac{1}{4} \cdot \frac{{\left(e^{-1}\right)}^{\left(\frac{r}{s}\right)}}{\color{blue}{\left(\left(\pi \cdot 2\right) \cdot s\right)} \cdot r}\right) \]

   3. associate-*l* N/A

      \[\leadsto \mathsf{fma}\left(\frac{3}{4}, \frac{e^{\frac{\frac{-r}{3}}{s}}}{\left(\pi \cdot 6\right) \cdot \left(s \cdot r\right)}, \frac{1}{4} \cdot \frac{{\left(e^{-1}\right)}^{\left(\frac{r}{s}\right)}}{\color{blue}{\left(\pi \cdot 2\right) \cdot \left(s \cdot r\right)}}\right) \]

   4. lift-*.f32 N/A

      \[\leadsto \mathsf{fma}\left(\frac{3}{4}, \frac{e^{\frac{\frac{-r}{3}}{s}}}{\left(\pi \cdot 6\right) \cdot \left(s \cdot r\right)}, \frac{1}{4} \cdot \frac{{\left(e^{-1}\right)}^{\left(\frac{r}{s}\right)}}{\left(\pi \cdot 2\right) \cdot \color{blue}{\left(s \cdot r\right)}}\right) \]

   5. lower-*.f32 99.9

      \[\leadsto \mathsf{fma}\left(0.75, \frac{e^{\frac{\frac{-r}{3}}{s}}}{\left(\pi \cdot 6\right) \cdot \left(s \cdot r\right)}, 0.25 \cdot \frac{{\left(e^{-1}\right)}^{\left(\frac{r}{s}\right)}}{\color{blue}{\left(\pi \cdot 2\right) \cdot \left(s \cdot r\right)}}\right) \]

10. Applied rewrites 99.9%

    \[\leadsto \mathsf{fma}\left(0.75, \frac{e^{\frac{\frac{-r}{3}}{s}}}{\left(\pi \cdot 6\right) \cdot \left(s \cdot r\right)}, 0.25 \cdot \frac{{\left(e^{-1}\right)}^{\left(\frac{r}{s}\right)}}{\color{blue}{\left(\pi \cdot 2\right) \cdot \left(s \cdot r\right)}}\right) \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025059 
(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))))