Disney BSSRDF, PDF of scattering profile

Average Error: 0.1 → 0.1

Time: 13.0s

Precision: binary32

Cost: 13728

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

Math

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

↓

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

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))))

↓

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

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));
}

↓

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

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

↓

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

↓

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

Derivation

1. Initial program 0.1

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

2. Simplified 0.1

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

   [Start] 0.1	\[ \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
   rational_best_oopsla_all_46_json_45_simplify-74 [=>] 0.1	\[ \frac{0.25 \cdot e^{\frac{-r}{s}}}{\color{blue}{r \cdot \left(\left(2 \cdot \pi\right) \cdot s\right)}} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
   rational_best_oopsla_all_46_json_45_simplify-74 [=>] 0.1	\[ \frac{0.25 \cdot e^{\frac{-r}{s}}}{r \cdot \color{blue}{\left(s \cdot \left(2 \cdot \pi\right)\right)}} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
   rational_best_oopsla_all_46_json_45_simplify-74 [=>] 0.1	\[ \frac{0.25 \cdot e^{\frac{-r}{s}}}{r \cdot \left(s \cdot \color{blue}{\left(\pi \cdot 2\right)}\right)} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
   rational_best_oopsla_all_46_json_45_simplify-7 [=>] 0.1	\[ \frac{0.25 \cdot e^{\frac{-r}{s}}}{r \cdot \color{blue}{\left(\pi \cdot \left(s \cdot 2\right)\right)}} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
   rational_best_oopsla_all_46_json_45_simplify-7 [=>] 0.1	\[ \frac{0.25 \cdot e^{\frac{-r}{s}}}{\color{blue}{\pi \cdot \left(r \cdot \left(s \cdot 2\right)\right)}} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
   rational_best_oopsla_all_46_json_45_simplify-74 [=>] 0.1	\[ \frac{0.25 \cdot e^{\frac{-r}{s}}}{\pi \cdot \left(r \cdot \left(s \cdot 2\right)\right)} + \frac{0.75 \cdot e^{\frac{-r}{\color{blue}{s \cdot 3}}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
   rational_best_oopsla_all_46_json_45_simplify-74 [=>] 0.1	\[ \frac{0.25 \cdot e^{\frac{-r}{s}}}{\pi \cdot \left(r \cdot \left(s \cdot 2\right)\right)} + \frac{0.75 \cdot e^{\frac{-r}{s \cdot 3}}}{\color{blue}{r \cdot \left(\left(6 \cdot \pi\right) \cdot s\right)}} \]
   rational_best_oopsla_all_46_json_45_simplify-74 [=>] 0.1	\[ \frac{0.25 \cdot e^{\frac{-r}{s}}}{\pi \cdot \left(r \cdot \left(s \cdot 2\right)\right)} + \frac{0.75 \cdot e^{\frac{-r}{s \cdot 3}}}{r \cdot \color{blue}{\left(s \cdot \left(6 \cdot \pi\right)\right)}} \]
   rational_best_oopsla_all_46_json_45_simplify-74 [=>] 0.1	\[ \frac{0.25 \cdot e^{\frac{-r}{s}}}{\pi \cdot \left(r \cdot \left(s \cdot 2\right)\right)} + \frac{0.75 \cdot e^{\frac{-r}{s \cdot 3}}}{r \cdot \left(s \cdot \color{blue}{\left(\pi \cdot 6\right)}\right)} \]

3. Final simplification 0.1

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

Alternatives

Alternative 1
Error	0.1
Cost	13632

\[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\pi \cdot \left(r \cdot \left(s \cdot 2\right)\right)} + 0.125 \cdot \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r \cdot \left(s \cdot \pi\right)} \]

Alternative 2
Error	0.1
Cost	13632

\[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\pi \cdot \left(r \cdot \left(s \cdot 2\right)\right)} + 0.125 \cdot \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{s \cdot \left(r \cdot \pi\right)} \]

Alternative 3
Error	0.1
Cost	13632

\[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(r \cdot \pi\right) \cdot \left(2 \cdot s\right)} + 0.125 \cdot \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{s \cdot \left(r \cdot \pi\right)} \]

Alternative 4
Error	29.0
Cost	10240

\[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\pi \cdot \left(r \cdot \left(s \cdot 2\right)\right)} + \frac{0.125}{r \cdot \left(s \cdot \pi\right)} \]

Alternative 5
Error	29.1
Cost	6880

\[\frac{0.25}{\pi \cdot \left(r \cdot \left(s \cdot 2\right)\right)} + \frac{0.125}{r \cdot \left(s \cdot \pi\right)} \]

Alternative 6
Error	29.1
Cost	6816

\[\frac{0.125}{s \cdot \left(r \cdot \pi\right)} + \frac{0.125}{\pi \cdot \left(s \cdot r\right)} \]

Reproduce

herbie shell --seed 2023090 
(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))))