Disney BSSRDF, PDF of scattering profile

Average Error: 0.1 → 0.1

Time: 11.5s

Precision: binary32

Cost: 13696

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

Math

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

↓

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

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))) (* r (* s (* 2.0 PI))))
  (/ (* 0.75 (exp (/ (/ r -3.0) s))) (* s (* r (* 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))) / (r * (s * (2.0f * ((float) M_PI))))) + ((0.75f * expf(((r / -3.0f) / s))) / (s * (r * (((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(r * Float32(s * Float32(Float32(2.0) * Float32(pi))))) + Float32(Float32(Float32(0.75) * exp(Float32(Float32(r / Float32(-3.0)) / s))) / Float32(s * Float32(r * 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))) / (r * (s * (single(2.0) * single(pi))))) + ((single(0.75) * exp(((r / single(-3.0)) / s))) / (s * (r * (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. Taylor expanded in r around 0 0.1

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\color{blue}{-0.3333333333333333 \cdot \frac{r}{s}}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

3. Simplified 0.1

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\color{blue}{\frac{\frac{r}{-3}}{s}}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
   Proof
   (/.f32 (/.f32 r -3) s): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate-/r*_binary32 (/.f32 r (*.f32 -3 s))): 42 points increase in error, 59 points decrease in error
   (/.f32 (Rewrite<= *-lft-identity_binary32 (*.f32 1 r)) (*.f32 -3 s)): 0 points increase in error, 0 points decrease in error
   (Rewrite=> times-frac_binary32 (*.f32 (/.f32 1 -3) (/.f32 r s))): 104 points increase in error, 20 points decrease in error
   (*.f32 (Rewrite=> metadata-eval -1/3) (/.f32 r s)): 0 points increase in error, 0 points decrease in error

4. Taylor expanded in s around 0 0.1

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

5. Simplified 0.1

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{\frac{r}{-3}}{s}}}{\color{blue}{s \cdot \left(r \cdot \left(\pi \cdot 6\right)\right)}} \]
   Proof
   (*.f32 s (*.f32 r (*.f32 (PI.f32) 6))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate-*l*_binary32 (*.f32 (*.f32 s r) (*.f32 (PI.f32) 6))): 56 points increase in error, 49 points decrease in error
   (Rewrite<= associate-*l*_binary32 (*.f32 (*.f32 (*.f32 s r) (PI.f32)) 6)): 82 points increase in error, 21 points decrease in error
   (*.f32 (Rewrite<= associate-*r*_binary32 (*.f32 s (*.f32 r (PI.f32)))) 6): 43 points increase in error, 47 points decrease in error
   (Rewrite<= *-commutative_binary32 (*.f32 6 (*.f32 s (*.f32 r (PI.f32))))): 0 points increase in error, 0 points decrease in error

6. Final simplification 0.1

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

Alternatives

Alternative 1
Error	0.1
Cost	13696

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

Alternative 2
Error	0.1
Cost	10144

\[\frac{0.125}{r \cdot \pi} \cdot \frac{e^{\frac{-r}{s}} + e^{-0.3333333333333333 \cdot \frac{r}{s}}}{s} \]

Alternative 3
Error	18.0
Cost	9792

\[\frac{0.25}{s \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(r \cdot \pi\right)\right)} \]

Alternative 4
Error	29.0
Cost	3520

\[1 + \left(\frac{0.25}{s \cdot \left(r \cdot \pi\right)} + -1\right) \]

Alternative 5
Error	29.0
Cost	3520

\[\left(1 + \frac{\frac{0.25}{r \cdot \pi}}{s}\right) + -1 \]

Alternative 6
Error	29.2
Cost	3392

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

Reproduce

herbie shell --seed 2022326 
(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))))