Disney BSSRDF, PDF of scattering profile

Average Error: 0.2 → 0.2

Time: 11.3s

Precision: binary32

Cost: 13600

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

Math

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

↓

\[\frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{-r}{s}}}{r} + \frac{\frac{0.125}{s}}{\pi} \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r} \]

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
 (+
  (* (/ 0.125 (* s PI)) (/ (exp (/ (- r) s)) r))
  (* (/ (/ 0.125 s) PI) (/ (exp (/ (- r) (* s 3.0))) 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));
}

↓

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

↓

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

↓

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

Derivation

1. Initial program 0.2

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

2. Simplified 0.2

   \[\leadsto \color{blue}{\frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{\frac{-r}{s}}}{r} + \frac{0.75}{6 \cdot \left(\pi \cdot s\right)} \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r}} \]
   Proof
   (+.f32 (*.f32 (/.f32 1/4 (*.f32 s (*.f32 2 (PI.f32)))) (/.f32 (exp.f32 (/.f32 (neg.f32 r) s)) r)) (*.f32 (/.f32 3/4 (*.f32 6 (*.f32 (PI.f32) s))) (/.f32 (exp.f32 (/.f32 (neg.f32 r) (*.f32 s 3))) r))): 0 points increase in error, 0 points decrease in error
   (+.f32 (*.f32 (/.f32 1/4 (Rewrite<= *-commutative_binary32 (*.f32 (*.f32 2 (PI.f32)) s))) (/.f32 (exp.f32 (/.f32 (neg.f32 r) s)) r)) (*.f32 (/.f32 3/4 (*.f32 6 (*.f32 (PI.f32) s))) (/.f32 (exp.f32 (/.f32 (neg.f32 r) (*.f32 s 3))) r))): 0 points increase in error, 0 points decrease in error
   (+.f32 (Rewrite<= times-frac_binary32 (/.f32 (*.f32 1/4 (exp.f32 (/.f32 (neg.f32 r) s))) (*.f32 (*.f32 (*.f32 2 (PI.f32)) s) r))) (*.f32 (/.f32 3/4 (*.f32 6 (*.f32 (PI.f32) s))) (/.f32 (exp.f32 (/.f32 (neg.f32 r) (*.f32 s 3))) r))): 4 points increase in error, 2 points decrease in error
   (+.f32 (/.f32 (*.f32 1/4 (exp.f32 (/.f32 (neg.f32 r) s))) (*.f32 (*.f32 (*.f32 2 (PI.f32)) s) r)) (*.f32 (/.f32 3/4 (Rewrite<= associate-*l*_binary32 (*.f32 (*.f32 6 (PI.f32)) s))) (/.f32 (exp.f32 (/.f32 (neg.f32 r) (*.f32 s 3))) r))): 3 points increase in error, 7 points decrease in error
   (+.f32 (/.f32 (*.f32 1/4 (exp.f32 (/.f32 (neg.f32 r) s))) (*.f32 (*.f32 (*.f32 2 (PI.f32)) s) r)) (*.f32 (/.f32 3/4 (*.f32 (*.f32 6 (PI.f32)) s)) (/.f32 (exp.f32 (/.f32 (neg.f32 r) (Rewrite<= *-commutative_binary32 (*.f32 3 s)))) r))): 0 points increase in error, 0 points decrease in error
   (+.f32 (/.f32 (*.f32 1/4 (exp.f32 (/.f32 (neg.f32 r) s))) (*.f32 (*.f32 (*.f32 2 (PI.f32)) s) r)) (Rewrite<= times-frac_binary32 (/.f32 (*.f32 3/4 (exp.f32 (/.f32 (neg.f32 r) (*.f32 3 s)))) (*.f32 (*.f32 (*.f32 6 (PI.f32)) s) r)))): 9 points increase in error, 6 points decrease in error

3. Taylor expanded in s around 0 0.2

   \[\leadsto \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{\frac{-r}{s}}}{r} + \color{blue}{\frac{0.125}{s \cdot \pi}} \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r} \]

4. Simplified 0.2

   \[\leadsto \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{\frac{-r}{s}}}{r} + \color{blue}{\frac{\frac{0.125}{s}}{\pi}} \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r} \]
   Proof
   (/.f32 (/.f32 1/8 s) (PI.f32)): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate-/r*_binary32 (/.f32 1/8 (*.f32 s (PI.f32)))): 58 points increase in error, 48 points decrease in error

5. Taylor expanded in s around 0 0.2

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

6. Final simplification 0.2

   \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{-r}{s}}}{r} + \frac{\frac{0.125}{s}}{\pi} \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r} \]

Alternatives

Alternative 1
Error	0.2
Cost	13600

\[\begin{array}{l} t_0 := \frac{0.125}{s \cdot \pi}\\ t_0 \cdot \frac{e^{\frac{-r}{s}}}{r} + t_0 \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r} \end{array} \]

Alternative 2
Error	0.2
Cost	13568

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

Alternative 3
Error	0.2
Cost	10208

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

Alternative 4
Error	0.2
Cost	10144

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

Alternative 5
Error	0.2
Cost	10144

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

Alternative 6
Error	18.0
Cost	9792

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

Alternative 7
Error	29.0
Cost	6880

\[\frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{-r}{s}}}{r} + \frac{1}{r}\right) \]

Alternative 8
Error	29.1
Cost	6848

\[0.125 \cdot \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}} + 1}{s \cdot \left(\pi \cdot r\right)} \]

Alternative 9
Error	29.1
Cost	6848

\[0.125 \cdot \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}} + 1}{\pi \cdot \left(s \cdot r\right)} \]

Alternative 10
Error	29.1
Cost	3456

\[\frac{0.25}{s \cdot r} \cdot \frac{1}{\pi} \]

Alternative 11
Error	29.1
Cost	3392

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

Alternative 12
Error	29.1
Cost	3392

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

Alternative 13
Error	29.1
Cost	3392

\[\frac{\frac{0.25}{s}}{\pi \cdot r} \]

Alternative 14
Error	29.1
Cost	3392

\[\frac{\frac{0.25}{\pi}}{s \cdot r} \]

Reproduce

herbie shell --seed 2022329 
(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))))