Disney BSSRDF, PDF of scattering profile

Average Error: 0.1 → 0.1

Time: 8.7s

Precision: binary32

Cost: 13376

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

Math

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

↓

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

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
 (*
  0.125
  (/
   (+ (/ 1.0 (exp (/ r s))) (pow E (* (/ r s) -0.3333333333333333)))
   (* s (* r PI)))))

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 * (((1.0f / expf((r / s))) + powf(((float) M_E), ((r / s) * -0.3333333333333333f))) / (s * (r * ((float) M_PI))));
}

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(0.125) * Float32(Float32(Float32(Float32(1.0) / exp(Float32(r / s))) + (Float32(exp(1)) ^ Float32(Float32(r / s) * Float32(-0.3333333333333333)))) / Float32(s * Float32(r * Float32(pi)))))
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) * (((single(1.0) / exp((r / s))) + (single(2.71828182845904523536) ^ ((r / s) * single(-0.3333333333333333)))) / (s * (r * single(pi))));
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 1.3

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

3. Taylor expanded in s around 0 0.1

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

4. Applied egg-rr 0.1

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

5. Applied egg-rr 0.1

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

6. Final simplification 0.1

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

Alternatives

Alternative 1
Error	0.2
Cost	10176

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

Alternative 2
Error	0.7
Cost	10144

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

Alternative 3
Error	0.7
Cost	10144

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

Alternative 4
Error	0.2
Cost	10144

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

Alternative 5
Error	17.9
Cost	9792

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

Alternative 6
Error	28.9
Cost	6816

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

Alternative 7
Error	29.0
Cost	3520

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

Alternative 8
Error	29.1
Cost	3456

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

Alternative 9
Error	29.1
Cost	3456

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

Alternative 10
Error	29.1
Cost	3392

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

Alternative 11
Error	29.1
Cost	3392

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

Alternative 12
Error	29.1
Cost	3392

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

Alternative 13
Error	29.1
Cost	3392

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

Reproduce

herbie shell --seed 2022217 
(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))))