Disney BSSRDF, sample scattering profile, upper

Average Error: 1.3 → 1.1

Time: 56.5s

Precision: binary32

Cost: 6848

\[\left(0 \leq s \land s \leq 256\right) \land \left(0.25 \leq u \land u \leq 1\right)\]

Math

\[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]

↓

\[\frac{s \cdot \log \left(\frac{1}{{\left(1 - 1.3333333333333333 \cdot \left(u - 0.25\right)\right)}^{12}}\right)}{4} \]

FPCore

(FPCore (s u)
 :precision binary32
 (* (* 3.0 s) (log (/ 1.0 (- 1.0 (/ (- u 0.25) 0.75))))))

↓

(FPCore (s u)
 :precision binary32
 (/
  (* s (log (/ 1.0 (pow (- 1.0 (* 1.3333333333333333 (- u 0.25))) 12.0))))
  4.0))

C

float code(float s, float u) {
	return (3.0f * s) * logf((1.0f / (1.0f - ((u - 0.25f) / 0.75f))));
}

↓

float code(float s, float u) {
	return (s * logf((1.0f / powf((1.0f - (1.3333333333333333f * (u - 0.25f))), 12.0f)))) / 4.0f;
}

Fortran

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = (3.0e0 * s) * log((1.0e0 / (1.0e0 - ((u - 0.25e0) / 0.75e0))))
end function

↓

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    code = (s * log((1.0e0 / ((1.0e0 - (1.3333333333333333e0 * (u - 0.25e0))) ** 12.0e0)))) / 4.0e0
end function

Julia

function code(s, u)
	return Float32(Float32(Float32(3.0) * s) * log(Float32(Float32(1.0) / Float32(Float32(1.0) - Float32(Float32(u - Float32(0.25)) / Float32(0.75))))))
end

↓

function code(s, u)
	return Float32(Float32(s * log(Float32(Float32(1.0) / (Float32(Float32(1.0) - Float32(Float32(1.3333333333333333) * Float32(u - Float32(0.25)))) ^ Float32(12.0))))) / Float32(4.0))
end

MATLAB

function tmp = code(s, u)
	tmp = (single(3.0) * s) * log((single(1.0) / (single(1.0) - ((u - single(0.25)) / single(0.75)))));
end

↓

function tmp = code(s, u)
	tmp = (s * log((single(1.0) / ((single(1.0) - (single(1.3333333333333333) * (u - single(0.25)))) ^ single(12.0))))) / single(4.0);
end

Derivation

1. Initial program 1.3

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]

2. Taylor expanded in s around 0 1.4

   \[\leadsto \color{blue}{3 \cdot \left(s \cdot \log \left(\frac{1}{1 - 1.3333333333333333 \cdot \left(u - 0.25\right)}\right)\right)} \]

3. Applied egg-rr 23.8

   \[\leadsto \color{blue}{\frac{\log \left({\left(\frac{1}{1 - 1.3333333333333333 \cdot \left(u - 0.25\right)}\right)}^{s}\right) \cdot 12}{4}} \]

4. Simplified 1.4

   \[\leadsto \color{blue}{\frac{s \cdot \log \left({\left(\frac{1}{1 - 1.3333333333333333 \cdot \left(u - 0.25\right)}\right)}^{12}\right)}{4}} \]
   Proof

   [Start] 23.8	\[ \frac{\log \left({\left(\frac{1}{1 - 1.3333333333333333 \cdot \left(u - 0.25\right)}\right)}^{s}\right) \cdot 12}{4} \]
   rational_best-simplify-1 [=>] 23.8	\[ \frac{\color{blue}{12 \cdot \log \left({\left(\frac{1}{1 - 1.3333333333333333 \cdot \left(u - 0.25\right)}\right)}^{s}\right)}}{4} \]
   exponential-simplify-30 [=>] 1.4	\[ \frac{\color{blue}{s \cdot \log \left({\left(\frac{1}{1 - 1.3333333333333333 \cdot \left(u - 0.25\right)}\right)}^{12}\right)}}{4} \]

5. Taylor expanded in s around 0 1.1

   \[\leadsto \frac{\color{blue}{s \cdot \log \left(\frac{1}{{\left(1 - 1.3333333333333333 \cdot \left(u - 0.25\right)\right)}^{12}}\right)}}{4} \]

6. Final simplification 1.1

   \[\leadsto \frac{s \cdot \log \left(\frac{1}{{\left(1 - 1.3333333333333333 \cdot \left(u - 0.25\right)\right)}^{12}}\right)}{4} \]

Alternatives

Alternative 1
Error	1.2
Cost	6784

\[\frac{s \cdot \log \left(\frac{1}{{\left(-1.3333333333333333 \cdot u + 1.3333333333333333\right)}^{12}}\right)}{4} \]

Alternative 2
Error	1.2
Cost	3744

\[3 \cdot \left(s \cdot \log \left(\frac{1}{\left(u - 0.25\right) \cdot -0.3333333333333333 - \left(\left(u - 0.25\right) + -1\right)}\right)\right) \]

Alternative 3
Error	1.4
Cost	3616

\[3 \cdot \left(s \cdot \log \left(\frac{1}{1 - 1.3333333333333333 \cdot \left(u - 0.25\right)}\right)\right) \]

Alternative 4
Error	1.3
Cost	3616

\[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]

Alternative 5
Error	1.8
Cost	3552

\[3 \cdot \left(s \cdot \log \left(\frac{1}{1.3333333333333333 + \frac{u}{-0.75}}\right)\right) \]

Alternative 6
Error	1.4
Cost	3552

\[3 \cdot \left(s \cdot \log \left(\frac{2}{u \cdot -2.6666666666666665 + 2.6666666666666665}\right)\right) \]

Alternative 7
Error	22.3
Cost	672

\[s \cdot \left(u + u\right) + \left(s \cdot \left(u \cdot 0.5\right) - \left(1 + \left(-1 + \frac{s \cdot u}{-2}\right)\right)\right) \]

Alternative 8
Error	22.3
Cost	608

\[\left(s \cdot u + s \cdot \left(u \cdot 1.5\right)\right) + \left(1 - \left(1 + \frac{s \cdot u}{-2}\right)\right) \]

Alternative 9
Error	22.4
Cost	480

\[\frac{s \cdot u}{-2} + \left(s \cdot u - \left(s \cdot u\right) \cdot -2.5\right) \]

Alternative 10
Error	22.4
Cost	224

\[\frac{s \cdot \left(u \cdot 6\right)}{2} \]

Alternative 11
Error	22.4
Cost	160

\[3 \cdot \left(s \cdot u\right) \]

Alternative 12
Error	22.4
Cost	160

\[s \cdot \left(u \cdot 3\right) \]

Reproduce

herbie shell --seed 2023099 
(FPCore (s u)
  :name "Disney BSSRDF, sample scattering profile, upper"
  :precision binary32
  :pre (and (and (<= 0.0 s) (<= s 256.0)) (and (<= 0.25 u) (<= u 1.0)))
  (* (* 3.0 s) (log (/ 1.0 (- 1.0 (/ (- u 0.25) 0.75))))))