Disney BSSRDF, sample scattering profile, lower

Average Error: 12.4 → 0.4

Time: 28.9s

Precision: binary32

Cost: 10436

\[\left(0 \leq s \land s \leq 256\right) \land \left(2.328306437 \cdot 10^{-10} \leq u \land u \leq 0.25\right)\]

Math

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

↓

\[\begin{array}{l} t_0 := 1 - 4 \cdot u\\ \mathbf{if}\;t_0 \leq 0.9549999833106995:\\ \;\;\;\;s \cdot \left(\frac{\log \left({\left(\frac{1}{t_0}\right)}^{2}\right)}{4} \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;s \cdot \left(\left(4 \cdot u + 64 \cdot {u}^{4}\right) + \left(8 \cdot {u}^{2} + 21.333333333333332 \cdot {u}^{3}\right)\right)\\ \end{array} \]

FPCore

(FPCore (s u) :precision binary32 (* s (log (/ 1.0 (- 1.0 (* 4.0 u))))))

↓

(FPCore (s u)
 :precision binary32
 (let* ((t_0 (- 1.0 (* 4.0 u))))
   (if (<= t_0 0.9549999833106995)
     (* s (* (/ (log (pow (/ 1.0 t_0) 2.0)) 4.0) 2.0))
     (*
      s
      (+
       (+ (* 4.0 u) (* 64.0 (pow u 4.0)))
       (+ (* 8.0 (pow u 2.0)) (* 21.333333333333332 (pow u 3.0))))))))

C

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

↓

float code(float s, float u) {
	float t_0 = 1.0f - (4.0f * u);
	float tmp;
	if (t_0 <= 0.9549999833106995f) {
		tmp = s * ((logf(powf((1.0f / t_0), 2.0f)) / 4.0f) * 2.0f);
	} else {
		tmp = s * (((4.0f * u) + (64.0f * powf(u, 4.0f))) + ((8.0f * powf(u, 2.0f)) + (21.333333333333332f * powf(u, 3.0f))));
	}
	return tmp;
}

Fortran

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

↓

real(4) function code(s, u)
    real(4), intent (in) :: s
    real(4), intent (in) :: u
    real(4) :: t_0
    real(4) :: tmp
    t_0 = 1.0e0 - (4.0e0 * u)
    if (t_0 <= 0.9549999833106995e0) then
        tmp = s * ((log(((1.0e0 / t_0) ** 2.0e0)) / 4.0e0) * 2.0e0)
    else
        tmp = s * (((4.0e0 * u) + (64.0e0 * (u ** 4.0e0))) + ((8.0e0 * (u ** 2.0e0)) + (21.333333333333332e0 * (u ** 3.0e0))))
    end if
    code = tmp
end function

Julia

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

↓

function code(s, u)
	t_0 = Float32(Float32(1.0) - Float32(Float32(4.0) * u))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.9549999833106995))
		tmp = Float32(s * Float32(Float32(log((Float32(Float32(1.0) / t_0) ^ Float32(2.0))) / Float32(4.0)) * Float32(2.0)));
	else
		tmp = Float32(s * Float32(Float32(Float32(Float32(4.0) * u) + Float32(Float32(64.0) * (u ^ Float32(4.0)))) + Float32(Float32(Float32(8.0) * (u ^ Float32(2.0))) + Float32(Float32(21.333333333333332) * (u ^ Float32(3.0))))));
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(s, u)
	t_0 = single(1.0) - (single(4.0) * u);
	tmp = single(0.0);
	if (t_0 <= single(0.9549999833106995))
		tmp = s * ((log(((single(1.0) / t_0) ^ single(2.0))) / single(4.0)) * single(2.0));
	else
		tmp = s * (((single(4.0) * u) + (single(64.0) * (u ^ single(4.0)))) + ((single(8.0) * (u ^ single(2.0))) + (single(21.333333333333332) * (u ^ single(3.0)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 1 (*.f32 4 u)) < 0.954999983

   1. Initial program 1.3

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

   2. Applied egg-rr 1.8

      \[\leadsto s \cdot \color{blue}{\left(\log \left(\sqrt{\frac{1}{1 - 4 \cdot u}}\right) \cdot 2\right)} \]

   3. Applied egg-rr 1.3

      \[\leadsto s \cdot \left(\color{blue}{\frac{\log \left({\left(\frac{1}{1 - 4 \cdot u}\right)}^{2}\right)}{4}} \cdot 2\right) \]

   if 0.954999983 < (-.f32 1 (*.f32 4 u))

   1. Initial program 14.3

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

   2. Taylor expanded in u around 0 0.3

      \[\leadsto s \cdot \color{blue}{\left(8 \cdot {u}^{2} + \left(64 \cdot {u}^{4} + \left(21.333333333333332 \cdot {u}^{3} + 4 \cdot u\right)\right)\right)} \]

   3. Simplified 0.3

      \[\leadsto s \cdot \color{blue}{\left(\left(4 \cdot u + 64 \cdot {u}^{4}\right) + \left(8 \cdot {u}^{2} + 21.333333333333332 \cdot {u}^{3}\right)\right)} \]
      Proof

      [Start] 0.3	\[ s \cdot \left(8 \cdot {u}^{2} + \left(64 \cdot {u}^{4} + \left(21.333333333333332 \cdot {u}^{3} + 4 \cdot u\right)\right)\right) \]
      rational.json-simplify-41 [=>] 0.3	\[ s \cdot \left(8 \cdot {u}^{2} + \color{blue}{\left(21.333333333333332 \cdot {u}^{3} + \left(4 \cdot u + 64 \cdot {u}^{4}\right)\right)}\right) \]
      rational.json-simplify-41 [<=] 0.3	\[ s \cdot \color{blue}{\left(\left(4 \cdot u + 64 \cdot {u}^{4}\right) + \left(8 \cdot {u}^{2} + 21.333333333333332 \cdot {u}^{3}\right)\right)} \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - 4 \cdot u \leq 0.9549999833106995:\\ \;\;\;\;s \cdot \left(\frac{\log \left({\left(\frac{1}{1 - 4 \cdot u}\right)}^{2}\right)}{4} \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;s \cdot \left(\left(4 \cdot u + 64 \cdot {u}^{4}\right) + \left(8 \cdot {u}^{2} + 21.333333333333332 \cdot {u}^{3}\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.4
Cost	10436

\[\begin{array}{l} t_0 := 1 - 4 \cdot u\\ \mathbf{if}\;t_0 \leq 0.9549999833106995:\\ \;\;\;\;s \cdot \left(\frac{\log \left({\left(\frac{1}{t_0}\right)}^{2}\right)}{4} \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;s \cdot \left(8 \cdot {u}^{2} + \left(u \cdot 4 + \left(21.333333333333332 \cdot {u}^{3} + 64 \cdot {u}^{4}\right)\right)\right)\\ \end{array} \]

Alternative 2
Error	0.6
Cost	7012

\[\begin{array}{l} \mathbf{if}\;4 \cdot u \leq 0.014499999582767487:\\ \;\;\;\;s \cdot \left(4 \cdot u + \left(8 \cdot {u}^{2} + 21.333333333333332 \cdot {u}^{3}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right)\\ \end{array} \]

Alternative 3
Error	1.1
Cost	6980

\[\begin{array}{l} \mathbf{if}\;4 \cdot u \leq 0.0036800000816583633:\\ \;\;\;\;u \cdot \left(4 \cdot s\right) + {u}^{2} \cdot \left(8 \cdot s\right)\\ \mathbf{else}:\\ \;\;\;\;s \cdot \left(\frac{\log \left({\left(\frac{1}{1 - 4 \cdot u}\right)}^{2}\right)}{4} \cdot 2\right)\\ \end{array} \]

Alternative 4
Error	1.1
Cost	3716

\[\begin{array}{l} \mathbf{if}\;4 \cdot u \leq 0.0036800000816583633:\\ \;\;\;\;\left(2 \cdot \left(u + \left(u + {\left(u + u\right)}^{2}\right)\right)\right) \cdot s\\ \mathbf{else}:\\ \;\;\;\;s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right)\\ \end{array} \]

Alternative 5
Error	1.1
Cost	3716

\[\begin{array}{l} \mathbf{if}\;4 \cdot u \leq 0.0036800000816583633:\\ \;\;\;\;u \cdot \left(4 \cdot s\right) + {u}^{2} \cdot \left(8 \cdot s\right)\\ \mathbf{else}:\\ \;\;\;\;s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right)\\ \end{array} \]

Alternative 6
Error	1.1
Cost	3652

\[\begin{array}{l} \mathbf{if}\;4 \cdot u \leq 0.0036800000816583633:\\ \;\;\;\;\left(4 \cdot u + 8 \cdot {u}^{2}\right) \cdot s\\ \mathbf{else}:\\ \;\;\;\;s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right)\\ \end{array} \]

Alternative 7
Error	3.5
Cost	3620

\[\begin{array}{l} \mathbf{if}\;4 \cdot u \leq 0.00011999999696854502:\\ \;\;\;\;s \cdot \left(4 \cdot u\right)\\ \mathbf{else}:\\ \;\;\;\;s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right)\\ \end{array} \]

Alternative 8
Error	8.3
Cost	160

\[4 \cdot \left(u \cdot s\right) \]

Alternative 9
Error	8.3
Cost	160

\[s \cdot \left(4 \cdot u\right) \]

Reproduce

herbie shell --seed 2023073 
(FPCore (s u)
  :name "Disney BSSRDF, sample scattering profile, lower"
  :precision binary32
  :pre (and (and (<= 0.0 s) (<= s 256.0)) (and (<= 2.328306437e-10 u) (<= u 0.25)))
  (* s (log (/ 1.0 (- 1.0 (* 4.0 u))))))