Disney BSSRDF, sample scattering profile, lower

Average Error: 12.4 → 0.4

Time: 37.2s

Precision: binary32

Cost: 10500

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

Math

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

↓

\[\begin{array}{l} t_0 := 1 - 4 \cdot u\\ t_1 := t_0 \cdot t_0\\ \mathbf{if}\;t_0 \leq 0.9559999704360962:\\ \;\;\;\;s \cdot \log \left(\frac{t_1}{t_0 \cdot t_1}\right)\\ \mathbf{else}:\\ \;\;\;\;u \cdot \left(4 \cdot s\right) + s \cdot \left(21.333333333333332 \cdot {u}^{3} + \left(8 \cdot {u}^{2} + 64 \cdot {u}^{4}\right)\right)\\ \end{array} \]

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))) (t_1 (* t_0 t_0)))
   (if (<= t_0 0.9559999704360962)
     (* s (log (/ t_1 (* t_0 t_1))))
     (+
      (* u (* 4.0 s))
      (*
       s
       (+
        (* 21.333333333333332 (pow u 3.0))
        (+ (* 8.0 (pow u 2.0)) (* 64.0 (pow u 4.0)))))))))

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 t_1 = t_0 * t_0;
	float tmp;
	if (t_0 <= 0.9559999704360962f) {
		tmp = s * logf((t_1 / (t_0 * t_1)));
	} else {
		tmp = (u * (4.0f * s)) + (s * ((21.333333333333332f * powf(u, 3.0f)) + ((8.0f * powf(u, 2.0f)) + (64.0f * powf(u, 4.0f)))));
	}
	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) :: t_1
    real(4) :: tmp
    t_0 = 1.0e0 - (4.0e0 * u)
    t_1 = t_0 * t_0
    if (t_0 <= 0.9559999704360962e0) then
        tmp = s * log((t_1 / (t_0 * t_1)))
    else
        tmp = (u * (4.0e0 * s)) + (s * ((21.333333333333332e0 * (u ** 3.0e0)) + ((8.0e0 * (u ** 2.0e0)) + (64.0e0 * (u ** 4.0e0)))))
    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))
	t_1 = Float32(t_0 * t_0)
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.9559999704360962))
		tmp = Float32(s * log(Float32(t_1 / Float32(t_0 * t_1))));
	else
		tmp = Float32(Float32(u * Float32(Float32(4.0) * s)) + Float32(s * Float32(Float32(Float32(21.333333333333332) * (u ^ Float32(3.0))) + Float32(Float32(Float32(8.0) * (u ^ Float32(2.0))) + Float32(Float32(64.0) * (u ^ Float32(4.0)))))));
	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);
	t_1 = t_0 * t_0;
	tmp = single(0.0);
	if (t_0 <= single(0.9559999704360962))
		tmp = s * log((t_1 / (t_0 * t_1)));
	else
		tmp = (u * (single(4.0) * s)) + (s * ((single(21.333333333333332) * (u ^ single(3.0))) + ((single(8.0) * (u ^ single(2.0))) + (single(64.0) * (u ^ single(4.0))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 1 (*.f32 4 u)) < 0.95599997

   1. Initial program 1.3

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

   2. Applied egg-rr 1.5

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

   3. Applied egg-rr 1.4

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

   if 0.95599997 < (-.f32 1 (*.f32 4 u))

   1. Initial program 14.4

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

   2. Taylor expanded in u around 0 0.4

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

   3. Simplified 0.3

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

      [Start] 0.4	\[ 21.333333333333332 \cdot \left(s \cdot {u}^{3}\right) + \left(4 \cdot \left(s \cdot u\right) + \left(8 \cdot \left(s \cdot {u}^{2}\right) + 64 \cdot \left(s \cdot {u}^{4}\right)\right)\right) \]
      rational.json-simplify-41 [=>] 0.4	\[ \color{blue}{4 \cdot \left(s \cdot u\right) + \left(\left(8 \cdot \left(s \cdot {u}^{2}\right) + 64 \cdot \left(s \cdot {u}^{4}\right)\right) + 21.333333333333332 \cdot \left(s \cdot {u}^{3}\right)\right)} \]
      rational.json-simplify-43 [<=] 0.3	\[ \color{blue}{u \cdot \left(4 \cdot s\right)} + \left(\left(8 \cdot \left(s \cdot {u}^{2}\right) + 64 \cdot \left(s \cdot {u}^{4}\right)\right) + 21.333333333333332 \cdot \left(s \cdot {u}^{3}\right)\right) \]
      rational.json-simplify-43 [=>] 0.3	\[ u \cdot \left(4 \cdot s\right) + \left(\left(\color{blue}{s \cdot \left({u}^{2} \cdot 8\right)} + 64 \cdot \left(s \cdot {u}^{4}\right)\right) + 21.333333333333332 \cdot \left(s \cdot {u}^{3}\right)\right) \]
      rational.json-simplify-2 [<=] 0.3	\[ u \cdot \left(4 \cdot s\right) + \left(\left(s \cdot \color{blue}{\left(8 \cdot {u}^{2}\right)} + 64 \cdot \left(s \cdot {u}^{4}\right)\right) + 21.333333333333332 \cdot \left(s \cdot {u}^{3}\right)\right) \]
      rational.json-simplify-43 [=>] 0.3	\[ u \cdot \left(4 \cdot s\right) + \left(\left(s \cdot \left(8 \cdot {u}^{2}\right) + \color{blue}{s \cdot \left({u}^{4} \cdot 64\right)}\right) + 21.333333333333332 \cdot \left(s \cdot {u}^{3}\right)\right) \]
      rational.json-simplify-2 [<=] 0.3	\[ u \cdot \left(4 \cdot s\right) + \left(\left(s \cdot \left(8 \cdot {u}^{2}\right) + s \cdot \color{blue}{\left(64 \cdot {u}^{4}\right)}\right) + 21.333333333333332 \cdot \left(s \cdot {u}^{3}\right)\right) \]
      rational.json-simplify-2 [=>] 0.3	\[ u \cdot \left(4 \cdot s\right) + \left(\left(s \cdot \left(8 \cdot {u}^{2}\right) + \color{blue}{\left(64 \cdot {u}^{4}\right) \cdot s}\right) + 21.333333333333332 \cdot \left(s \cdot {u}^{3}\right)\right) \]
      rational.json-simplify-51 [=>] 0.3	\[ u \cdot \left(4 \cdot s\right) + \left(\color{blue}{s \cdot \left(64 \cdot {u}^{4} + 8 \cdot {u}^{2}\right)} + 21.333333333333332 \cdot \left(s \cdot {u}^{3}\right)\right) \]
      rational.json-simplify-1 [<=] 0.3	\[ u \cdot \left(4 \cdot s\right) + \left(s \cdot \color{blue}{\left(8 \cdot {u}^{2} + 64 \cdot {u}^{4}\right)} + 21.333333333333332 \cdot \left(s \cdot {u}^{3}\right)\right) \]
      rational.json-simplify-43 [=>] 0.3	\[ u \cdot \left(4 \cdot s\right) + \left(s \cdot \left(8 \cdot {u}^{2} + 64 \cdot {u}^{4}\right) + \color{blue}{s \cdot \left({u}^{3} \cdot 21.333333333333332\right)}\right) \]
      rational.json-simplify-2 [<=] 0.3	\[ u \cdot \left(4 \cdot s\right) + \left(s \cdot \left(8 \cdot {u}^{2} + 64 \cdot {u}^{4}\right) + s \cdot \color{blue}{\left(21.333333333333332 \cdot {u}^{3}\right)}\right) \]
      rational.json-simplify-2 [=>] 0.3	\[ u \cdot \left(4 \cdot s\right) + \left(s \cdot \left(8 \cdot {u}^{2} + 64 \cdot {u}^{4}\right) + \color{blue}{\left(21.333333333333332 \cdot {u}^{3}\right) \cdot s}\right) \]
      rational.json-simplify-51 [=>] 0.3	\[ u \cdot \left(4 \cdot s\right) + \color{blue}{s \cdot \left(21.333333333333332 \cdot {u}^{3} + \left(8 \cdot {u}^{2} + 64 \cdot {u}^{4}\right)\right)} \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - 4 \cdot u \leq 0.9559999704360962:\\ \;\;\;\;s \cdot \log \left(\frac{\left(1 - 4 \cdot u\right) \cdot \left(1 - 4 \cdot u\right)}{\left(1 - 4 \cdot u\right) \cdot \left(\left(1 - 4 \cdot u\right) \cdot \left(1 - 4 \cdot u\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;u \cdot \left(4 \cdot s\right) + s \cdot \left(21.333333333333332 \cdot {u}^{3} + \left(8 \cdot {u}^{2} + 64 \cdot {u}^{4}\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.4
Cost	10436

\[\begin{array}{l} t_0 := 1 - 4 \cdot u\\ t_1 := t_0 \cdot t_0\\ \mathbf{if}\;t_0 \leq 0.9559999704360962:\\ \;\;\;\;s \cdot \log \left(\frac{t_1}{t_0 \cdot t_1}\right)\\ \mathbf{else}:\\ \;\;\;\;s \cdot \left(21.333333333333332 \cdot {u}^{3} + \left(\left(4 \cdot u + 8 \cdot {u}^{2}\right) + 64 \cdot {u}^{4}\right)\right)\\ \end{array} \]

Alternative 2
Error	0.4
Cost	10436

\[\begin{array}{l} t_0 := 1 - 4 \cdot u\\ t_1 := t_0 \cdot t_0\\ \mathbf{if}\;t_0 \leq 0.9559999704360962:\\ \;\;\;\;s \cdot \log \left(\frac{t_1}{t_0 \cdot t_1}\right)\\ \mathbf{else}:\\ \;\;\;\;s \cdot \left(\left(4 \cdot u + 8 \cdot {u}^{2}\right) + \left(21.333333333333332 \cdot {u}^{3} + 64 \cdot {u}^{4}\right)\right)\\ \end{array} \]

Alternative 3
Error	0.4
Cost	10436

\[\begin{array}{l} t_0 := 1 - 4 \cdot u\\ t_1 := t_0 \cdot t_0\\ \mathbf{if}\;t_0 \leq 0.9559999704360962:\\ \;\;\;\;s \cdot \log \left(\frac{t_1}{t_0 \cdot t_1}\right)\\ \mathbf{else}:\\ \;\;\;\;s \cdot \left(\left(4 \cdot u + 64 \cdot {u}^{4}\right) + \left(8 \cdot {u}^{2} + 21.333333333333332 \cdot {u}^{3}\right)\right)\\ \end{array} \]

Alternative 4
Error	0.6
Cost	7140

\[\begin{array}{l} t_0 := 1 - 4 \cdot u\\ \mathbf{if}\;t_0 \leq 0.9819999933242798:\\ \;\;\;\;s \cdot \log \left(\frac{t_0}{t_0 \cdot t_0}\right)\\ \mathbf{else}:\\ \;\;\;\;u \cdot \left(4 \cdot s\right) + s \cdot \left(8 \cdot {u}^{2} + 21.333333333333332 \cdot {u}^{3}\right)\\ \end{array} \]

Alternative 5
Error	0.6
Cost	7076

\[\begin{array}{l} t_0 := 1 - 4 \cdot u\\ \mathbf{if}\;4 \cdot u \leq 0.017999999225139618:\\ \;\;\;\;s \cdot \left(8 \cdot {u}^{2}\right) + s \cdot \left(21.333333333333332 \cdot {u}^{3} + u \cdot 4\right)\\ \mathbf{else}:\\ \;\;\;\;s \cdot \log \left(\frac{t_0}{t_0 \cdot t_0}\right)\\ \end{array} \]

Alternative 6
Error	0.6
Cost	7012

\[\begin{array}{l} t_0 := 1 - 4 \cdot u\\ \mathbf{if}\;4 \cdot u \leq 0.017999999225139618:\\ \;\;\;\;s \cdot \left(4 \cdot u + \left(21.333333333333332 \cdot {u}^{3} + 8 \cdot {u}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;s \cdot \log \left(\frac{t_0}{t_0 \cdot t_0}\right)\\ \end{array} \]

Alternative 7
Error	1.2
Cost	3940

\[\begin{array}{l} t_0 := 1 - 4 \cdot u\\ \mathbf{if}\;4 \cdot u \leq 0.005799999926239252:\\ \;\;\;\;u \cdot \left(4 \cdot s\right) + s \cdot \left(8 \cdot {u}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;s \cdot \log \left(\frac{t_0}{t_0 \cdot t_0}\right)\\ \end{array} \]

Alternative 8
Error	1.2
Cost	3716

\[\begin{array}{l} \mathbf{if}\;4 \cdot u \leq 0.005799999926239252:\\ \;\;\;\;u \cdot \left(4 \cdot s\right) + s \cdot \left(8 \cdot {u}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right)\\ \end{array} \]

Alternative 9
Error	3.5
Cost	3684

\[\begin{array}{l} t_0 := 1 - 4 \cdot u\\ \mathbf{if}\;t_0 \leq 0.9997400045394897:\\ \;\;\;\;s \cdot \log \left(\frac{1}{t_0}\right)\\ \mathbf{else}:\\ \;\;\;\;u \cdot \left(4 \cdot s\right)\\ \end{array} \]

Alternative 10
Error	1.3
Cost	3652

\[\begin{array}{l} \mathbf{if}\;4 \cdot u \leq 0.004000000189989805:\\ \;\;\;\;8 \cdot \left(s \cdot \left({u}^{2} + u \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right)\\ \end{array} \]

Alternative 11
Error	1.1
Cost	3652

\[\begin{array}{l} \mathbf{if}\;4 \cdot u \leq 0.005799999926239252:\\ \;\;\;\;\left(4 \cdot u + 8 \cdot {u}^{2}\right) \cdot s\\ \mathbf{else}:\\ \;\;\;\;s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right)\\ \end{array} \]

Alternative 12
Error	8.3
Cost	160

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

Alternative 13
Error	8.3
Cost	160

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

Reproduce

herbie shell --seed 2023074 
(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))))))