Disney BSSRDF, sample scattering profile, lower

Average Error: 12.4 → 0.4

Time: 1.1min

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

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

↓

\[\begin{array}{l} t_0 := 1 - 4 \cdot u\\ \mathbf{if}\;t_0 \leq 0.9700000286102295:\\ \;\;\;\;s \cdot \log \left(\frac{1}{t_0}\right)\\ \mathbf{else}:\\ \;\;\;\;s \cdot \left(\left(64 \cdot {u}^{4} + u \cdot 4\right) - \left({u}^{2} \cdot -8 - 21.333333333333332 \cdot {u}^{3}\right)\right)\\ \end{array} \]

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 0.9700000286102295)
     (* s (log (/ 1.0 t_0)))
     (*
      s
      (-
       (+ (* 64.0 (pow u 4.0)) (* u 4.0))
       (- (* (pow u 2.0) -8.0) (* 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 <= 0.9700000286102295f) {
		tmp = s * logf((1.0f / t_0));
	} else {
		tmp = s * (((64.0f * powf(u, 4.0f)) + (u * 4.0f)) - ((powf(u, 2.0f) * -8.0f) - (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 <= 0.9700000286102295e0) then
        tmp = s * log((1.0e0 / t_0))
    else
        tmp = s * (((64.0e0 * (u ** 4.0e0)) + (u * 4.0e0)) - (((u ** 2.0e0) * (-8.0e0)) - (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(0.9700000286102295))
		tmp = Float32(s * log(Float32(Float32(1.0) / t_0)));
	else
		tmp = Float32(s * Float32(Float32(Float32(Float32(64.0) * (u ^ Float32(4.0))) + Float32(u * Float32(4.0))) - Float32(Float32((u ^ Float32(2.0)) * Float32(-8.0)) - 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(0.9700000286102295))
		tmp = s * log((single(1.0) / t_0));
	else
		tmp = s * (((single(64.0) * (u ^ single(4.0))) + (u * single(4.0))) - (((u ^ single(2.0)) * single(-8.0)) - (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)) < 0.970000029

   1. Initial program 1.5

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

   if 0.970000029 < (-.f32 1 (*.f32 4 u))

   1. Initial program 14.6

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

   2. Taylor expanded in u around 0 0.2

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

   3. Simplified 0.2

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

      [Start] 0.2	\[ s \cdot \left(8 \cdot {u}^{2} + \left(64 \cdot {u}^{4} + \left(21.333333333333332 \cdot {u}^{3} + 4 \cdot u\right)\right)\right) \]
      rational_best-simplify-47 [=>] 0.2	\[ s \cdot \color{blue}{\left(\left(21.333333333333332 \cdot {u}^{3} + 4 \cdot u\right) + \left(64 \cdot {u}^{4} + 8 \cdot {u}^{2}\right)\right)} \]
      rational_best-simplify-3 [=>] 0.2	\[ s \cdot \color{blue}{\left(\left(64 \cdot {u}^{4} + 8 \cdot {u}^{2}\right) + \left(21.333333333333332 \cdot {u}^{3} + 4 \cdot u\right)\right)} \]
      rational_best-simplify-3 [=>] 0.2	\[ s \cdot \left(\left(64 \cdot {u}^{4} + 8 \cdot {u}^{2}\right) + \color{blue}{\left(4 \cdot u + 21.333333333333332 \cdot {u}^{3}\right)}\right) \]
      rational_best-simplify-47 [=>] 0.2	\[ s \cdot \color{blue}{\left(21.333333333333332 \cdot {u}^{3} + \left(4 \cdot u + \left(64 \cdot {u}^{4} + 8 \cdot {u}^{2}\right)\right)\right)} \]
      rational_best-simplify-3 [=>] 0.2	\[ s \cdot \left(21.333333333333332 \cdot {u}^{3} + \left(4 \cdot u + \color{blue}{\left(8 \cdot {u}^{2} + 64 \cdot {u}^{4}\right)}\right)\right) \]

   4. Applied egg-rr 0.2

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

   5. Simplified 0.2

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

      [Start] 0.2	\[ s \cdot \left(21.333333333333332 \cdot {u}^{3} - \left({u}^{2} \cdot -8 - \left(u \cdot 4 + 64 \cdot {u}^{4}\right)\right)\right) \]
      rational_best-simplify-51 [=>] 0.2	\[ s \cdot \color{blue}{\left(\left(u \cdot 4 + 64 \cdot {u}^{4}\right) - \left({u}^{2} \cdot -8 - 21.333333333333332 \cdot {u}^{3}\right)\right)} \]
      rational_best-simplify-3 [=>] 0.2	\[ s \cdot \left(\color{blue}{\left(64 \cdot {u}^{4} + u \cdot 4\right)} - \left({u}^{2} \cdot -8 - 21.333333333333332 \cdot {u}^{3}\right)\right) \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - 4 \cdot u \leq 0.9700000286102295:\\ \;\;\;\;s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right)\\ \mathbf{else}:\\ \;\;\;\;s \cdot \left(\left(64 \cdot {u}^{4} + u \cdot 4\right) - \left({u}^{2} \cdot -8 - 21.333333333333332 \cdot {u}^{3}\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.4
Cost	10436

\[\begin{array}{l} t_0 := 1 - 4 \cdot u\\ \mathbf{if}\;t_0 \leq 0.9700000286102295:\\ \;\;\;\;s \cdot \log \left(\frac{1}{t_0}\right)\\ \mathbf{else}:\\ \;\;\;\;s \cdot \left(8 \cdot {u}^{2} + \left(64 \cdot {u}^{4} + \left(21.333333333333332 \cdot {u}^{3} + 4 \cdot u\right)\right)\right)\\ \end{array} \]

Alternative 2
Error	0.4
Cost	10436

\[\begin{array}{l} t_0 := 1 - 4 \cdot u\\ \mathbf{if}\;t_0 \leq 0.9700000286102295:\\ \;\;\;\;s \cdot \log \left(\frac{1}{t_0}\right)\\ \mathbf{else}:\\ \;\;\;\;s \cdot \left(21.333333333333332 \cdot {u}^{3} + \left(4 \cdot u + \left(8 \cdot {u}^{2} + 64 \cdot {u}^{4}\right)\right)\right)\\ \end{array} \]

Alternative 3
Error	0.6
Cost	7076

\[\begin{array}{l} t_0 := 1 - 4 \cdot u\\ \mathbf{if}\;t_0 \leq 0.9800000190734863:\\ \;\;\;\;s \cdot \log \left(\frac{1}{t_0}\right)\\ \mathbf{else}:\\ \;\;\;\;s \cdot \left(4 \cdot u + \left(8 \cdot {u}^{2} + 21.333333333333332 \cdot {u}^{3}\right)\right)\\ \end{array} \]

Alternative 4
Error	1.1
Cost	3652

\[\begin{array}{l} \mathbf{if}\;4 \cdot u \leq 0.003000000026077032:\\ \;\;\;\;s \cdot \left(8 \cdot {u}^{2} + 4 \cdot u\right)\\ \mathbf{else}:\\ \;\;\;\;s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right)\\ \end{array} \]

Alternative 5
Error	3.4
Cost	3620

\[\begin{array}{l} \mathbf{if}\;4 \cdot u \leq 0.0002800000074785203:\\ \;\;\;\;s \cdot \left(4 \cdot u\right)\\ \mathbf{else}:\\ \;\;\;\;s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right)\\ \end{array} \]

Alternative 6
Error	8.3
Cost	160

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

Alternative 7
Error	8.2
Cost	160

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

Reproduce

herbie shell --seed 2023099 
(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))))))