Disney BSSRDF, sample scattering profile, lower

Average Error: 12.3 → 0.4

Time: 14.5s

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

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

↓

\[\begin{array}{l} t_0 := \frac{1}{u \cdot -4 + 1}\\ \mathbf{if}\;1 - 4 \cdot u \leq 0.9580000042915344:\\ \;\;\;\;s \cdot \log \left(t_0 \cdot \left(t_0 \cdot \frac{1}{t_0}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;s \cdot \left(\left(21.333333333333332 \cdot {u}^{3} + \left(64 \cdot {u}^{4} + 8 \cdot {u}^{2}\right)\right) - u \cdot -4\right)\\ \end{array} \]

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 (+ (* u -4.0) 1.0))))
   (if (<= (- 1.0 (* 4.0 u)) 0.9580000042915344)
     (* s (log (* t_0 (* t_0 (/ 1.0 t_0)))))
     (*
      s
      (-
       (+
        (* 21.333333333333332 (pow u 3.0))
        (+ (* 64.0 (pow u 4.0)) (* 8.0 (pow u 2.0))))
       (* 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 / ((u * -4.0f) + 1.0f);
	float tmp;
	if ((1.0f - (4.0f * u)) <= 0.9580000042915344f) {
		tmp = s * logf((t_0 * (t_0 * (1.0f / t_0))));
	} else {
		tmp = s * (((21.333333333333332f * powf(u, 3.0f)) + ((64.0f * powf(u, 4.0f)) + (8.0f * powf(u, 2.0f)))) - (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) :: tmp
    t_0 = 1.0e0 / ((u * (-4.0e0)) + 1.0e0)
    if ((1.0e0 - (4.0e0 * u)) <= 0.9580000042915344e0) then
        tmp = s * log((t_0 * (t_0 * (1.0e0 / t_0))))
    else
        tmp = s * (((21.333333333333332e0 * (u ** 3.0e0)) + ((64.0e0 * (u ** 4.0e0)) + (8.0e0 * (u ** 2.0e0)))) - (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(u * Float32(-4.0)) + Float32(1.0)))
	tmp = Float32(0.0)
	if (Float32(Float32(1.0) - Float32(Float32(4.0) * u)) <= Float32(0.9580000042915344))
		tmp = Float32(s * log(Float32(t_0 * Float32(t_0 * Float32(Float32(1.0) / t_0)))));
	else
		tmp = Float32(s * Float32(Float32(Float32(Float32(21.333333333333332) * (u ^ Float32(3.0))) + Float32(Float32(Float32(64.0) * (u ^ Float32(4.0))) + Float32(Float32(8.0) * (u ^ Float32(2.0))))) - Float32(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) / ((u * single(-4.0)) + single(1.0));
	tmp = single(0.0);
	if ((single(1.0) - (single(4.0) * u)) <= single(0.9580000042915344))
		tmp = s * log((t_0 * (t_0 * (single(1.0) / t_0))));
	else
		tmp = s * (((single(21.333333333333332) * (u ^ single(3.0))) + ((single(64.0) * (u ^ single(4.0))) + (single(8.0) * (u ^ single(2.0))))) - (u * single(-4.0)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (-.f32 1 (*.f32 4 u)) < 0.958000004

   1. Initial program 1.3

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

   2. Applied egg-rr 1.4

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

   if 0.958000004 < (-.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 0.3

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

   3. Simplified 0.3

      \[\leadsto s \cdot \color{blue}{\left(21.333333333333332 \cdot {u}^{3} + \left(64 \cdot {u}^{4} + \left(4 \cdot u + 8 \cdot {u}^{2}\right)\right)\right)} \]
      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_best_oopsla_all_46_json_45_simplify-82 [=>] 0.3	\[ s \cdot \color{blue}{\left(64 \cdot {u}^{4} + \left(8 \cdot {u}^{2} + \left(21.333333333333332 \cdot {u}^{3} + 4 \cdot u\right)\right)\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-82 [=>] 0.3	\[ s \cdot \left(64 \cdot {u}^{4} + \color{blue}{\left(21.333333333333332 \cdot {u}^{3} + \left(8 \cdot {u}^{2} + 4 \cdot u\right)\right)}\right) \]
      rational_best_oopsla_all_46_json_45_simplify-82 [=>] 0.3	\[ s \cdot \color{blue}{\left(21.333333333333332 \cdot {u}^{3} + \left(64 \cdot {u}^{4} + \left(8 \cdot {u}^{2} + 4 \cdot u\right)\right)\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-35 [=>] 0.3	\[ s \cdot \left(21.333333333333332 \cdot {u}^{3} + \left(64 \cdot {u}^{4} + \color{blue}{\left(4 \cdot u + 8 \cdot {u}^{2}\right)}\right)\right) \]

   4. Applied egg-rr 0.2

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

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - 4 \cdot u \leq 0.9580000042915344:\\ \;\;\;\;s \cdot \log \left(\frac{1}{u \cdot -4 + 1} \cdot \left(\frac{1}{u \cdot -4 + 1} \cdot \frac{1}{\frac{1}{u \cdot -4 + 1}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;s \cdot \left(\left(21.333333333333332 \cdot {u}^{3} + \left(64 \cdot {u}^{4} + 8 \cdot {u}^{2}\right)\right) - u \cdot -4\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.4
Cost	10436

\[\begin{array}{l} t_0 := \frac{1}{u \cdot -4 + 1}\\ \mathbf{if}\;1 - 4 \cdot u \leq 0.9580000042915344:\\ \;\;\;\;s \cdot \log \left(t_0 \cdot \left(t_0 \cdot \frac{1}{t_0}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;s \cdot \left(21.333333333333332 \cdot {u}^{3} + \left(64 \cdot {u}^{4} + \left(4 \cdot u + 8 \cdot {u}^{2}\right)\right)\right)\\ \end{array} \]

Alternative 2
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 3
Error	0.6
Cost	7076

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

Alternative 4
Error	3.4
Cost	3684

\[\begin{array}{l} t_0 := 1 - 4 \cdot u\\ \mathbf{if}\;t_0 \leq 0.9998000264167786:\\ \;\;\;\;s \cdot \log \left(\frac{1}{t_0}\right)\\ \mathbf{else}:\\ \;\;\;\;s \cdot \left(4 \cdot u\right)\\ \end{array} \]

Alternative 5
Error	1.1
Cost	3652

\[\begin{array}{l} \mathbf{if}\;4 \cdot u \leq 0.003100000089034438:\\ \;\;\;\;s \cdot \left(8 \cdot {u}^{2} + 4 \cdot u\right)\\ \mathbf{else}:\\ \;\;\;\;s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right)\\ \end{array} \]

Alternative 6
Error	8.5
Cost	160

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

Alternative 7
Error	8.5
Cost	160

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

Reproduce

herbie shell --seed 2023090 
(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))))))