?

Average Error: 12.3 → 0.4
Time: 15.3s
Precision: binary32
Cost: 10660

?

\[\left(0 \leq s \land s \leq 256\right) \land \left(2.328306437 \cdot 10^{-10} \leq u \land u \leq 0.25\right)\]
\[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
↓
\[\begin{array}{l} t_0 := 1 - 4 \cdot u\\ t_1 := \log \left(\frac{1}{t_0}\right)\\ \mathbf{if}\;t_0 \leq 0.9570000171661377:\\ \;\;\;\;s \cdot \left(\left(t_1 \cdot \frac{1}{t_1}\right) \cdot t_1\right)\\ \mathbf{else}:\\ \;\;\;\;s \cdot \left(\left(u \cdot 4 - {u}^{4} \cdot -64\right) + \left(21.333333333333332 \cdot {u}^{3} - {u}^{2} \cdot -8\right)\right)\\ \end{array} \]
(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 (log (/ 1.0 t_0))))
   (if (<= t_0 0.9570000171661377)
     (* s (* (* t_1 (/ 1.0 t_1)) t_1))
     (*
      s
      (+
       (- (* u 4.0) (* (pow u 4.0) -64.0))
       (- (* 21.333333333333332 (pow u 3.0)) (* (pow u 2.0) -8.0)))))))
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 = logf((1.0f / t_0));
	float tmp;
	if (t_0 <= 0.9570000171661377f) {
		tmp = s * ((t_1 * (1.0f / t_1)) * t_1);
	} else {
		tmp = s * (((u * 4.0f) - (powf(u, 4.0f) * -64.0f)) + ((21.333333333333332f * powf(u, 3.0f)) - (powf(u, 2.0f) * -8.0f)));
	}
	return tmp;
}

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 = log((1.0e0 / t_0))
    if (t_0 <= 0.9570000171661377e0) then
        tmp = s * ((t_1 * (1.0e0 / t_1)) * t_1)
    else
        tmp = s * (((u * 4.0e0) - ((u ** 4.0e0) * (-64.0e0))) + ((21.333333333333332e0 * (u ** 3.0e0)) - ((u ** 2.0e0) * (-8.0e0))))
    end if
    code = tmp
end function

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 = log(Float32(Float32(1.0) / t_0))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.9570000171661377))
		tmp = Float32(s * Float32(Float32(t_1 * Float32(Float32(1.0) / t_1)) * t_1));
	else
		tmp = Float32(s * Float32(Float32(Float32(u * Float32(4.0)) - Float32((u ^ Float32(4.0)) * Float32(-64.0))) + Float32(Float32(Float32(21.333333333333332) * (u ^ Float32(3.0))) - Float32((u ^ Float32(2.0)) * Float32(-8.0)))));
	end
	return tmp
end

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 = log((single(1.0) / t_0));
	tmp = single(0.0);
	if (t_0 <= single(0.9570000171661377))
		tmp = s * ((t_1 * (single(1.0) / t_1)) * t_1);
	else
		tmp = s * (((u * single(4.0)) - ((u ^ single(4.0)) * single(-64.0))) + ((single(21.333333333333332) * (u ^ single(3.0))) - ((u ^ single(2.0)) * single(-8.0))));
	end
	tmp_2 = tmp;
end

s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right)
↓
\begin{array}{l}
t_0 := 1 - 4 \cdot u\\
t_1 := \log \left(\frac{1}{t_0}\right)\\
\mathbf{if}\;t_0 \leq 0.9570000171661377:\\
\;\;\;\;s \cdot \left(\left(t_1 \cdot \frac{1}{t_1}\right) \cdot t_1\right)\\

\mathbf{else}:\\
\;\;\;\;s \cdot \left(\left(u \cdot 4 - {u}^{4} \cdot -64\right) + \left(21.333333333333332 \cdot {u}^{3} - {u}^{2} \cdot -8\right)\right)\\


\end{array}

Error?

Try it out?

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation?

  1. Split input into 2 regimes

  2. if (-.f32 1 (*.f32 4 u)) < 0.957000017

    1. Initial program 1.2

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

    2. Applied egg-rr1.3

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

    if 0.957000017 < (-.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. Simplified0.3

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

      [Start]0.3

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

      rational_best_45_simplify-80 [=>]0.3

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

      rational_best_45_simplify-80 [=>]0.3

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

      rational_best_45_simplify-73 [=>]0.3

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

      rational_best_45_simplify-80 [=>]0.3

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

    4. Applied egg-rr0.3

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

    5. Simplified0.3

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

      [Start]0.3

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

      rational_best_45_simplify-109 [=>]0.3

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;1 - 4 \cdot u \leq 0.9570000171661377:\\ \;\;\;\;s \cdot \left(\left(\log \left(\frac{1}{1 - 4 \cdot u}\right) \cdot \frac{1}{\log \left(\frac{1}{1 - 4 \cdot u}\right)}\right) \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;s \cdot \left(\left(u \cdot 4 - {u}^{4} \cdot -64\right) + \left(21.333333333333332 \cdot {u}^{3} - {u}^{2} \cdot -8\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error0.4
Cost10436
\[\begin{array}{l} t_0 := 1 - 4 \cdot u\\ \mathbf{if}\;t_0 \leq 0.9570000171661377:\\ \;\;\;\;s \cdot \log \left(\frac{1}{t_0}\right)\\ \mathbf{else}:\\ \;\;\;\;s \cdot \left(21.333333333333332 \cdot {u}^{3} + \left(4 \cdot u + \left(8 \cdot {u}^{2} + 64 \cdot {u}^{4}\right)\right)\right)\\ \end{array} \]
Alternative 2
Error0.4
Cost10436
\[\begin{array}{l} t_0 := 1 - 4 \cdot u\\ \mathbf{if}\;t_0 \leq 0.9570000171661377:\\ \;\;\;\;s \cdot \log \left(\frac{1}{t_0}\right)\\ \mathbf{else}:\\ \;\;\;\;s \cdot \left(\left(u \cdot 4 - {u}^{4} \cdot -64\right) + \left(21.333333333333332 \cdot {u}^{3} - {u}^{2} \cdot -8\right)\right)\\ \end{array} \]
Alternative 3
Error0.4
Cost10372
\[\begin{array}{l} \mathbf{if}\;4 \cdot u \leq 0.0430000014603138:\\ \;\;\;\;s \cdot \left(21.333333333333332 \cdot {u}^{3} + \left(8 \cdot {u}^{2} + \left(4 \cdot u + 64 \cdot {u}^{4}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right)\\ \end{array} \]
Alternative 4
Error0.6
Cost7140
\[\begin{array}{l} t_0 := 1 - 4 \cdot u\\ \mathbf{if}\;t_0 \leq 0.9865000247955322:\\ \;\;\;\;s \cdot \log \left(\frac{1}{t_0}\right)\\ \mathbf{else}:\\ \;\;\;\;{u}^{3} \cdot \left(s \cdot 21.333333333333332\right) + s \cdot \left(8 \cdot {u}^{2} + u \cdot 4\right)\\ \end{array} \]
Alternative 5
Error0.6
Cost7076
\[\begin{array}{l} t_0 := 1 - 4 \cdot u\\ \mathbf{if}\;t_0 \leq 0.9865000247955322:\\ \;\;\;\;s \cdot \log \left(\frac{1}{t_0}\right)\\ \mathbf{else}:\\ \;\;\;\;s \cdot \left(4 \cdot u + \left(8 \cdot {u}^{2} + 21.333333333333332 \cdot {u}^{3}\right)\right)\\ \end{array} \]
Alternative 6
Error1.1
Cost3716
\[\begin{array}{l} t_0 := 1 - 4 \cdot u\\ \mathbf{if}\;t_0 \leq 0.9951000213623047:\\ \;\;\;\;s \cdot \log \left(\frac{1}{t_0}\right)\\ \mathbf{else}:\\ \;\;\;\;s \cdot \left(8 \cdot {u}^{2} + 4 \cdot u\right)\\ \end{array} \]
Alternative 7
Error3.5
Cost3684
\[\begin{array}{l} t_0 := 1 - 4 \cdot u\\ \mathbf{if}\;t_0 \leq 0.9997599720954895:\\ \;\;\;\;s \cdot \log \left(\frac{1}{t_0}\right)\\ \mathbf{else}:\\ \;\;\;\;s \cdot \left(4 \cdot u\right)\\ \end{array} \]
Alternative 8
Error8.5
Cost160
\[4 \cdot \left(u \cdot s\right) \]
Alternative 9
Error8.4
Cost160
\[s \cdot \left(4 \cdot u\right) \]

Error

Reproduce?

herbie shell --seed 2023098 
(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))))))