Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.5% → 99.5%
Time: 16.0s
Alternatives: 21
Speedup: N/A×

Specification

?
\[\left(0 \leq s \land s \leq 256\right) \land \left(10^{-6} < r \land r < 1000000\right)\]
\[\begin{array}{l} \\ \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \end{array} \]
(FPCore (s r)
 :precision binary32
 (+
  (/ (* 0.25 (exp (/ (- r) s))) (* (* (* 2.0 PI) s) r))
  (/ (* 0.75 (exp (/ (- r) (* 3.0 s)))) (* (* (* 6.0 PI) s) r))))
float code(float s, float r) {
	return ((0.25f * expf((-r / s))) / (((2.0f * ((float) M_PI)) * s) * r)) + ((0.75f * expf((-r / (3.0f * s)))) / (((6.0f * ((float) M_PI)) * s) * r));
}

function code(s, r)
	return Float32(Float32(Float32(Float32(0.25) * exp(Float32(Float32(-r) / s))) / Float32(Float32(Float32(Float32(2.0) * Float32(pi)) * s) * r)) + Float32(Float32(Float32(0.75) * exp(Float32(Float32(-r) / Float32(Float32(3.0) * s)))) / Float32(Float32(Float32(Float32(6.0) * Float32(pi)) * s) * r)))
end

function tmp = code(s, r)
	tmp = ((single(0.25) * exp((-r / s))) / (((single(2.0) * single(pi)) * s) * r)) + ((single(0.75) * exp((-r / (single(3.0) * s)))) / (((single(6.0) * single(pi)) * s) * r));
end

\begin{array}{l}

\\
\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r}
\end{array}

Sampling outcomes in binary32 precision:

Local Percentage Accuracy vs ?

The average percentage accuracy by input value. Horizontal axis shows value of an input variable; the variable is choosen in the title. Vertical axis is accuracy; higher is better. Red represent the original program, while blue represents Herbie's suggestion. These can be toggled with buttons below the plot. The line is an average while dots represent individual samples.

Accuracy vs Speed?

Herbie found 21 alternatives:

AlternativeAccuracySpeedup
The accuracy (vertical axis) and speed (horizontal axis) of each alternatives. Up and to the right is better. The red square shows the initial program, and each blue circle shows an alternative.The line shows the best available speed-accuracy tradeoffs.

Initial Program: 99.5% accurate, 1.0× speedup?

\[\begin{array}{l} \\ \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \end{array} \]
(FPCore (s r)
 :precision binary32
 (+
  (/ (* 0.25 (exp (/ (- r) s))) (* (* (* 2.0 PI) s) r))
  (/ (* 0.75 (exp (/ (- r) (* 3.0 s)))) (* (* (* 6.0 PI) s) r))))
float code(float s, float r) {
	return ((0.25f * expf((-r / s))) / (((2.0f * ((float) M_PI)) * s) * r)) + ((0.75f * expf((-r / (3.0f * s)))) / (((6.0f * ((float) M_PI)) * s) * r));
}

function code(s, r)
	return Float32(Float32(Float32(Float32(0.25) * exp(Float32(Float32(-r) / s))) / Float32(Float32(Float32(Float32(2.0) * Float32(pi)) * s) * r)) + Float32(Float32(Float32(0.75) * exp(Float32(Float32(-r) / Float32(Float32(3.0) * s)))) / Float32(Float32(Float32(Float32(6.0) * Float32(pi)) * s) * r)))
end

function tmp = code(s, r)
	tmp = ((single(0.25) * exp((-r / s))) / (((single(2.0) * single(pi)) * s) * r)) + ((single(0.75) * exp((-r / (single(3.0) * s)))) / (((single(6.0) * single(pi)) * s) * r));
end

\begin{array}{l}

\\
\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r}
\end{array}

Alternative 1: 99.5% accurate, 1.1× speedup?

\[\begin{array}{l} \\ \frac{e^{\frac{r}{-s}} + e^{\frac{r}{s \cdot -3}}}{r \cdot \pi} \cdot \frac{0.125}{s} \end{array} \]
(FPCore (s r)
 :precision binary32
 (* (/ (+ (exp (/ r (- s))) (exp (/ r (* s -3.0)))) (* r PI)) (/ 0.125 s)))
float code(float s, float r) {
	return ((expf((r / -s)) + expf((r / (s * -3.0f)))) / (r * ((float) M_PI))) * (0.125f / s);
}

function code(s, r)
	return Float32(Float32(Float32(exp(Float32(r / Float32(-s))) + exp(Float32(r / Float32(s * Float32(-3.0))))) / Float32(r * Float32(pi))) * Float32(Float32(0.125) / s))
end

function tmp = code(s, r)
	tmp = ((exp((r / -s)) + exp((r / (s * single(-3.0))))) / (r * single(pi))) * (single(0.125) / s);
end

\begin{array}{l}

\\
\frac{e^{\frac{r}{-s}} + e^{\frac{r}{s \cdot -3}}}{r \cdot \pi} \cdot \frac{0.125}{s}
\end{array}
Derivation

  1. Initial program 99.5%

    \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
  2. Add Preprocessing

  3. Taylor expanded in s around 0

    \[\leadsto \color{blue}{\frac{\frac{1}{8} \cdot \frac{e^{-1 \cdot \frac{r}{s}}}{r \cdot \mathsf{PI}\left(\right)} + \frac{1}{8} \cdot \frac{e^{\frac{-1}{3} \cdot \frac{r}{s}}}{r \cdot \mathsf{PI}\left(\right)}}{s}} \]
  4. Step-by-step derivation

    1. distribute-lft-outN/A

      \[\leadsto \frac{\color{blue}{\frac{1}{8} \cdot \left(\frac{e^{-1 \cdot \frac{r}{s}}}{r \cdot \mathsf{PI}\left(\right)} + \frac{e^{\frac{-1}{3} \cdot \frac{r}{s}}}{r \cdot \mathsf{PI}\left(\right)}\right)}}{s} \]

    2. *-commutativeN/A

      \[\leadsto \frac{\color{blue}{\left(\frac{e^{-1 \cdot \frac{r}{s}}}{r \cdot \mathsf{PI}\left(\right)} + \frac{e^{\frac{-1}{3} \cdot \frac{r}{s}}}{r \cdot \mathsf{PI}\left(\right)}\right) \cdot \frac{1}{8}}}{s} \]

    3. associate-/l*N/A

      \[\leadsto \color{blue}{\left(\frac{e^{-1 \cdot \frac{r}{s}}}{r \cdot \mathsf{PI}\left(\right)} + \frac{e^{\frac{-1}{3} \cdot \frac{r}{s}}}{r \cdot \mathsf{PI}\left(\right)}\right) \cdot \frac{\frac{1}{8}}{s}} \]

    4. lower-*.f32N/A

      \[\leadsto \color{blue}{\left(\frac{e^{-1 \cdot \frac{r}{s}}}{r \cdot \mathsf{PI}\left(\right)} + \frac{e^{\frac{-1}{3} \cdot \frac{r}{s}}}{r \cdot \mathsf{PI}\left(\right)}\right) \cdot \frac{\frac{1}{8}}{s}} \]

  5. Applied rewrites99.5%

    \[\leadsto \color{blue}{\left(\frac{e^{\frac{r}{-s}}}{r \cdot \pi} + \frac{e^{\frac{r}{s} \cdot -0.3333333333333333}}{r \cdot \pi}\right) \cdot \frac{0.125}{s}} \]
  6. Step-by-step derivation

    1. Applied rewrites99.5%

      \[\leadsto \frac{0.125}{s} \cdot \color{blue}{\left(\frac{1}{r \cdot \pi} \cdot \left(e^{\frac{r}{-s}} + e^{\frac{r}{s \cdot -3}}\right)\right)} \]
    2. Step-by-step derivation

      1. Applied rewrites99.5%

        \[\leadsto \frac{e^{\frac{r}{-s}} + e^{\frac{r}{s \cdot -3}}}{r \cdot \pi} \cdot \color{blue}{\frac{0.125}{s}} \]
      2. Add Preprocessing

      Alternative 2: 99.5% accurate, 1.1× speedup?

      \[\begin{array}{l} \\ \frac{0.125 \cdot \left(e^{\frac{r}{-s}} + e^{\frac{r}{s} \cdot -0.3333333333333333}\right)}{r \cdot \left(s \cdot \pi\right)} \end{array} \]
      (FPCore (s r)
 :precision binary32
 (/
  (* 0.125 (+ (exp (/ r (- s))) (exp (* (/ r s) -0.3333333333333333))))
  (* r (* s PI))))
      float code(float s, float r) {
	return (0.125f * (expf((r / -s)) + expf(((r / s) * -0.3333333333333333f)))) / (r * (s * ((float) M_PI)));
}

      function code(s, r)
	return Float32(Float32(Float32(0.125) * Float32(exp(Float32(r / Float32(-s))) + exp(Float32(Float32(r / s) * Float32(-0.3333333333333333))))) / Float32(r * Float32(s * Float32(pi))))
end

      function tmp = code(s, r)
	tmp = (single(0.125) * (exp((r / -s)) + exp(((r / s) * single(-0.3333333333333333))))) / (r * (s * single(pi)));
end

      \begin{array}{l}

\\
\frac{0.125 \cdot \left(e^{\frac{r}{-s}} + e^{\frac{r}{s} \cdot -0.3333333333333333}\right)}{r \cdot \left(s \cdot \pi\right)}
\end{array}
      Derivation

      1. Initial program 99.5%

        \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
      2. Add Preprocessing

      3. Taylor expanded in s around 0

        \[\leadsto \color{blue}{\frac{\frac{1}{8} \cdot \frac{e^{-1 \cdot \frac{r}{s}}}{r \cdot \mathsf{PI}\left(\right)} + \frac{1}{8} \cdot \frac{e^{\frac{-1}{3} \cdot \frac{r}{s}}}{r \cdot \mathsf{PI}\left(\right)}}{s}} \]
      4. Step-by-step derivation

        1. distribute-lft-outN/A

          \[\leadsto \frac{\color{blue}{\frac{1}{8} \cdot \left(\frac{e^{-1 \cdot \frac{r}{s}}}{r \cdot \mathsf{PI}\left(\right)} + \frac{e^{\frac{-1}{3} \cdot \frac{r}{s}}}{r \cdot \mathsf{PI}\left(\right)}\right)}}{s} \]

        2. *-commutativeN/A

          \[\leadsto \frac{\color{blue}{\left(\frac{e^{-1 \cdot \frac{r}{s}}}{r \cdot \mathsf{PI}\left(\right)} + \frac{e^{\frac{-1}{3} \cdot \frac{r}{s}}}{r \cdot \mathsf{PI}\left(\right)}\right) \cdot \frac{1}{8}}}{s} \]

        3. associate-/l*N/A

          \[\leadsto \color{blue}{\left(\frac{e^{-1 \cdot \frac{r}{s}}}{r \cdot \mathsf{PI}\left(\right)} + \frac{e^{\frac{-1}{3} \cdot \frac{r}{s}}}{r \cdot \mathsf{PI}\left(\right)}\right) \cdot \frac{\frac{1}{8}}{s}} \]

        4. lower-*.f32N/A

          \[\leadsto \color{blue}{\left(\frac{e^{-1 \cdot \frac{r}{s}}}{r \cdot \mathsf{PI}\left(\right)} + \frac{e^{\frac{-1}{3} \cdot \frac{r}{s}}}{r \cdot \mathsf{PI}\left(\right)}\right) \cdot \frac{\frac{1}{8}}{s}} \]

      5. Applied rewrites99.4%

        \[\leadsto \color{blue}{\left(\frac{e^{\frac{r}{-s}}}{r \cdot \pi} + \frac{e^{\frac{r}{s} \cdot -0.3333333333333333}}{r \cdot \pi}\right) \cdot \frac{0.125}{s}} \]
      6. Step-by-step derivation

        1. Applied rewrites99.5%

          \[\leadsto \frac{0.125}{s} \cdot \color{blue}{\left(\frac{1}{r \cdot \pi} \cdot \left(e^{\frac{r}{-s}} + e^{\frac{r}{s \cdot -3}}\right)\right)} \]

        2. Taylor expanded in s around 0

          \[\leadsto \frac{1}{8} \cdot \color{blue}{\frac{e^{-1 \cdot \frac{r}{s}} + e^{\frac{-1}{3} \cdot \frac{r}{s}}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}} \]
        3. Step-by-step derivation

          1. Applied rewrites99.5%

            \[\leadsto \frac{0.125 \cdot \left(e^{\frac{r}{-s}} + e^{\frac{r}{s} \cdot -0.3333333333333333}\right)}{\color{blue}{r \cdot \left(s \cdot \pi\right)}} \]
          2. Add Preprocessing

          Reproduce

          ?
          herbie shell --seed 2024222 
(FPCore (s r)
  :name "Disney BSSRDF, PDF of scattering profile"
  :precision binary32
  :pre (and (and (<= 0.0 s) (<= s 256.0)) (and (< 1e-6 r) (< r 1000000.0)))
  (+ (/ (* 0.25 (exp (/ (- r) s))) (* (* (* 2.0 PI) s) r)) (/ (* 0.75 (exp (/ (- r) (* 3.0 s)))) (* (* (* 6.0 PI) s) r))))