Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.5% → 99.5%

Time: 13.4s

Alternatives: 8

Speedup: 1.0×

Specification

\[\left(0 \leq s \land s \leq 256\right) \land \left(10^{-6} < r \land r < 1000000\right)\]

Math

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

FPCore

(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))))

C

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));
}

Julia

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

MATLAB

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

(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))))

C

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));
}

Julia

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

MATLAB

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

Alternative 1: 99.5% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (s r)
 :precision binary32
 (/
  (*
   0.125
   (+
    (/ (exp (/ r (- s))) (* r PI))
    (/ (pow E (* r (/ -0.3333333333333333 s))) (* r PI))))
  s))

C

float code(float s, float r) {
	return (0.125f * ((expf((r / -s)) / (r * ((float) M_PI))) + (powf(((float) M_E), (r * (-0.3333333333333333f / s))) / (r * ((float) M_PI))))) / s;
}

Julia

function code(s, r)
	return Float32(Float32(Float32(0.125) * Float32(Float32(exp(Float32(r / Float32(-s))) / Float32(r * Float32(pi))) + Float32((Float32(exp(1)) ^ Float32(r * Float32(Float32(-0.3333333333333333) / s))) / Float32(r * Float32(pi))))) / s)
end

MATLAB

function tmp = code(s, r)
	tmp = (single(0.125) * ((exp((r / -s)) / (r * single(pi))) + ((single(2.71828182845904523536) ^ (r * (single(-0.3333333333333333) / s))) / (r * single(pi))))) / s;
end

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. Step-by-step derivation

   1. +-commutative 99.5%

      \[\leadsto \color{blue}{\frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}} \]

   2. times-frac 99.5%

      \[\leadsto \color{blue}{\frac{0.75}{\left(6 \cdot \pi\right) \cdot s} \cdot \frac{e^{\frac{-r}{3 \cdot s}}}{r}} + \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} \]

   3. fma-define 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0.75}{\left(6 \cdot \pi\right) \cdot s}, \frac{e^{\frac{-r}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right)} \]

   4. associate-*l* 99.5%

      \[\leadsto \mathsf{fma}\left(\frac{0.75}{\color{blue}{6 \cdot \left(\pi \cdot s\right)}}, \frac{e^{\frac{-r}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   5. associate-/r* 99.5%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{0.75}{6}}{\pi \cdot s}}, \frac{e^{\frac{-r}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   6. metadata-eval 99.5%

      \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{0.125}}{\pi \cdot s}, \frac{e^{\frac{-r}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   7. *-commutative 99.5%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{\color{blue}{s \cdot \pi}}, \frac{e^{\frac{-r}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   8. neg-mul-1 99.5%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\frac{\color{blue}{-1 \cdot r}}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   9. times-frac 99.5%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\color{blue}{\frac{-1}{3} \cdot \frac{r}{s}}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   10. metadata-eval 99.5%

       \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\color{blue}{-0.3333333333333333} \cdot \frac{r}{s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   11. times-frac 99.5%

       \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r}, \color{blue}{\frac{0.25}{\left(2 \cdot \pi\right) \cdot s} \cdot \frac{e^{\frac{-r}{s}}}{r}}\right) \]

3. Simplified 99.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{r}{-s}}}{r}\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. *-un-lft-identity 99.5%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\color{blue}{1 \cdot \left(-0.3333333333333333 \cdot \frac{r}{s}\right)}}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{r}{-s}}}{r}\right) \]

   2. exp-prod 99.6%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{\color{blue}{{\left(e^{1}\right)}^{\left(-0.3333333333333333 \cdot \frac{r}{s}\right)}}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{r}{-s}}}{r}\right) \]

   3. associate-*r/ 99.6%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{{\left(e^{1}\right)}^{\color{blue}{\left(\frac{-0.3333333333333333 \cdot r}{s}\right)}}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{r}{-s}}}{r}\right) \]

6. Applied egg-rr 99.6%

   \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{\color{blue}{{\left(e^{1}\right)}^{\left(\frac{-0.3333333333333333 \cdot r}{s}\right)}}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{r}{-s}}}{r}\right) \]

7. Taylor expanded in s around 0 99.5%

   \[\leadsto \color{blue}{\frac{0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}}}{r \cdot \pi} + 0.125 \cdot \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r \cdot \pi}}{s}} \]
8. Step-by-step derivation

   1. distribute-lft-out 99.5%

      \[\leadsto \frac{\color{blue}{0.125 \cdot \left(\frac{e^{-1 \cdot \frac{r}{s}}}{r \cdot \pi} + \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r \cdot \pi}\right)}}{s} \]

   2. mul-1-neg 99.5%

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

   3. associate-*r/ 99.5%

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

   4. *-commutative 99.5%

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

   5. associate-*r/ 99.6%

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

9. Simplified 99.6%

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

    1. associate-*r/ 99.5%

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

    2. *-un-lft-identity 99.5%

       \[\leadsto \frac{0.125 \cdot \left(\frac{e^{-\frac{r}{s}}}{r \cdot \pi} + \frac{e^{\color{blue}{1 \cdot \frac{r \cdot -0.3333333333333333}{s}}}}{r \cdot \pi}\right)}{s} \]

    3. exp-prod 99.6%

       \[\leadsto \frac{0.125 \cdot \left(\frac{e^{-\frac{r}{s}}}{r \cdot \pi} + \frac{\color{blue}{{\left(e^{1}\right)}^{\left(\frac{r \cdot -0.3333333333333333}{s}\right)}}}{r \cdot \pi}\right)}{s} \]

    4. associate-*r/ 99.6%

       \[\leadsto \frac{0.125 \cdot \left(\frac{e^{-\frac{r}{s}}}{r \cdot \pi} + \frac{{\left(e^{1}\right)}^{\color{blue}{\left(r \cdot \frac{-0.3333333333333333}{s}\right)}}}{r \cdot \pi}\right)}{s} \]

11. Applied egg-rr 99.6%

    \[\leadsto \frac{0.125 \cdot \left(\frac{e^{-\frac{r}{s}}}{r \cdot \pi} + \frac{\color{blue}{{\left(e^{1}\right)}^{\left(r \cdot \frac{-0.3333333333333333}{s}\right)}}}{r \cdot \pi}\right)}{s} \]

12. Final simplification 99.6%

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

Alternative 2: 99.5% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (s r)
 :precision binary32
 (/
  (*
   0.125
   (+
    (/ (exp (/ r (- s))) (* r PI))
    (/ (exp (* r (/ -0.3333333333333333 s))) (* r PI))))
  s))

C

float code(float s, float r) {
	return (0.125f * ((expf((r / -s)) / (r * ((float) M_PI))) + (expf((r * (-0.3333333333333333f / s))) / (r * ((float) M_PI))))) / s;
}

Julia

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

MATLAB

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

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. Step-by-step derivation

   1. +-commutative 99.5%

      \[\leadsto \color{blue}{\frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}} \]

   2. times-frac 99.5%

      \[\leadsto \color{blue}{\frac{0.75}{\left(6 \cdot \pi\right) \cdot s} \cdot \frac{e^{\frac{-r}{3 \cdot s}}}{r}} + \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} \]

   3. fma-define 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0.75}{\left(6 \cdot \pi\right) \cdot s}, \frac{e^{\frac{-r}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right)} \]

   4. associate-*l* 99.5%

      \[\leadsto \mathsf{fma}\left(\frac{0.75}{\color{blue}{6 \cdot \left(\pi \cdot s\right)}}, \frac{e^{\frac{-r}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   5. associate-/r* 99.5%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{0.75}{6}}{\pi \cdot s}}, \frac{e^{\frac{-r}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   6. metadata-eval 99.5%

      \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{0.125}}{\pi \cdot s}, \frac{e^{\frac{-r}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   7. *-commutative 99.5%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{\color{blue}{s \cdot \pi}}, \frac{e^{\frac{-r}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   8. neg-mul-1 99.5%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\frac{\color{blue}{-1 \cdot r}}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   9. times-frac 99.5%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\color{blue}{\frac{-1}{3} \cdot \frac{r}{s}}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   10. metadata-eval 99.5%

       \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\color{blue}{-0.3333333333333333} \cdot \frac{r}{s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   11. times-frac 99.5%

       \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r}, \color{blue}{\frac{0.25}{\left(2 \cdot \pi\right) \cdot s} \cdot \frac{e^{\frac{-r}{s}}}{r}}\right) \]

3. Simplified 99.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{r}{-s}}}{r}\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. *-un-lft-identity 99.5%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\color{blue}{1 \cdot \left(-0.3333333333333333 \cdot \frac{r}{s}\right)}}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{r}{-s}}}{r}\right) \]

   2. exp-prod 99.6%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{\color{blue}{{\left(e^{1}\right)}^{\left(-0.3333333333333333 \cdot \frac{r}{s}\right)}}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{r}{-s}}}{r}\right) \]

   3. associate-*r/ 99.6%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{{\left(e^{1}\right)}^{\color{blue}{\left(\frac{-0.3333333333333333 \cdot r}{s}\right)}}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{r}{-s}}}{r}\right) \]

6. Applied egg-rr 99.6%

   \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{\color{blue}{{\left(e^{1}\right)}^{\left(\frac{-0.3333333333333333 \cdot r}{s}\right)}}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{r}{-s}}}{r}\right) \]

7. Taylor expanded in s around 0 99.5%

   \[\leadsto \color{blue}{\frac{0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}}}{r \cdot \pi} + 0.125 \cdot \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r \cdot \pi}}{s}} \]
8. Step-by-step derivation

   1. distribute-lft-out 99.5%

      \[\leadsto \frac{\color{blue}{0.125 \cdot \left(\frac{e^{-1 \cdot \frac{r}{s}}}{r \cdot \pi} + \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r \cdot \pi}\right)}}{s} \]

   2. mul-1-neg 99.5%

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

   3. associate-*r/ 99.5%

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

   4. *-commutative 99.5%

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

   5. associate-*r/ 99.6%

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

9. Simplified 99.6%

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

10. Final simplification 99.6%

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

Alternative 3: 99.5% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (s r)
 :precision binary32
 (*
  0.125
  (/
   (+ (exp (* r (/ -0.3333333333333333 s))) (exp (/ r (- s))))
   (* s (* r PI)))))

C

float code(float s, float r) {
	return 0.125f * ((expf((r * (-0.3333333333333333f / s))) + expf((r / -s))) / (s * (r * ((float) M_PI))));
}

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Simplified 99.1%

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

   1. add-sqr-sqrt 99.1%

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

   2. sqrt-unprod 98.3%

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

   3. pow-prod-down 98.4%

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

   4. prod-exp 98.7%

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

   5. metadata-eval 98.7%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\sqrt{{\left(e^{\color{blue}{-0.6666666666666666}}\right)}^{\left(\frac{r}{s}\right)}}}{r}\right) \]

6. Applied egg-rr 98.7%

   \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\color{blue}{\sqrt{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s}\right)}}}}{r}\right) \]

7. Taylor expanded in r around inf 98.6%

   \[\leadsto \color{blue}{0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}} + \sqrt{e^{-0.6666666666666666 \cdot \frac{r}{s}}}}{r \cdot \left(s \cdot \pi\right)}} \]
8. Step-by-step derivation

   1. neg-mul-1 98.6%

      \[\leadsto 0.125 \cdot \frac{e^{\color{blue}{-\frac{r}{s}}} + \sqrt{e^{-0.6666666666666666 \cdot \frac{r}{s}}}}{r \cdot \left(s \cdot \pi\right)} \]

   2. distribute-frac-neg2 98.6%

      \[\leadsto 0.125 \cdot \frac{e^{\color{blue}{\frac{r}{-s}}} + \sqrt{e^{-0.6666666666666666 \cdot \frac{r}{s}}}}{r \cdot \left(s \cdot \pi\right)} \]

   3. exp-prod 98.7%

      \[\leadsto 0.125 \cdot \frac{e^{\frac{r}{-s}} + \sqrt{\color{blue}{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s}\right)}}}}{r \cdot \left(s \cdot \pi\right)} \]

   4. unpow1/2 98.7%

      \[\leadsto 0.125 \cdot \frac{e^{\frac{r}{-s}} + \color{blue}{{\left({\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s}\right)}\right)}^{0.5}}}{r \cdot \left(s \cdot \pi\right)} \]

   5. exp-prod 98.6%

      \[\leadsto 0.125 \cdot \frac{e^{\frac{r}{-s}} + {\color{blue}{\left(e^{-0.6666666666666666 \cdot \frac{r}{s}}\right)}}^{0.5}}{r \cdot \left(s \cdot \pi\right)} \]

   6. exp-prod 99.5%

      \[\leadsto 0.125 \cdot \frac{e^{\frac{r}{-s}} + \color{blue}{e^{\left(-0.6666666666666666 \cdot \frac{r}{s}\right) \cdot 0.5}}}{r \cdot \left(s \cdot \pi\right)} \]

   7. *-commutative 99.5%

      \[\leadsto 0.125 \cdot \frac{e^{\frac{r}{-s}} + e^{\color{blue}{\left(\frac{r}{s} \cdot -0.6666666666666666\right)} \cdot 0.5}}{r \cdot \left(s \cdot \pi\right)} \]

   8. associate-*l* 99.5%

      \[\leadsto 0.125 \cdot \frac{e^{\frac{r}{-s}} + e^{\color{blue}{\frac{r}{s} \cdot \left(-0.6666666666666666 \cdot 0.5\right)}}}{r \cdot \left(s \cdot \pi\right)} \]

   9. metadata-eval 99.5%

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

   10. associate-*l/ 99.5%

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

   11. associate-*r/ 99.5%

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

   12. *-commutative 99.5%

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

   13. associate-*r* 99.5%

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

9. Simplified 99.5%

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

10. Final simplification 99.5%

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

Alternative 4: 9.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.0625 \cdot \frac{r}{s \cdot \pi} - \frac{0.125}{\pi}}{s} + \frac{0.125}{r \cdot \pi}}{s} + 0.75 \cdot \frac{e^{\frac{r}{-s \cdot 3}}}{r \cdot \left(\left(s \cdot \pi\right) \cdot 6\right)} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/ (+ (/ (- (* 0.0625 (/ r (* s PI))) (/ 0.125 PI)) s) (/ 0.125 (* r PI))) s)
  (* 0.75 (/ (exp (/ r (- (* s 3.0)))) (* r (* (* s PI) 6.0))))))

C

float code(float s, float r) {
	return (((((0.0625f * (r / (s * ((float) M_PI)))) - (0.125f / ((float) M_PI))) / s) + (0.125f / (r * ((float) M_PI)))) / s) + (0.75f * (expf((r / -(s * 3.0f))) / (r * ((s * ((float) M_PI)) * 6.0f))));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(Float32(Float32(Float32(0.0625) * Float32(r / Float32(s * Float32(pi)))) - Float32(Float32(0.125) / Float32(pi))) / s) + Float32(Float32(0.125) / Float32(r * Float32(pi)))) / s) + Float32(Float32(0.75) * Float32(exp(Float32(r / Float32(-Float32(s * Float32(3.0))))) / Float32(r * Float32(Float32(s * Float32(pi)) * Float32(6.0))))))
end

MATLAB

function tmp = code(s, r)
	tmp = (((((single(0.0625) * (r / (s * single(pi)))) - (single(0.125) / single(pi))) / s) + (single(0.125) / (r * single(pi)))) / s) + (single(0.75) * (exp((r / -(s * single(3.0)))) / (r * ((s * single(pi)) * single(6.0)))));
end

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. Step-by-step derivation

   1. times-frac 99.5%

      \[\leadsto \color{blue}{\frac{0.25}{\left(2 \cdot \pi\right) \cdot s} \cdot \frac{e^{\frac{-r}{s}}}{r}} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   2. *-commutative 99.5%

      \[\leadsto \frac{0.25}{\color{blue}{s \cdot \left(2 \cdot \pi\right)}} \cdot \frac{e^{\frac{-r}{s}}}{r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   3. distribute-frac-neg 99.5%

      \[\leadsto \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{\color{blue}{-\frac{r}{s}}}}{r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   4. associate-/l* 99.5%

      \[\leadsto \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{-\frac{r}{s}}}{r} + \color{blue}{0.75 \cdot \frac{e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r}} \]

   5. *-commutative 99.5%

      \[\leadsto \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{-\frac{r}{s}}}{r} + 0.75 \cdot \frac{e^{\frac{-r}{\color{blue}{s \cdot 3}}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   6. *-commutative 99.5%

      \[\leadsto \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{-\frac{r}{s}}}{r} + 0.75 \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{\color{blue}{r \cdot \left(\left(6 \cdot \pi\right) \cdot s\right)}} \]

   7. associate-*l* 99.5%

      \[\leadsto \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{-\frac{r}{s}}}{r} + 0.75 \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r \cdot \color{blue}{\left(6 \cdot \left(\pi \cdot s\right)\right)}} \]

3. Simplified 99.5%

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

5. Taylor expanded in s around -inf 10.8%

   \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{0.0625 \cdot \frac{r}{s \cdot \pi} - 0.125 \cdot \frac{1}{\pi}}{s} - 0.125 \cdot \frac{1}{r \cdot \pi}}{s}} + 0.75 \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r \cdot \left(6 \cdot \left(\pi \cdot s\right)\right)} \]
6. Step-by-step derivation

   1. mul-1-neg 10.8%

      \[\leadsto \color{blue}{\left(-\frac{-1 \cdot \frac{0.0625 \cdot \frac{r}{s \cdot \pi} - 0.125 \cdot \frac{1}{\pi}}{s} - 0.125 \cdot \frac{1}{r \cdot \pi}}{s}\right)} + 0.75 \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r \cdot \left(6 \cdot \left(\pi \cdot s\right)\right)} \]

   2. mul-1-neg 10.8%

      \[\leadsto \left(-\frac{\color{blue}{\left(-\frac{0.0625 \cdot \frac{r}{s \cdot \pi} - 0.125 \cdot \frac{1}{\pi}}{s}\right)} - 0.125 \cdot \frac{1}{r \cdot \pi}}{s}\right) + 0.75 \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r \cdot \left(6 \cdot \left(\pi \cdot s\right)\right)} \]

   3. associate-*r/ 10.8%

      \[\leadsto \left(-\frac{\left(-\frac{0.0625 \cdot \frac{r}{s \cdot \pi} - \color{blue}{\frac{0.125 \cdot 1}{\pi}}}{s}\right) - 0.125 \cdot \frac{1}{r \cdot \pi}}{s}\right) + 0.75 \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r \cdot \left(6 \cdot \left(\pi \cdot s\right)\right)} \]

   4. metadata-eval 10.8%

      \[\leadsto \left(-\frac{\left(-\frac{0.0625 \cdot \frac{r}{s \cdot \pi} - \frac{\color{blue}{0.125}}{\pi}}{s}\right) - 0.125 \cdot \frac{1}{r \cdot \pi}}{s}\right) + 0.75 \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r \cdot \left(6 \cdot \left(\pi \cdot s\right)\right)} \]

   5. associate-*r/ 10.8%

      \[\leadsto \left(-\frac{\left(-\frac{0.0625 \cdot \frac{r}{s \cdot \pi} - \frac{0.125}{\pi}}{s}\right) - \color{blue}{\frac{0.125 \cdot 1}{r \cdot \pi}}}{s}\right) + 0.75 \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r \cdot \left(6 \cdot \left(\pi \cdot s\right)\right)} \]

   6. metadata-eval 10.8%

      \[\leadsto \left(-\frac{\left(-\frac{0.0625 \cdot \frac{r}{s \cdot \pi} - \frac{0.125}{\pi}}{s}\right) - \frac{\color{blue}{0.125}}{r \cdot \pi}}{s}\right) + 0.75 \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r \cdot \left(6 \cdot \left(\pi \cdot s\right)\right)} \]

7. Simplified 10.8%

   \[\leadsto \color{blue}{\left(-\frac{\left(-\frac{0.0625 \cdot \frac{r}{s \cdot \pi} - \frac{0.125}{\pi}}{s}\right) - \frac{0.125}{r \cdot \pi}}{s}\right)} + 0.75 \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r \cdot \left(6 \cdot \left(\pi \cdot s\right)\right)} \]

8. Final simplification 10.8%

   \[\leadsto \frac{\frac{0.0625 \cdot \frac{r}{s \cdot \pi} - \frac{0.125}{\pi}}{s} + \frac{0.125}{r \cdot \pi}}{s} + 0.75 \cdot \frac{e^{\frac{r}{-s \cdot 3}}}{r \cdot \left(\left(s \cdot \pi\right) \cdot 6\right)} \]
9. Add Preprocessing

Alternative 5: 9.9% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.125 \cdot \frac{0.05555555555555555 \cdot \frac{r}{\pi} + \frac{r}{\pi} \cdot 0.5}{s} + 0.16666666666666666 \cdot \frac{-1}{\pi}}{s} + 0.25 \cdot \frac{1}{r \cdot \pi}}{s} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (/
  (+
   (/
    (+
     (* 0.125 (/ (+ (* 0.05555555555555555 (/ r PI)) (* (/ r PI) 0.5)) s))
     (* 0.16666666666666666 (/ -1.0 PI)))
    s)
   (* 0.25 (/ 1.0 (* r PI))))
  s))

C

float code(float s, float r) {
	return ((((0.125f * (((0.05555555555555555f * (r / ((float) M_PI))) + ((r / ((float) M_PI)) * 0.5f)) / s)) + (0.16666666666666666f * (-1.0f / ((float) M_PI)))) / s) + (0.25f * (1.0f / (r * ((float) M_PI))))) / s;
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(Float32(Float32(0.125) * Float32(Float32(Float32(Float32(0.05555555555555555) * Float32(r / Float32(pi))) + Float32(Float32(r / Float32(pi)) * Float32(0.5))) / s)) + Float32(Float32(0.16666666666666666) * Float32(Float32(-1.0) / Float32(pi)))) / s) + Float32(Float32(0.25) * Float32(Float32(1.0) / Float32(r * Float32(pi))))) / s)
end

MATLAB

function tmp = code(s, r)
	tmp = ((((single(0.125) * (((single(0.05555555555555555) * (r / single(pi))) + ((r / single(pi)) * single(0.5))) / s)) + (single(0.16666666666666666) * (single(-1.0) / single(pi)))) / s) + (single(0.25) * (single(1.0) / (r * single(pi))))) / s;
end

Derivation

1. Initial program 99.5%

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

3. Simplified 99.1%

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

5. Taylor expanded in s around -inf 10.8%

   \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{0.125 \cdot \frac{0.05555555555555555 \cdot \frac{r}{\pi} + 0.5 \cdot \frac{r}{\pi}}{s} - 0.16666666666666666 \cdot \frac{1}{\pi}}{s} - 0.25 \cdot \frac{1}{r \cdot \pi}}{s}} \]

6. Final simplification 10.8%

   \[\leadsto \frac{\frac{0.125 \cdot \frac{0.05555555555555555 \cdot \frac{r}{\pi} + \frac{r}{\pi} \cdot 0.5}{s} + 0.16666666666666666 \cdot \frac{-1}{\pi}}{s} + 0.25 \cdot \frac{1}{r \cdot \pi}}{s} \]
7. Add Preprocessing

Alternative 6: 9.9% accurate, 10.0× speedup

Math

\[\begin{array}{l} \\ \frac{0.125 \cdot \left(\frac{\frac{2}{\pi}}{r} + \frac{\frac{\frac{r \cdot 0.5555555555555556}{\pi}}{s} - \frac{1.3333333333333333}{\pi}}{s}\right)}{s} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (/
  (*
   0.125
   (+
    (/ (/ 2.0 PI) r)
    (/ (- (/ (/ (* r 0.5555555555555556) PI) s) (/ 1.3333333333333333 PI)) s)))
  s))

C

float code(float s, float r) {
	return (0.125f * (((2.0f / ((float) M_PI)) / r) + (((((r * 0.5555555555555556f) / ((float) M_PI)) / s) - (1.3333333333333333f / ((float) M_PI))) / s))) / s;
}

Julia

function code(s, r)
	return Float32(Float32(Float32(0.125) * Float32(Float32(Float32(Float32(2.0) / Float32(pi)) / r) + Float32(Float32(Float32(Float32(Float32(r * Float32(0.5555555555555556)) / Float32(pi)) / s) - Float32(Float32(1.3333333333333333) / Float32(pi))) / s))) / s)
end

MATLAB

function tmp = code(s, r)
	tmp = (single(0.125) * (((single(2.0) / single(pi)) / r) + (((((r * single(0.5555555555555556)) / single(pi)) / s) - (single(1.3333333333333333) / single(pi))) / s))) / s;
end

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. Step-by-step derivation

   1. +-commutative 99.5%

      \[\leadsto \color{blue}{\frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}} \]

   2. times-frac 99.5%

      \[\leadsto \color{blue}{\frac{0.75}{\left(6 \cdot \pi\right) \cdot s} \cdot \frac{e^{\frac{-r}{3 \cdot s}}}{r}} + \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} \]

   3. fma-define 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0.75}{\left(6 \cdot \pi\right) \cdot s}, \frac{e^{\frac{-r}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right)} \]

   4. associate-*l* 99.5%

      \[\leadsto \mathsf{fma}\left(\frac{0.75}{\color{blue}{6 \cdot \left(\pi \cdot s\right)}}, \frac{e^{\frac{-r}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   5. associate-/r* 99.5%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{0.75}{6}}{\pi \cdot s}}, \frac{e^{\frac{-r}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   6. metadata-eval 99.5%

      \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{0.125}}{\pi \cdot s}, \frac{e^{\frac{-r}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   7. *-commutative 99.5%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{\color{blue}{s \cdot \pi}}, \frac{e^{\frac{-r}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   8. neg-mul-1 99.5%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\frac{\color{blue}{-1 \cdot r}}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   9. times-frac 99.5%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\color{blue}{\frac{-1}{3} \cdot \frac{r}{s}}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   10. metadata-eval 99.5%

       \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\color{blue}{-0.3333333333333333} \cdot \frac{r}{s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   11. times-frac 99.5%

       \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r}, \color{blue}{\frac{0.25}{\left(2 \cdot \pi\right) \cdot s} \cdot \frac{e^{\frac{-r}{s}}}{r}}\right) \]

3. Simplified 99.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{r}{-s}}}{r}\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. *-un-lft-identity 99.5%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\color{blue}{1 \cdot \left(-0.3333333333333333 \cdot \frac{r}{s}\right)}}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{r}{-s}}}{r}\right) \]

   2. exp-prod 99.6%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{\color{blue}{{\left(e^{1}\right)}^{\left(-0.3333333333333333 \cdot \frac{r}{s}\right)}}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{r}{-s}}}{r}\right) \]

   3. associate-*r/ 99.6%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{{\left(e^{1}\right)}^{\color{blue}{\left(\frac{-0.3333333333333333 \cdot r}{s}\right)}}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{r}{-s}}}{r}\right) \]

6. Applied egg-rr 99.6%

   \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{\color{blue}{{\left(e^{1}\right)}^{\left(\frac{-0.3333333333333333 \cdot r}{s}\right)}}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{r}{-s}}}{r}\right) \]

7. Taylor expanded in s around 0 99.5%

   \[\leadsto \color{blue}{\frac{0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}}}{r \cdot \pi} + 0.125 \cdot \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r \cdot \pi}}{s}} \]
8. Step-by-step derivation

   1. distribute-lft-out 99.5%

      \[\leadsto \frac{\color{blue}{0.125 \cdot \left(\frac{e^{-1 \cdot \frac{r}{s}}}{r \cdot \pi} + \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r \cdot \pi}\right)}}{s} \]

   2. mul-1-neg 99.5%

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

   3. associate-*r/ 99.5%

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

   4. *-commutative 99.5%

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

   5. associate-*r/ 99.6%

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

9. Simplified 99.6%

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

10. Taylor expanded in s around -inf 10.8%

    \[\leadsto \frac{0.125 \cdot \color{blue}{\left(-1 \cdot \frac{-1 \cdot \frac{0.05555555555555555 \cdot \frac{r}{\pi} + 0.5 \cdot \frac{r}{\pi}}{s} + 1.3333333333333333 \cdot \frac{1}{\pi}}{s} + 2 \cdot \frac{1}{r \cdot \pi}\right)}}{s} \]
11. Step-by-step derivation

    1. +-commutative 10.8%

       \[\leadsto \frac{0.125 \cdot \color{blue}{\left(2 \cdot \frac{1}{r \cdot \pi} + -1 \cdot \frac{-1 \cdot \frac{0.05555555555555555 \cdot \frac{r}{\pi} + 0.5 \cdot \frac{r}{\pi}}{s} + 1.3333333333333333 \cdot \frac{1}{\pi}}{s}\right)}}{s} \]

    2. mul-1-neg 10.8%

       \[\leadsto \frac{0.125 \cdot \left(2 \cdot \frac{1}{r \cdot \pi} + \color{blue}{\left(-\frac{-1 \cdot \frac{0.05555555555555555 \cdot \frac{r}{\pi} + 0.5 \cdot \frac{r}{\pi}}{s} + 1.3333333333333333 \cdot \frac{1}{\pi}}{s}\right)}\right)}{s} \]

    3. unsub-neg 10.8%

       \[\leadsto \frac{0.125 \cdot \color{blue}{\left(2 \cdot \frac{1}{r \cdot \pi} - \frac{-1 \cdot \frac{0.05555555555555555 \cdot \frac{r}{\pi} + 0.5 \cdot \frac{r}{\pi}}{s} + 1.3333333333333333 \cdot \frac{1}{\pi}}{s}\right)}}{s} \]

    4. associate-*r/ 10.8%

       \[\leadsto \frac{0.125 \cdot \left(\color{blue}{\frac{2 \cdot 1}{r \cdot \pi}} - \frac{-1 \cdot \frac{0.05555555555555555 \cdot \frac{r}{\pi} + 0.5 \cdot \frac{r}{\pi}}{s} + 1.3333333333333333 \cdot \frac{1}{\pi}}{s}\right)}{s} \]

    5. metadata-eval 10.8%

       \[\leadsto \frac{0.125 \cdot \left(\frac{\color{blue}{2}}{r \cdot \pi} - \frac{-1 \cdot \frac{0.05555555555555555 \cdot \frac{r}{\pi} + 0.5 \cdot \frac{r}{\pi}}{s} + 1.3333333333333333 \cdot \frac{1}{\pi}}{s}\right)}{s} \]

    6. *-commutative 10.8%

       \[\leadsto \frac{0.125 \cdot \left(\frac{2}{\color{blue}{\pi \cdot r}} - \frac{-1 \cdot \frac{0.05555555555555555 \cdot \frac{r}{\pi} + 0.5 \cdot \frac{r}{\pi}}{s} + 1.3333333333333333 \cdot \frac{1}{\pi}}{s}\right)}{s} \]

    7. associate-/r* 10.8%

       \[\leadsto \frac{0.125 \cdot \left(\color{blue}{\frac{\frac{2}{\pi}}{r}} - \frac{-1 \cdot \frac{0.05555555555555555 \cdot \frac{r}{\pi} + 0.5 \cdot \frac{r}{\pi}}{s} + 1.3333333333333333 \cdot \frac{1}{\pi}}{s}\right)}{s} \]

12. Simplified 10.8%

    \[\leadsto \frac{0.125 \cdot \color{blue}{\left(\frac{\frac{2}{\pi}}{r} - \frac{\frac{1.3333333333333333}{\pi} - \frac{\frac{r \cdot 0.5555555555555556}{\pi}}{s}}{s}\right)}}{s} \]

13. Final simplification 10.8%

    \[\leadsto \frac{0.125 \cdot \left(\frac{\frac{2}{\pi}}{r} + \frac{\frac{\frac{r \cdot 0.5555555555555556}{\pi}}{s} - \frac{1.3333333333333333}{\pi}}{s}\right)}{s} \]
14. Add Preprocessing

Alternative 7: 9.9% accurate, 11.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.25}{r \cdot \pi} - \frac{\frac{\frac{r}{\pi} \cdot -0.06944444444444445}{s} + \frac{0.16666666666666666}{\pi}}{s}}{s} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (/
  (-
   (/ 0.25 (* r PI))
   (/
    (+ (/ (* (/ r PI) -0.06944444444444445) s) (/ 0.16666666666666666 PI))
    s))
  s))

C

float code(float s, float r) {
	return ((0.25f / (r * ((float) M_PI))) - (((((r / ((float) M_PI)) * -0.06944444444444445f) / s) + (0.16666666666666666f / ((float) M_PI))) / s)) / s;
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.25) / Float32(r * Float32(pi))) - Float32(Float32(Float32(Float32(Float32(r / Float32(pi)) * Float32(-0.06944444444444445)) / s) + Float32(Float32(0.16666666666666666) / Float32(pi))) / s)) / s)
end

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.25) / (r * single(pi))) - (((((r / single(pi)) * single(-0.06944444444444445)) / s) + (single(0.16666666666666666) / single(pi))) / s)) / s;
end

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. Step-by-step derivation

   1. +-commutative 99.5%

      \[\leadsto \color{blue}{\frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}} \]

   2. times-frac 99.5%

      \[\leadsto \color{blue}{\frac{0.75}{\left(6 \cdot \pi\right) \cdot s} \cdot \frac{e^{\frac{-r}{3 \cdot s}}}{r}} + \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} \]

   3. fma-define 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0.75}{\left(6 \cdot \pi\right) \cdot s}, \frac{e^{\frac{-r}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right)} \]

   4. associate-*l* 99.5%

      \[\leadsto \mathsf{fma}\left(\frac{0.75}{\color{blue}{6 \cdot \left(\pi \cdot s\right)}}, \frac{e^{\frac{-r}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   5. associate-/r* 99.5%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{0.75}{6}}{\pi \cdot s}}, \frac{e^{\frac{-r}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   6. metadata-eval 99.5%

      \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{0.125}}{\pi \cdot s}, \frac{e^{\frac{-r}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   7. *-commutative 99.5%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{\color{blue}{s \cdot \pi}}, \frac{e^{\frac{-r}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   8. neg-mul-1 99.5%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\frac{\color{blue}{-1 \cdot r}}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   9. times-frac 99.5%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\color{blue}{\frac{-1}{3} \cdot \frac{r}{s}}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   10. metadata-eval 99.5%

       \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\color{blue}{-0.3333333333333333} \cdot \frac{r}{s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   11. times-frac 99.5%

       \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r}, \color{blue}{\frac{0.25}{\left(2 \cdot \pi\right) \cdot s} \cdot \frac{e^{\frac{-r}{s}}}{r}}\right) \]

3. Simplified 99.5%

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

5. Taylor expanded in s around -inf 10.8%

   \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{-0.0625 \cdot \frac{r}{\pi} + -0.006944444444444444 \cdot \frac{r}{\pi}}{s} - 0.16666666666666666 \cdot \frac{1}{\pi}}{s} - 0.25 \cdot \frac{1}{r \cdot \pi}}{s}} \]
6. Step-by-step derivation

   1. mul-1-neg 10.8%

      \[\leadsto \color{blue}{-\frac{-1 \cdot \frac{-1 \cdot \frac{-0.0625 \cdot \frac{r}{\pi} + -0.006944444444444444 \cdot \frac{r}{\pi}}{s} - 0.16666666666666666 \cdot \frac{1}{\pi}}{s} - 0.25 \cdot \frac{1}{r \cdot \pi}}{s}} \]

7. Simplified 10.8%

   \[\leadsto \color{blue}{-\frac{\left(-\frac{\left(-\frac{\frac{r}{\pi} \cdot -0.06944444444444445}{s}\right) - \frac{0.16666666666666666}{\pi}}{s}\right) - \frac{0.25}{r \cdot \pi}}{s}} \]

8. Final simplification 10.8%

   \[\leadsto \frac{\frac{0.25}{r \cdot \pi} - \frac{\frac{\frac{r}{\pi} \cdot -0.06944444444444445}{s} + \frac{0.16666666666666666}{\pi}}{s}}{s} \]
9. Add Preprocessing

Alternative 8: 8.9% accurate, 33.0× speedup

Math

\[\begin{array}{l} \\ \frac{0.25}{r \cdot \left(s \cdot \pi\right)} \end{array} \]

FPCore

(FPCore (s r) :precision binary32 (/ 0.25 (* r (* s PI))))

C

float code(float s, float r) {
	return 0.25f / (r * (s * ((float) M_PI)));
}

Julia

function code(s, r)
	return Float32(Float32(0.25) / Float32(r * Float32(s * Float32(pi))))
end

MATLAB

function tmp = code(s, r)
	tmp = single(0.25) / (r * (s * single(pi)));
end

Derivation

1. Initial program 99.5%

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

3. Simplified 99.1%

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

5. Taylor expanded in s around inf 9.8%

   \[\leadsto \color{blue}{\frac{0.25}{r \cdot \left(s \cdot \pi\right)}} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024146 
(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))))