Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.6%

Time: 5.8s

Alternatives: 6

Speedup: N/A×

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: 99.6% 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: 99.6% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{0.25}{\left(2 \cdot \pi\right) \cdot s}, \frac{e^{\frac{-1 \cdot r}{s}}}{r}, \frac{0.75}{\left(6 \cdot \pi\right) \cdot s} \cdot \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r}\right) \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (fma
  (/ 0.25 (* (* 2.0 PI) s))
  (/ (exp (/ (* -1.0 r) s)) r)
  (* (/ 0.75 (* (* 6.0 PI) s)) (/ (exp (* -0.3333333333333333 (/ r s))) r))))

C

float code(float s, float r) {
	return fmaf((0.25f / ((2.0f * ((float) M_PI)) * s)), (expf(((-1.0f * r) / s)) / r), ((0.75f / ((6.0f * ((float) M_PI)) * s)) * (expf((-0.3333333333333333f * (r / s))) / r)));
}

Julia

function code(s, r)
	return fma(Float32(Float32(0.25) / Float32(Float32(Float32(2.0) * Float32(pi)) * s)), Float32(exp(Float32(Float32(Float32(-1.0) * r) / s)) / r), Float32(Float32(Float32(0.75) / Float32(Float32(Float32(6.0) * Float32(pi)) * s)) * Float32(exp(Float32(Float32(-0.3333333333333333) * Float32(r / s))) / r)))
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. Add Preprocessing

3. Taylor expanded in s around 0

   \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\frac{3}{4} \cdot e^{\color{blue}{\frac{-1}{3} \cdot \frac{r}{s}}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
4. Step-by-step derivation

   1. lower-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\frac{3}{4} \cdot e^{\frac{-1}{3} \cdot \color{blue}{\frac{r}{s}}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   2. lower-/.f32 99.5

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{-0.3333333333333333 \cdot \frac{r}{\color{blue}{s}}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

5. Applied rewrites 99.5%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\color{blue}{-0.3333333333333333 \cdot \frac{r}{s}}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
6. Step-by-step derivation

   1. lift-+.f32 N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\frac{3}{4} \cdot e^{\frac{-1}{3} \cdot \frac{r}{s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r}} \]

   2. lift-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}} + \frac{\frac{3}{4} \cdot e^{\frac{-1}{3} \cdot \frac{r}{s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   3. lift-*.f32 N/A

      \[\leadsto \frac{\color{blue}{\frac{1}{4} \cdot e^{\frac{-r}{s}}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\frac{3}{4} \cdot e^{\frac{-1}{3} \cdot \frac{r}{s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   4. lift-exp.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot \color{blue}{e^{\frac{-r}{s}}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\frac{3}{4} \cdot e^{\frac{-1}{3} \cdot \frac{r}{s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   5. lift-neg.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\color{blue}{\mathsf{neg}\left(r\right)}}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\frac{3}{4} \cdot e^{\frac{-1}{3} \cdot \frac{r}{s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   6. lift-/.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\color{blue}{\frac{\mathsf{neg}\left(r\right)}{s}}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\frac{3}{4} \cdot e^{\frac{-1}{3} \cdot \frac{r}{s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   7. lift-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\color{blue}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}} + \frac{\frac{3}{4} \cdot e^{\frac{-1}{3} \cdot \frac{r}{s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   8. times-frac N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{4}}{\left(2 \cdot \pi\right) \cdot s} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{r}} + \frac{\frac{3}{4} \cdot e^{\frac{-1}{3} \cdot \frac{r}{s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   9. lower-fma.f32 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\frac{1}{4}}{\left(2 \cdot \pi\right) \cdot s}, \frac{e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{r}, \frac{\frac{3}{4} \cdot e^{\frac{-1}{3} \cdot \frac{r}{s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r}\right)} \]

7. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0.25}{\left(2 \cdot \pi\right) \cdot s}, \frac{e^{\frac{-r}{s}}}{r}, \frac{0.75}{\left(6 \cdot \pi\right) \cdot s} \cdot \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r}\right)} \]

8. Final simplification 99.5%

   \[\leadsto \mathsf{fma}\left(\frac{0.25}{\left(2 \cdot \pi\right) \cdot s}, \frac{e^{\frac{-1 \cdot r}{s}}}{r}, \frac{0.75}{\left(6 \cdot \pi\right) \cdot s} \cdot \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r}\right) \]
9. Add Preprocessing

Alternative 2: 99.6% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{{\left(e^{-1}\right)}^{\left(\frac{r}{s}\right)} \cdot 0.25}{\left(\pi \cdot 2\right) \cdot s}}{r} + \frac{0.75 \cdot e^{-1 \cdot \frac{r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/ (/ (* (pow (exp -1.0) (/ r s)) 0.25) (* (* PI 2.0) s)) r)
  (/ (* 0.75 (exp (* -1.0 (/ r (* 3.0 s))))) (* (* (* 6.0 PI) s) r))))

C

float code(float s, float r) {
	return (((powf(expf(-1.0f), (r / s)) * 0.25f) / ((((float) M_PI) * 2.0f) * s)) / r) + ((0.75f * expf((-1.0f * (r / (3.0f * s))))) / (((6.0f * ((float) M_PI)) * s) * r));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32((exp(Float32(-1.0)) ^ Float32(r / s)) * Float32(0.25)) / Float32(Float32(Float32(pi) * Float32(2.0)) * s)) / r) + Float32(Float32(Float32(0.75) * exp(Float32(Float32(-1.0) * 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 = ((((exp(single(-1.0)) ^ (r / s)) * single(0.25)) / ((single(pi) * single(2.0)) * s)) / r) + ((single(0.75) * exp((single(-1.0) * (r / (single(3.0) * s))))) / (((single(6.0) * single(pi)) * s) * r));
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. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}} + \frac{\frac{3}{4} \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   2. lift-*.f32 N/A

      \[\leadsto \frac{\color{blue}{\frac{1}{4} \cdot e^{\frac{-r}{s}}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\frac{3}{4} \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   3. lift-exp.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot \color{blue}{e^{\frac{-r}{s}}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\frac{3}{4} \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   4. lift-neg.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\color{blue}{\mathsf{neg}\left(r\right)}}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\frac{3}{4} \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   5. lift-/.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\color{blue}{\frac{\mathsf{neg}\left(r\right)}{s}}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\frac{3}{4} \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   6. lift-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\color{blue}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}} + \frac{\frac{3}{4} \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   7. associate-/r* N/A

      \[\leadsto \color{blue}{\frac{\frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(2 \cdot \pi\right) \cdot s}}{r}} + \frac{\frac{3}{4} \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   8. lower-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(2 \cdot \pi\right) \cdot s}}{r}} + \frac{\frac{3}{4} \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

4. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\frac{\frac{{\left(e^{-1}\right)}^{\left(\frac{r}{s}\right)} \cdot 0.25}{\left(\pi \cdot 2\right) \cdot s}}{r}} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

5. Final simplification 99.5%

   \[\leadsto \frac{\frac{{\left(e^{-1}\right)}^{\left(\frac{r}{s}\right)} \cdot 0.25}{\left(\pi \cdot 2\right) \cdot s}}{r} + \frac{0.75 \cdot e^{-1 \cdot \frac{r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
6. Add Preprocessing

Alternative 3: 99.5% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \frac{0.25 \cdot e^{\frac{-1 \cdot r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{-0.3333333333333333 \cdot \frac{r}{s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/ (* 0.25 (exp (/ (* -1.0 r) s))) (* (* (* 2.0 PI) s) r))
  (/ (* 0.75 (exp (* -0.3333333333333333 (/ r s)))) (* (* (* 6.0 PI) s) r))))

C

float code(float s, float r) {
	return ((0.25f * expf(((-1.0f * r) / s))) / (((2.0f * ((float) M_PI)) * s) * r)) + ((0.75f * expf((-0.3333333333333333f * (r / s)))) / (((6.0f * ((float) M_PI)) * s) * r));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.25) * exp(Float32(Float32(Float32(-1.0) * r) / s))) / Float32(Float32(Float32(Float32(2.0) * Float32(pi)) * s) * r)) + Float32(Float32(Float32(0.75) * exp(Float32(Float32(-0.3333333333333333) * Float32(r / s)))) / Float32(Float32(Float32(Float32(6.0) * Float32(pi)) * s) * r)))
end

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.25) * exp(((single(-1.0) * r) / s))) / (((single(2.0) * single(pi)) * s) * r)) + ((single(0.75) * exp((single(-0.3333333333333333) * (r / s)))) / (((single(6.0) * single(pi)) * s) * r));
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. Add Preprocessing

3. Taylor expanded in s around 0

   \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\frac{3}{4} \cdot e^{\color{blue}{\frac{-1}{3} \cdot \frac{r}{s}}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
4. Step-by-step derivation

   1. lower-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\frac{3}{4} \cdot e^{\frac{-1}{3} \cdot \color{blue}{\frac{r}{s}}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   2. lower-/.f32 99.5

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{-0.3333333333333333 \cdot \frac{r}{\color{blue}{s}}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

5. Applied rewrites 99.5%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\color{blue}{-0.3333333333333333 \cdot \frac{r}{s}}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

6. Final simplification 99.5%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-1 \cdot r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{-0.3333333333333333 \cdot \frac{r}{s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
7. Add Preprocessing

Alternative 4: 99.3% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \frac{0.25 \cdot e^{\frac{-1 \cdot r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{{\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}}{\left(\pi \cdot s\right) \cdot r} \cdot 0.125 \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/ (* 0.25 (exp (/ (* -1.0 r) s))) (* (* (* 2.0 PI) s) r))
  (* (/ (pow (exp -0.3333333333333333) (/ r s)) (* (* PI s) r)) 0.125)))

C

float code(float s, float r) {
	return ((0.25f * expf(((-1.0f * r) / s))) / (((2.0f * ((float) M_PI)) * s) * r)) + ((powf(expf(-0.3333333333333333f), (r / s)) / ((((float) M_PI) * s) * r)) * 0.125f);
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.25) * exp(Float32(Float32(Float32(-1.0) * r) / s))) / Float32(Float32(Float32(Float32(2.0) * Float32(pi)) * s) * r)) + Float32(Float32((exp(Float32(-0.3333333333333333)) ^ Float32(r / s)) / Float32(Float32(Float32(pi) * s) * r)) * Float32(0.125)))
end

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.25) * exp(((single(-1.0) * r) / s))) / (((single(2.0) * single(pi)) * s) * r)) + (((exp(single(-0.3333333333333333)) ^ (r / s)) / ((single(pi) * s) * r)) * single(0.125));
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. Add Preprocessing

3. Taylor expanded in s around 0

   \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \color{blue}{\frac{1}{8} \cdot \frac{e^{\frac{-1}{3} \cdot \frac{r}{s}}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{e^{\frac{-1}{3} \cdot \frac{r}{s}}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} \cdot \color{blue}{\frac{1}{8}} \]

   2. lower-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{e^{\frac{-1}{3} \cdot \frac{r}{s}}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} \cdot \color{blue}{\frac{1}{8}} \]

   3. lower-/.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{e^{\frac{-1}{3} \cdot \frac{r}{s}}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} \cdot \frac{1}{8} \]

   4. exp-prod N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{{\left(e^{\frac{-1}{3}}\right)}^{\left(\frac{r}{s}\right)}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} \cdot \frac{1}{8} \]

   5. lower-pow.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{{\left(e^{\frac{-1}{3}}\right)}^{\left(\frac{r}{s}\right)}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} \cdot \frac{1}{8} \]

   6. lower-exp.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{{\left(e^{\frac{-1}{3}}\right)}^{\left(\frac{r}{s}\right)}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} \cdot \frac{1}{8} \]

   7. lower-/.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{{\left(e^{\frac{-1}{3}}\right)}^{\left(\frac{r}{s}\right)}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} \cdot \frac{1}{8} \]

   8. *-commutative N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{{\left(e^{\frac{-1}{3}}\right)}^{\left(\frac{r}{s}\right)}}{\left(s \cdot \mathsf{PI}\left(\right)\right) \cdot r} \cdot \frac{1}{8} \]

   9. lower-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{{\left(e^{\frac{-1}{3}}\right)}^{\left(\frac{r}{s}\right)}}{\left(s \cdot \mathsf{PI}\left(\right)\right) \cdot r} \cdot \frac{1}{8} \]

   10. *-commutative N/A

       \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{{\left(e^{\frac{-1}{3}}\right)}^{\left(\frac{r}{s}\right)}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} \cdot \frac{1}{8} \]

   11. lower-*.f32 N/A

       \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{{\left(e^{\frac{-1}{3}}\right)}^{\left(\frac{r}{s}\right)}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} \cdot \frac{1}{8} \]

   12. lift-PI.f32 99.3

       \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{{\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}}{\left(\pi \cdot s\right) \cdot r} \cdot 0.125 \]

5. Applied rewrites 99.3%

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

6. Final simplification 99.3%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-1 \cdot r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{{\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}}{\left(\pi \cdot s\right) \cdot r} \cdot 0.125 \]
7. Add Preprocessing

Alternative 5: 9.6% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ 12 \cdot \left(\frac{s \cdot s}{r} \cdot \frac{\left(\pi \cdot \pi\right) \cdot \left(0.015625 \cdot \frac{e^{\frac{-1 \cdot r}{s} \cdot 2}}{{\left(s \cdot \pi\right)}^{2}} - 0.015625 \cdot \frac{1}{\left(s \cdot s\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot {\left(e^{0.6666666666666666}\right)}^{\left(\frac{r}{s}\right)}\right)}\right)}{1.5 \cdot \left(s \cdot \left(\pi \cdot {\left(e^{-1}\right)}^{\left(\frac{r}{s}\right)}\right)\right) - 1.5 \cdot \frac{s \cdot \pi}{{\left(e^{0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}}}\right) \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (*
  12.0
  (*
   (/ (* s s) r)
   (/
    (*
     (* PI PI)
     (-
      (* 0.015625 (/ (exp (* (/ (* -1.0 r) s) 2.0)) (pow (* s PI) 2.0)))
      (*
       0.015625
       (/
        1.0
        (* (* s s) (* (* PI PI) (pow (exp 0.6666666666666666) (/ r s))))))))
    (-
     (* 1.5 (* s (* PI (pow (exp -1.0) (/ r s)))))
     (* 1.5 (/ (* s PI) (pow (exp 0.3333333333333333) (/ r s)))))))))

C

float code(float s, float r) {
	return 12.0f * (((s * s) / r) * (((((float) M_PI) * ((float) M_PI)) * ((0.015625f * (expf((((-1.0f * r) / s) * 2.0f)) / powf((s * ((float) M_PI)), 2.0f))) - (0.015625f * (1.0f / ((s * s) * ((((float) M_PI) * ((float) M_PI)) * powf(expf(0.6666666666666666f), (r / s)))))))) / ((1.5f * (s * (((float) M_PI) * powf(expf(-1.0f), (r / s))))) - (1.5f * ((s * ((float) M_PI)) / powf(expf(0.3333333333333333f), (r / s)))))));
}

Julia

function code(s, r)
	return Float32(Float32(12.0) * Float32(Float32(Float32(s * s) / r) * Float32(Float32(Float32(Float32(pi) * Float32(pi)) * Float32(Float32(Float32(0.015625) * Float32(exp(Float32(Float32(Float32(Float32(-1.0) * r) / s) * Float32(2.0))) / (Float32(s * Float32(pi)) ^ Float32(2.0)))) - Float32(Float32(0.015625) * Float32(Float32(1.0) / Float32(Float32(s * s) * Float32(Float32(Float32(pi) * Float32(pi)) * (exp(Float32(0.6666666666666666)) ^ Float32(r / s)))))))) / Float32(Float32(Float32(1.5) * Float32(s * Float32(Float32(pi) * (exp(Float32(-1.0)) ^ Float32(r / s))))) - Float32(Float32(1.5) * Float32(Float32(s * Float32(pi)) / (exp(Float32(0.3333333333333333)) ^ Float32(r / s))))))))
end

MATLAB

function tmp = code(s, r)
	tmp = single(12.0) * (((s * s) / r) * (((single(pi) * single(pi)) * ((single(0.015625) * (exp((((single(-1.0) * r) / s) * single(2.0))) / ((s * single(pi)) ^ single(2.0)))) - (single(0.015625) * (single(1.0) / ((s * s) * ((single(pi) * single(pi)) * (exp(single(0.6666666666666666)) ^ (r / s)))))))) / ((single(1.5) * (s * (single(pi) * (exp(single(-1.0)) ^ (r / s))))) - (single(1.5) * ((s * single(pi)) / (exp(single(0.3333333333333333)) ^ (r / 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. Add Preprocessing

4. Applied rewrites 9.8%

   \[\leadsto \color{blue}{\frac{\left(0.25 \cdot \frac{{\left(e^{-1}\right)}^{\left(\frac{r}{s}\right)}}{\left(\left(\pi \cdot 2\right) \cdot s\right) \cdot r}\right) \cdot \left(0.25 \cdot \frac{{\left(e^{-1}\right)}^{\left(\frac{r}{s}\right)}}{\left(\left(\pi \cdot 2\right) \cdot s\right) \cdot r}\right) - \left(\frac{0.75}{\left(\pi \cdot 6\right) \cdot s} \cdot \frac{\frac{1}{e^{\frac{r}{3 \cdot s}}}}{r}\right) \cdot \left(\frac{0.75}{\left(\pi \cdot 6\right) \cdot s} \cdot \frac{\frac{1}{e^{\frac{r}{3 \cdot s}}}}{r}\right)}{\frac{\left({\left(e^{-1}\right)}^{\left(\frac{r}{s}\right)} \cdot 0.25\right) \cdot \left(\left(\left(\pi \cdot 6\right) \cdot s\right) \cdot r\right) - \left(\left(\left(\pi \cdot 2\right) \cdot s\right) \cdot r\right) \cdot \left(\frac{1}{e^{\frac{r}{3 \cdot s}}} \cdot 0.75\right)}{\left(\left(\left(\pi \cdot 2\right) \cdot s\right) \cdot r\right) \cdot \left(\left(\left(\pi \cdot 6\right) \cdot s\right) \cdot r\right)}}} \]

5. Taylor expanded in r around inf

   \[\leadsto \color{blue}{12 \cdot \frac{{s}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \left(\frac{1}{64} \cdot \frac{{\left(e^{-1 \cdot \frac{r}{s}}\right)}^{2}}{{s}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}} - \frac{1}{64} \cdot \frac{1}{{s}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot {\left(e^{\frac{1}{3} \cdot \frac{r}{s}}\right)}^{2}\right)}\right)\right)}{r \cdot \left(\frac{3}{2} \cdot \left(s \cdot \left(\mathsf{PI}\left(\right) \cdot e^{-1 \cdot \frac{r}{s}}\right)\right) - \frac{3}{2} \cdot \frac{s \cdot \mathsf{PI}\left(\right)}{e^{\frac{1}{3} \cdot \frac{r}{s}}}\right)}} \]

7. Applied rewrites 10.0%

   \[\leadsto \color{blue}{12 \cdot \left(\frac{s \cdot s}{r} \cdot \frac{\left(\pi \cdot \pi\right) \cdot \left(0.015625 \cdot \frac{e^{\left(-1 \cdot \frac{r}{s}\right) \cdot 2}}{{\left(s \cdot \pi\right)}^{2}} - 0.015625 \cdot \frac{1}{\left(s \cdot s\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot e^{\left(0.3333333333333333 \cdot \frac{r}{s}\right) \cdot 2}\right)}\right)}{1.5 \cdot \left(s \cdot \left(\pi \cdot {\left(e^{-1}\right)}^{\left(\frac{r}{s}\right)}\right)\right) - 1.5 \cdot \frac{s \cdot \pi}{{\left(e^{0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}}}\right)} \]

8. Taylor expanded in s around 0

   \[\leadsto 12 \cdot \left(\frac{s \cdot s}{r} \cdot \frac{\left(\pi \cdot \pi\right) \cdot \left(\frac{1}{64} \cdot \frac{e^{\left(-1 \cdot \frac{r}{s}\right) \cdot 2}}{{\left(s \cdot \pi\right)}^{2}} - \frac{1}{64} \cdot \frac{1}{\left(s \cdot s\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot e^{\frac{2}{3} \cdot \frac{r}{s}}\right)}\right)}{\frac{3}{2} \cdot \left(s \cdot \left(\pi \cdot {\left(e^{-1}\right)}^{\left(\frac{r}{s}\right)}\right)\right) - \frac{3}{2} \cdot \frac{s \cdot \pi}{{\left(e^{\frac{1}{3}}\right)}^{\left(\frac{r}{s}\right)}}}\right) \]
9. Step-by-step derivation

   1. exp-prod N/A

      \[\leadsto 12 \cdot \left(\frac{s \cdot s}{r} \cdot \frac{\left(\pi \cdot \pi\right) \cdot \left(\frac{1}{64} \cdot \frac{e^{\left(-1 \cdot \frac{r}{s}\right) \cdot 2}}{{\left(s \cdot \pi\right)}^{2}} - \frac{1}{64} \cdot \frac{1}{\left(s \cdot s\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot {\left(e^{\frac{2}{3}}\right)}^{\left(\frac{r}{s}\right)}\right)}\right)}{\frac{3}{2} \cdot \left(s \cdot \left(\pi \cdot {\left(e^{-1}\right)}^{\left(\frac{r}{s}\right)}\right)\right) - \frac{3}{2} \cdot \frac{s \cdot \pi}{{\left(e^{\frac{1}{3}}\right)}^{\left(\frac{r}{s}\right)}}}\right) \]

   2. lower-pow.f32 N/A

      \[\leadsto 12 \cdot \left(\frac{s \cdot s}{r} \cdot \frac{\left(\pi \cdot \pi\right) \cdot \left(\frac{1}{64} \cdot \frac{e^{\left(-1 \cdot \frac{r}{s}\right) \cdot 2}}{{\left(s \cdot \pi\right)}^{2}} - \frac{1}{64} \cdot \frac{1}{\left(s \cdot s\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot {\left(e^{\frac{2}{3}}\right)}^{\left(\frac{r}{s}\right)}\right)}\right)}{\frac{3}{2} \cdot \left(s \cdot \left(\pi \cdot {\left(e^{-1}\right)}^{\left(\frac{r}{s}\right)}\right)\right) - \frac{3}{2} \cdot \frac{s \cdot \pi}{{\left(e^{\frac{1}{3}}\right)}^{\left(\frac{r}{s}\right)}}}\right) \]

   3. lower-exp.f32 N/A

      \[\leadsto 12 \cdot \left(\frac{s \cdot s}{r} \cdot \frac{\left(\pi \cdot \pi\right) \cdot \left(\frac{1}{64} \cdot \frac{e^{\left(-1 \cdot \frac{r}{s}\right) \cdot 2}}{{\left(s \cdot \pi\right)}^{2}} - \frac{1}{64} \cdot \frac{1}{\left(s \cdot s\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot {\left(e^{\frac{2}{3}}\right)}^{\left(\frac{r}{s}\right)}\right)}\right)}{\frac{3}{2} \cdot \left(s \cdot \left(\pi \cdot {\left(e^{-1}\right)}^{\left(\frac{r}{s}\right)}\right)\right) - \frac{3}{2} \cdot \frac{s \cdot \pi}{{\left(e^{\frac{1}{3}}\right)}^{\left(\frac{r}{s}\right)}}}\right) \]

   4. lift-/.f32 10.1

      \[\leadsto 12 \cdot \left(\frac{s \cdot s}{r} \cdot \frac{\left(\pi \cdot \pi\right) \cdot \left(0.015625 \cdot \frac{e^{\left(-1 \cdot \frac{r}{s}\right) \cdot 2}}{{\left(s \cdot \pi\right)}^{2}} - 0.015625 \cdot \frac{1}{\left(s \cdot s\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot {\left(e^{0.6666666666666666}\right)}^{\left(\frac{r}{s}\right)}\right)}\right)}{1.5 \cdot \left(s \cdot \left(\pi \cdot {\left(e^{-1}\right)}^{\left(\frac{r}{s}\right)}\right)\right) - 1.5 \cdot \frac{s \cdot \pi}{{\left(e^{0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}}}\right) \]

10. Applied rewrites 10.1%

    \[\leadsto 12 \cdot \left(\frac{s \cdot s}{r} \cdot \frac{\left(\pi \cdot \pi\right) \cdot \left(0.015625 \cdot \frac{e^{\left(-1 \cdot \frac{r}{s}\right) \cdot 2}}{{\left(s \cdot \pi\right)}^{2}} - 0.015625 \cdot \frac{1}{\left(s \cdot s\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot {\left(e^{0.6666666666666666}\right)}^{\left(\frac{r}{s}\right)}\right)}\right)}{1.5 \cdot \left(s \cdot \left(\pi \cdot {\left(e^{-1}\right)}^{\left(\frac{r}{s}\right)}\right)\right) - 1.5 \cdot \frac{s \cdot \pi}{{\left(e^{0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}}}\right) \]

11. Final simplification 10.1%

    \[\leadsto 12 \cdot \left(\frac{s \cdot s}{r} \cdot \frac{\left(\pi \cdot \pi\right) \cdot \left(0.015625 \cdot \frac{e^{\frac{-1 \cdot r}{s} \cdot 2}}{{\left(s \cdot \pi\right)}^{2}} - 0.015625 \cdot \frac{1}{\left(s \cdot s\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot {\left(e^{0.6666666666666666}\right)}^{\left(\frac{r}{s}\right)}\right)}\right)}{1.5 \cdot \left(s \cdot \left(\pi \cdot {\left(e^{-1}\right)}^{\left(\frac{r}{s}\right)}\right)\right) - 1.5 \cdot \frac{s \cdot \pi}{{\left(e^{0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}}}\right) \]
12. Add Preprocessing

Alternative 6: 9.5% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ 12 \cdot \left(\frac{e^{\log s \cdot 2}}{r} \cdot \frac{\left(\pi \cdot \pi\right) \cdot \left(0.015625 \cdot \frac{e^{\frac{-1 \cdot r}{s} \cdot 2}}{{\left(s \cdot \pi\right)}^{2}} - 0.015625 \cdot \frac{1}{\left(s \cdot s\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot {\left(e^{0.6666666666666666}\right)}^{\left(\frac{r}{s}\right)}\right)}\right)}{1.5 \cdot \left(s \cdot \left(\pi \cdot {\left(e^{-1}\right)}^{\left(\frac{r}{s}\right)}\right)\right) - 1.5 \cdot \frac{s \cdot \pi}{{\left(e^{0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}}}\right) \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (*
  12.0
  (*
   (/ (exp (* (log s) 2.0)) r)
   (/
    (*
     (* PI PI)
     (-
      (* 0.015625 (/ (exp (* (/ (* -1.0 r) s) 2.0)) (pow (* s PI) 2.0)))
      (*
       0.015625
       (/
        1.0
        (* (* s s) (* (* PI PI) (pow (exp 0.6666666666666666) (/ r s))))))))
    (-
     (* 1.5 (* s (* PI (pow (exp -1.0) (/ r s)))))
     (* 1.5 (/ (* s PI) (pow (exp 0.3333333333333333) (/ r s)))))))))

C

float code(float s, float r) {
	return 12.0f * ((expf((logf(s) * 2.0f)) / r) * (((((float) M_PI) * ((float) M_PI)) * ((0.015625f * (expf((((-1.0f * r) / s) * 2.0f)) / powf((s * ((float) M_PI)), 2.0f))) - (0.015625f * (1.0f / ((s * s) * ((((float) M_PI) * ((float) M_PI)) * powf(expf(0.6666666666666666f), (r / s)))))))) / ((1.5f * (s * (((float) M_PI) * powf(expf(-1.0f), (r / s))))) - (1.5f * ((s * ((float) M_PI)) / powf(expf(0.3333333333333333f), (r / s)))))));
}

Julia

function code(s, r)
	return Float32(Float32(12.0) * Float32(Float32(exp(Float32(log(s) * Float32(2.0))) / r) * Float32(Float32(Float32(Float32(pi) * Float32(pi)) * Float32(Float32(Float32(0.015625) * Float32(exp(Float32(Float32(Float32(Float32(-1.0) * r) / s) * Float32(2.0))) / (Float32(s * Float32(pi)) ^ Float32(2.0)))) - Float32(Float32(0.015625) * Float32(Float32(1.0) / Float32(Float32(s * s) * Float32(Float32(Float32(pi) * Float32(pi)) * (exp(Float32(0.6666666666666666)) ^ Float32(r / s)))))))) / Float32(Float32(Float32(1.5) * Float32(s * Float32(Float32(pi) * (exp(Float32(-1.0)) ^ Float32(r / s))))) - Float32(Float32(1.5) * Float32(Float32(s * Float32(pi)) / (exp(Float32(0.3333333333333333)) ^ Float32(r / s))))))))
end

MATLAB

function tmp = code(s, r)
	tmp = single(12.0) * ((exp((log(s) * single(2.0))) / r) * (((single(pi) * single(pi)) * ((single(0.015625) * (exp((((single(-1.0) * r) / s) * single(2.0))) / ((s * single(pi)) ^ single(2.0)))) - (single(0.015625) * (single(1.0) / ((s * s) * ((single(pi) * single(pi)) * (exp(single(0.6666666666666666)) ^ (r / s)))))))) / ((single(1.5) * (s * (single(pi) * (exp(single(-1.0)) ^ (r / s))))) - (single(1.5) * ((s * single(pi)) / (exp(single(0.3333333333333333)) ^ (r / 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. Add Preprocessing

4. Applied rewrites 9.8%

   \[\leadsto \color{blue}{\frac{\left(0.25 \cdot \frac{{\left(e^{-1}\right)}^{\left(\frac{r}{s}\right)}}{\left(\left(\pi \cdot 2\right) \cdot s\right) \cdot r}\right) \cdot \left(0.25 \cdot \frac{{\left(e^{-1}\right)}^{\left(\frac{r}{s}\right)}}{\left(\left(\pi \cdot 2\right) \cdot s\right) \cdot r}\right) - \left(\frac{0.75}{\left(\pi \cdot 6\right) \cdot s} \cdot \frac{\frac{1}{e^{\frac{r}{3 \cdot s}}}}{r}\right) \cdot \left(\frac{0.75}{\left(\pi \cdot 6\right) \cdot s} \cdot \frac{\frac{1}{e^{\frac{r}{3 \cdot s}}}}{r}\right)}{\frac{\left({\left(e^{-1}\right)}^{\left(\frac{r}{s}\right)} \cdot 0.25\right) \cdot \left(\left(\left(\pi \cdot 6\right) \cdot s\right) \cdot r\right) - \left(\left(\left(\pi \cdot 2\right) \cdot s\right) \cdot r\right) \cdot \left(\frac{1}{e^{\frac{r}{3 \cdot s}}} \cdot 0.75\right)}{\left(\left(\left(\pi \cdot 2\right) \cdot s\right) \cdot r\right) \cdot \left(\left(\left(\pi \cdot 6\right) \cdot s\right) \cdot r\right)}}} \]

5. Taylor expanded in r around inf

   \[\leadsto \color{blue}{12 \cdot \frac{{s}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \left(\frac{1}{64} \cdot \frac{{\left(e^{-1 \cdot \frac{r}{s}}\right)}^{2}}{{s}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}} - \frac{1}{64} \cdot \frac{1}{{s}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot {\left(e^{\frac{1}{3} \cdot \frac{r}{s}}\right)}^{2}\right)}\right)\right)}{r \cdot \left(\frac{3}{2} \cdot \left(s \cdot \left(\mathsf{PI}\left(\right) \cdot e^{-1 \cdot \frac{r}{s}}\right)\right) - \frac{3}{2} \cdot \frac{s \cdot \mathsf{PI}\left(\right)}{e^{\frac{1}{3} \cdot \frac{r}{s}}}\right)}} \]

7. Applied rewrites 10.0%

   \[\leadsto \color{blue}{12 \cdot \left(\frac{s \cdot s}{r} \cdot \frac{\left(\pi \cdot \pi\right) \cdot \left(0.015625 \cdot \frac{e^{\left(-1 \cdot \frac{r}{s}\right) \cdot 2}}{{\left(s \cdot \pi\right)}^{2}} - 0.015625 \cdot \frac{1}{\left(s \cdot s\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot e^{\left(0.3333333333333333 \cdot \frac{r}{s}\right) \cdot 2}\right)}\right)}{1.5 \cdot \left(s \cdot \left(\pi \cdot {\left(e^{-1}\right)}^{\left(\frac{r}{s}\right)}\right)\right) - 1.5 \cdot \frac{s \cdot \pi}{{\left(e^{0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}}}\right)} \]

8. Taylor expanded in s around 0

   \[\leadsto 12 \cdot \left(\frac{s \cdot s}{r} \cdot \frac{\left(\pi \cdot \pi\right) \cdot \left(\frac{1}{64} \cdot \frac{e^{\left(-1 \cdot \frac{r}{s}\right) \cdot 2}}{{\left(s \cdot \pi\right)}^{2}} - \frac{1}{64} \cdot \frac{1}{\left(s \cdot s\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot e^{\frac{2}{3} \cdot \frac{r}{s}}\right)}\right)}{\frac{3}{2} \cdot \left(s \cdot \left(\pi \cdot {\left(e^{-1}\right)}^{\left(\frac{r}{s}\right)}\right)\right) - \frac{3}{2} \cdot \frac{s \cdot \pi}{{\left(e^{\frac{1}{3}}\right)}^{\left(\frac{r}{s}\right)}}}\right) \]
9. Step-by-step derivation

   1. exp-prod N/A

      \[\leadsto 12 \cdot \left(\frac{s \cdot s}{r} \cdot \frac{\left(\pi \cdot \pi\right) \cdot \left(\frac{1}{64} \cdot \frac{e^{\left(-1 \cdot \frac{r}{s}\right) \cdot 2}}{{\left(s \cdot \pi\right)}^{2}} - \frac{1}{64} \cdot \frac{1}{\left(s \cdot s\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot {\left(e^{\frac{2}{3}}\right)}^{\left(\frac{r}{s}\right)}\right)}\right)}{\frac{3}{2} \cdot \left(s \cdot \left(\pi \cdot {\left(e^{-1}\right)}^{\left(\frac{r}{s}\right)}\right)\right) - \frac{3}{2} \cdot \frac{s \cdot \pi}{{\left(e^{\frac{1}{3}}\right)}^{\left(\frac{r}{s}\right)}}}\right) \]

   2. lower-pow.f32 N/A

      \[\leadsto 12 \cdot \left(\frac{s \cdot s}{r} \cdot \frac{\left(\pi \cdot \pi\right) \cdot \left(\frac{1}{64} \cdot \frac{e^{\left(-1 \cdot \frac{r}{s}\right) \cdot 2}}{{\left(s \cdot \pi\right)}^{2}} - \frac{1}{64} \cdot \frac{1}{\left(s \cdot s\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot {\left(e^{\frac{2}{3}}\right)}^{\left(\frac{r}{s}\right)}\right)}\right)}{\frac{3}{2} \cdot \left(s \cdot \left(\pi \cdot {\left(e^{-1}\right)}^{\left(\frac{r}{s}\right)}\right)\right) - \frac{3}{2} \cdot \frac{s \cdot \pi}{{\left(e^{\frac{1}{3}}\right)}^{\left(\frac{r}{s}\right)}}}\right) \]

   3. lower-exp.f32 N/A

      \[\leadsto 12 \cdot \left(\frac{s \cdot s}{r} \cdot \frac{\left(\pi \cdot \pi\right) \cdot \left(\frac{1}{64} \cdot \frac{e^{\left(-1 \cdot \frac{r}{s}\right) \cdot 2}}{{\left(s \cdot \pi\right)}^{2}} - \frac{1}{64} \cdot \frac{1}{\left(s \cdot s\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot {\left(e^{\frac{2}{3}}\right)}^{\left(\frac{r}{s}\right)}\right)}\right)}{\frac{3}{2} \cdot \left(s \cdot \left(\pi \cdot {\left(e^{-1}\right)}^{\left(\frac{r}{s}\right)}\right)\right) - \frac{3}{2} \cdot \frac{s \cdot \pi}{{\left(e^{\frac{1}{3}}\right)}^{\left(\frac{r}{s}\right)}}}\right) \]

   4. lift-/.f32 10.1

      \[\leadsto 12 \cdot \left(\frac{s \cdot s}{r} \cdot \frac{\left(\pi \cdot \pi\right) \cdot \left(0.015625 \cdot \frac{e^{\left(-1 \cdot \frac{r}{s}\right) \cdot 2}}{{\left(s \cdot \pi\right)}^{2}} - 0.015625 \cdot \frac{1}{\left(s \cdot s\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot {\left(e^{0.6666666666666666}\right)}^{\left(\frac{r}{s}\right)}\right)}\right)}{1.5 \cdot \left(s \cdot \left(\pi \cdot {\left(e^{-1}\right)}^{\left(\frac{r}{s}\right)}\right)\right) - 1.5 \cdot \frac{s \cdot \pi}{{\left(e^{0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}}}\right) \]

10. Applied rewrites 10.1%

    \[\leadsto 12 \cdot \left(\frac{s \cdot s}{r} \cdot \frac{\left(\pi \cdot \pi\right) \cdot \left(0.015625 \cdot \frac{e^{\left(-1 \cdot \frac{r}{s}\right) \cdot 2}}{{\left(s \cdot \pi\right)}^{2}} - 0.015625 \cdot \frac{1}{\left(s \cdot s\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot {\left(e^{0.6666666666666666}\right)}^{\left(\frac{r}{s}\right)}\right)}\right)}{1.5 \cdot \left(s \cdot \left(\pi \cdot {\left(e^{-1}\right)}^{\left(\frac{r}{s}\right)}\right)\right) - 1.5 \cdot \frac{s \cdot \pi}{{\left(e^{0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}}}\right) \]
11. Step-by-step derivation

    1. lift-*.f32 N/A

       \[\leadsto 12 \cdot \left(\frac{s \cdot s}{r} \cdot \frac{\color{blue}{\left(\pi \cdot \pi\right)} \cdot \left(\frac{1}{64} \cdot \frac{e^{\left(-1 \cdot \frac{r}{s}\right) \cdot 2}}{{\left(s \cdot \pi\right)}^{2}} - \frac{1}{64} \cdot \frac{1}{\left(s \cdot s\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot {\left(e^{\frac{2}{3}}\right)}^{\left(\frac{r}{s}\right)}\right)}\right)}{\frac{3}{2} \cdot \left(s \cdot \left(\pi \cdot {\left(e^{-1}\right)}^{\left(\frac{r}{s}\right)}\right)\right) - \frac{3}{2} \cdot \frac{s \cdot \pi}{{\left(e^{\frac{1}{3}}\right)}^{\left(\frac{r}{s}\right)}}}\right) \]

    2. pow2 N/A

       \[\leadsto 12 \cdot \left(\frac{{s}^{2}}{r} \cdot \frac{\color{blue}{\left(\pi \cdot \pi\right)} \cdot \left(\frac{1}{64} \cdot \frac{e^{\left(-1 \cdot \frac{r}{s}\right) \cdot 2}}{{\left(s \cdot \pi\right)}^{2}} - \frac{1}{64} \cdot \frac{1}{\left(s \cdot s\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot {\left(e^{\frac{2}{3}}\right)}^{\left(\frac{r}{s}\right)}\right)}\right)}{\frac{3}{2} \cdot \left(s \cdot \left(\pi \cdot {\left(e^{-1}\right)}^{\left(\frac{r}{s}\right)}\right)\right) - \frac{3}{2} \cdot \frac{s \cdot \pi}{{\left(e^{\frac{1}{3}}\right)}^{\left(\frac{r}{s}\right)}}}\right) \]

    3. pow-to-exp N/A

       \[\leadsto 12 \cdot \left(\frac{e^{\log s \cdot 2}}{r} \cdot \frac{\color{blue}{\left(\pi \cdot \pi\right)} \cdot \left(\frac{1}{64} \cdot \frac{e^{\left(-1 \cdot \frac{r}{s}\right) \cdot 2}}{{\left(s \cdot \pi\right)}^{2}} - \frac{1}{64} \cdot \frac{1}{\left(s \cdot s\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot {\left(e^{\frac{2}{3}}\right)}^{\left(\frac{r}{s}\right)}\right)}\right)}{\frac{3}{2} \cdot \left(s \cdot \left(\pi \cdot {\left(e^{-1}\right)}^{\left(\frac{r}{s}\right)}\right)\right) - \frac{3}{2} \cdot \frac{s \cdot \pi}{{\left(e^{\frac{1}{3}}\right)}^{\left(\frac{r}{s}\right)}}}\right) \]

    4. lower-exp.f32 N/A

       \[\leadsto 12 \cdot \left(\frac{e^{\log s \cdot 2}}{r} \cdot \frac{\color{blue}{\left(\pi \cdot \pi\right)} \cdot \left(\frac{1}{64} \cdot \frac{e^{\left(-1 \cdot \frac{r}{s}\right) \cdot 2}}{{\left(s \cdot \pi\right)}^{2}} - \frac{1}{64} \cdot \frac{1}{\left(s \cdot s\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot {\left(e^{\frac{2}{3}}\right)}^{\left(\frac{r}{s}\right)}\right)}\right)}{\frac{3}{2} \cdot \left(s \cdot \left(\pi \cdot {\left(e^{-1}\right)}^{\left(\frac{r}{s}\right)}\right)\right) - \frac{3}{2} \cdot \frac{s \cdot \pi}{{\left(e^{\frac{1}{3}}\right)}^{\left(\frac{r}{s}\right)}}}\right) \]

    5. lower-*.f32 N/A

       \[\leadsto 12 \cdot \left(\frac{e^{\log s \cdot 2}}{r} \cdot \frac{\left(\color{blue}{\pi} \cdot \pi\right) \cdot \left(\frac{1}{64} \cdot \frac{e^{\left(-1 \cdot \frac{r}{s}\right) \cdot 2}}{{\left(s \cdot \pi\right)}^{2}} - \frac{1}{64} \cdot \frac{1}{\left(s \cdot s\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot {\left(e^{\frac{2}{3}}\right)}^{\left(\frac{r}{s}\right)}\right)}\right)}{\frac{3}{2} \cdot \left(s \cdot \left(\pi \cdot {\left(e^{-1}\right)}^{\left(\frac{r}{s}\right)}\right)\right) - \frac{3}{2} \cdot \frac{s \cdot \pi}{{\left(e^{\frac{1}{3}}\right)}^{\left(\frac{r}{s}\right)}}}\right) \]

    6. lower-log.f32 10.0

       \[\leadsto 12 \cdot \left(\frac{e^{\log s \cdot 2}}{r} \cdot \frac{\left(\pi \cdot \pi\right) \cdot \left(0.015625 \cdot \frac{e^{\left(-1 \cdot \frac{r}{s}\right) \cdot 2}}{{\left(s \cdot \pi\right)}^{2}} - 0.015625 \cdot \frac{1}{\left(s \cdot s\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot {\left(e^{0.6666666666666666}\right)}^{\left(\frac{r}{s}\right)}\right)}\right)}{1.5 \cdot \left(s \cdot \left(\pi \cdot {\left(e^{-1}\right)}^{\left(\frac{r}{s}\right)}\right)\right) - 1.5 \cdot \frac{s \cdot \pi}{{\left(e^{0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}}}\right) \]

12. Applied rewrites 10.0%

    \[\leadsto 12 \cdot \left(\frac{e^{\log s \cdot 2}}{r} \cdot \frac{\color{blue}{\left(\pi \cdot \pi\right)} \cdot \left(0.015625 \cdot \frac{e^{\left(-1 \cdot \frac{r}{s}\right) \cdot 2}}{{\left(s \cdot \pi\right)}^{2}} - 0.015625 \cdot \frac{1}{\left(s \cdot s\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot {\left(e^{0.6666666666666666}\right)}^{\left(\frac{r}{s}\right)}\right)}\right)}{1.5 \cdot \left(s \cdot \left(\pi \cdot {\left(e^{-1}\right)}^{\left(\frac{r}{s}\right)}\right)\right) - 1.5 \cdot \frac{s \cdot \pi}{{\left(e^{0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}}}\right) \]

13. Final simplification 10.0%

    \[\leadsto 12 \cdot \left(\frac{e^{\log s \cdot 2}}{r} \cdot \frac{\left(\pi \cdot \pi\right) \cdot \left(0.015625 \cdot \frac{e^{\frac{-1 \cdot r}{s} \cdot 2}}{{\left(s \cdot \pi\right)}^{2}} - 0.015625 \cdot \frac{1}{\left(s \cdot s\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot {\left(e^{0.6666666666666666}\right)}^{\left(\frac{r}{s}\right)}\right)}\right)}{1.5 \cdot \left(s \cdot \left(\pi \cdot {\left(e^{-1}\right)}^{\left(\frac{r}{s}\right)}\right)\right) - 1.5 \cdot \frac{s \cdot \pi}{{\left(e^{0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}}}\right) \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2025066 
(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))))