Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.6%

Time: 18.3s

Alternatives: 7

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: 99.6% 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, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in r around 0 99.5%

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

   1. associate-*r/ 99.6%

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

   2. associate-*l/ 99.6%

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

   3. *-commutative 99.6%

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

5. Simplified 99.6%

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

   1. clear-num 99.6%

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

   2. div-inv 99.6%

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

   3. div-inv 99.6%

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

   4. associate-/r* 99.6%

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

   5. metadata-eval 99.6%

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

7. Applied egg-rr 99.6%

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

8. Taylor expanded in s around 0 99.6%

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

   1. *-commutative 99.6%

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

   2. associate-*r* 99.6%

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

   3. *-commutative 99.6%

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

   4. associate-*l* 99.6%

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

   5. associate-*l* 99.6%

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

   6. *-commutative 99.6%

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

   7. *-commutative 99.6%

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

   8. associate-*l* 99.6%

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

10. Simplified 99.6%

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

11. Final simplification 99.6%

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

Alternative 2: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in r around 0 99.5%

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

   1. *-commutative 99.5%

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

   2. associate-/r/ 99.6%

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

5. Simplified 99.6%

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

6. Taylor expanded in s around 0 99.6%

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

   1. *-commutative 99.6%

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

   2. associate-*r* 99.6%

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

   3. *-commutative 99.6%

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

   4. associate-*l* 99.6%

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

   5. associate-*l* 99.6%

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

   6. *-commutative 99.6%

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

   7. *-commutative 99.6%

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

   8. associate-*l* 99.6%

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

8. Simplified 99.6%

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

9. Final simplification 99.6%

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

Alternative 3: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

   1. times-frac 99.6%

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

   2. fma-def 99.6%

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

   3. associate-*l* 99.6%

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

   4. associate-/r* 99.6%

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

   5. metadata-eval 99.6%

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

   6. metadata-eval 99.6%

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

   7. associate-/r* 99.6%

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

   8. associate-*l* 99.6%

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

   9. /-rgt-identity 99.6%

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

   10. fma-def 99.6%

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

3. Simplified 99.5%

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

5. Taylor expanded in s around 0 99.5%

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

6. Final simplification 99.5%

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

Alternative 4: 12.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{0.25}{\mathsf{log1p}\left(\mathsf{expm1}\left(r \cdot \left(s \cdot \pi\right)\right)\right)} \end{array} \]

FPCore

(FPCore (s r) :precision binary32 (/ 0.25 (log1p (expm1 (* r (* s PI))))))

C

float code(float s, float r) {
	return 0.25f / log1pf(expm1f((r * (s * ((float) M_PI)))));
}

Julia

function code(s, r)
	return Float32(Float32(0.25) / log1p(expm1(Float32(r * Float32(s * Float32(pi))))))
end

Derivation

1. Initial program 99.6%

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

3. Simplified 99.5%

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

5. Taylor expanded in r around 0 8.7%

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

6. Taylor expanded in s around inf 8.3%

   \[\leadsto \color{blue}{\frac{0.25}{r \cdot \left(s \cdot \pi\right)}} \]
7. Step-by-step derivation

   1. log1p-expm1-u 12.0%

      \[\leadsto \frac{0.25}{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(r \cdot \left(s \cdot \pi\right)\right)\right)}} \]

8. Applied egg-rr 12.0%

   \[\leadsto \frac{0.25}{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(r \cdot \left(s \cdot \pi\right)\right)\right)}} \]

9. Final simplification 12.0%

   \[\leadsto \frac{0.25}{\mathsf{log1p}\left(\mathsf{expm1}\left(r \cdot \left(s \cdot \pi\right)\right)\right)} \]
10. Add Preprocessing

Alternative 5: 9.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.125}{\pi} + \frac{0.125}{\pi \cdot e^{\frac{r}{s}}}}{r \cdot s} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Simplified 99.5%

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

5. Taylor expanded in r around 0 8.7%

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

6. Taylor expanded in r around -inf 8.7%

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

   1. mul-1-neg 8.7%

      \[\leadsto \color{blue}{-\frac{-0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}}}{s \cdot \pi} - 0.125 \cdot \frac{1}{s \cdot \pi}}{r}} \]

   2. *-commutative 8.7%

      \[\leadsto -\frac{\color{blue}{\frac{e^{-1 \cdot \frac{r}{s}}}{s \cdot \pi} \cdot -0.125} - 0.125 \cdot \frac{1}{s \cdot \pi}}{r} \]

   3. mul-1-neg 8.7%

      \[\leadsto -\frac{\frac{e^{\color{blue}{-\frac{r}{s}}}}{s \cdot \pi} \cdot -0.125 - 0.125 \cdot \frac{1}{s \cdot \pi}}{r} \]

   4. rec-exp 8.7%

      \[\leadsto -\frac{\frac{\color{blue}{\frac{1}{e^{\frac{r}{s}}}}}{s \cdot \pi} \cdot -0.125 - 0.125 \cdot \frac{1}{s \cdot \pi}}{r} \]

   5. reciprocal-define 8.6%

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

   6. associate-*r/ 8.6%

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

   7. metadata-eval 8.6%

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

8. Simplified 8.6%

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

9. Taylor expanded in s around 0 8.7%

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

    1. mul-1-neg 8.7%

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

    2. associate-*r/ 8.7%

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

    3. metadata-eval 8.7%

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

    4. associate-*r/ 8.7%

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

    5. metadata-eval 8.7%

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

    6. *-commutative 8.7%

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

11. Simplified 8.7%

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

12. Final simplification 8.7%

    \[\leadsto \frac{\frac{0.125}{\pi} + \frac{0.125}{\pi \cdot e^{\frac{r}{s}}}}{r \cdot s} \]
13. Add Preprocessing

Alternative 6: 9.0% 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 99.6%

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

3. Simplified 99.5%

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

5. Taylor expanded in r around 0 8.7%

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

6. Taylor expanded in s around inf 8.3%

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

7. Final simplification 8.3%

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

Alternative 7: 9.0% accurate, 33.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\frac{0.25}{r}}{s}}{\pi} \end{array} \]

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(Float32(Float32(0.25) / r) / s) / Float32(pi))
end

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Simplified 99.5%

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

5. Taylor expanded in r around 0 8.7%

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

6. Taylor expanded in s around inf 8.3%

   \[\leadsto \color{blue}{\frac{0.25}{r \cdot \left(s \cdot \pi\right)}} \]
7. Step-by-step derivation

   1. associate-*r* 8.3%

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

8. Simplified 8.3%

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

9. Taylor expanded in r around 0 8.3%

   \[\leadsto \color{blue}{\frac{0.25}{r \cdot \left(s \cdot \pi\right)}} \]
10. Step-by-step derivation

    1. associate-/r* 8.3%

       \[\leadsto \color{blue}{\frac{\frac{0.25}{r}}{s \cdot \pi}} \]

    2. associate-/r* 8.3%

       \[\leadsto \color{blue}{\frac{\frac{\frac{0.25}{r}}{s}}{\pi}} \]

11. Simplified 8.3%

    \[\leadsto \color{blue}{\frac{\frac{\frac{0.25}{r}}{s}}{\pi}} \]

12. Final simplification 8.3%

    \[\leadsto \frac{\frac{\frac{0.25}{r}}{s}}{\pi} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024024 
(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))))