Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.6%

Time: 9.9s

Alternatives: 13

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, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.125) * exp(Float32(-Float32(r / s)))) / Float32(Float32(r * s) * Float32(pi))) + 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(0.125) * exp(-(r / s))) / ((r * s) * single(pi))) + ((single(0.75) * exp((-r / (single(3.0) * s)))) / (((single(6.0) * single(pi)) * s) * 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. Taylor expanded in s around 0

   \[\leadsto \color{blue}{\frac{1}{8} \cdot \frac{e^{-1 \cdot \frac{r}{s}}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}} + \frac{\frac{3}{4} \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

4. Applied rewrites 99.6%

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

Alternative 2: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.125) * exp(-(r / s))) / ((r * s) * single(pi))) + ((single(0.75) * exp((-r / (single(3.0) * s)))) / ((single(6.0) * (s * single(pi))) * 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. Taylor expanded in s around 0

   \[\leadsto \color{blue}{\frac{1}{8} \cdot \frac{e^{-1 \cdot \frac{r}{s}}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}} + \frac{\frac{3}{4} \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

4. Applied rewrites 99.6%

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

5. Taylor expanded in s around 0

   \[\leadsto \frac{\frac{1}{8} \cdot e^{-\frac{r}{s}}}{\left(r \cdot s\right) \cdot \pi} + \frac{\frac{3}{4} \cdot e^{\frac{-r}{3 \cdot s}}}{\color{blue}{\left(6 \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot r} \]

7. Applied rewrites 99.5%

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

Alternative 3: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.125 \cdot e^{-\frac{r}{s}}}{s \cdot \pi}}{r} + \frac{0.75 \cdot e^{\frac{-0.3333333333333333 \cdot r}{s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/ (/ (* 0.125 (exp (- (/ r s)))) (* s PI)) r)
  (/ (* 0.75 (exp (/ (* -0.3333333333333333 r) s))) (* (* (* 6.0 PI) s) r))))

C

float code(float s, float r) {
	return (((0.125f * expf(-(r / s))) / (s * ((float) M_PI))) / 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(Float32(0.125) * exp(Float32(-Float32(r / s)))) / Float32(s * Float32(pi))) / r) + Float32(Float32(Float32(0.75) * exp(Float32(Float32(Float32(-0.3333333333333333) * r) / s))) / Float32(Float32(Float32(Float32(6.0) * Float32(pi)) * s) * r)))
end

MATLAB

function tmp = code(s, r)
	tmp = (((single(0.125) * exp(-(r / s))) / (s * single(pi))) / r) + ((single(0.75) * exp(((single(-0.3333333333333333) * r) / s))) / (((single(6.0) * single(pi)) * s) * r));
end

Derivation

1. Initial program 99.6%

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

3. Applied rewrites 99.6%

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

4. Taylor expanded in s around 0

   \[\leadsto \frac{\color{blue}{\frac{1}{8} \cdot \frac{e^{\mathsf{neg}\left(\frac{r}{s}\right)}}{s \cdot \mathsf{PI}\left(\right)}}}{r} + \frac{\frac{3}{4} \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

6. Applied rewrites 99.6%

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

7. Taylor expanded in s around 0

   \[\leadsto \frac{\frac{\frac{1}{8} \cdot e^{-\frac{r}{s}}}{s \cdot \pi}}{r} + \frac{\frac{3}{4} \cdot e^{\color{blue}{\frac{-1}{3} \cdot \frac{r}{s}}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

9. Applied rewrites 99.5%

   \[\leadsto \frac{\frac{0.125 \cdot e^{-\frac{r}{s}}}{s \cdot \pi}}{r} + \frac{0.75 \cdot e^{\color{blue}{\frac{-0.3333333333333333 \cdot r}{s}}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
10. Add Preprocessing

Alternative 4: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in s around 0

   \[\leadsto \color{blue}{\frac{1}{8} \cdot \frac{e^{-1 \cdot \frac{r}{s}}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}} + \frac{\frac{3}{4} \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

4. Applied rewrites 99.6%

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

5. Taylor expanded in s around 0

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

7. Applied rewrites 99.5%

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

Alternative 5: 99.5% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in s around 0

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

4. Applied rewrites 99.5%

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

Alternative 6: 10.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ -\left(\frac{\frac{\frac{\frac{0.06944444444444445 \cdot r}{\pi}}{s} - \frac{0.16666666666666666}{\pi}}{-s}}{s} - \frac{\frac{0.25}{\pi \cdot r}}{s}\right) \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in s around -inf

   \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{\frac{-1}{16} \cdot \frac{r}{\mathsf{PI}\left(\right)} + \frac{-1}{144} \cdot \frac{r}{\mathsf{PI}\left(\right)}}{s} - \frac{1}{6} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{s} - \frac{1}{4} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s}} \]

4. Applied rewrites 10.3%

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

6. Applied rewrites 10.3%

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

8. Applied rewrites 10.3%

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

9. Taylor expanded in r around 0

   \[\leadsto -\left(\frac{\frac{\frac{\frac{5}{72} \cdot \frac{r}{\mathsf{PI}\left(\right)}}{s} - \frac{\frac{1}{6}}{\pi}}{-s}}{s} - \frac{\frac{\frac{1}{4}}{\pi \cdot r}}{s}\right) \]

11. Applied rewrites 10.3%

    \[\leadsto -\left(\frac{\frac{\frac{\frac{0.06944444444444445 \cdot r}{\pi}}{s} - \frac{0.16666666666666666}{\pi}}{-s}}{s} - \frac{\frac{0.25}{\pi \cdot r}}{s}\right) \]
12. Add Preprocessing

Alternative 7: 10.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ -\frac{\left(-\frac{\left(-\frac{\frac{r}{\pi} \cdot -0.06944444444444445}{s}\right) - \frac{0.16666666666666666}{\pi}}{s}\right) - \frac{\frac{0.25}{r}}{\pi}}{s} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in s around -inf

   \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{\frac{-1}{16} \cdot \frac{r}{\mathsf{PI}\left(\right)} + \frac{-1}{144} \cdot \frac{r}{\mathsf{PI}\left(\right)}}{s} - \frac{1}{6} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{s} - \frac{1}{4} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s}} \]

4. Applied rewrites 10.3%

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

6. Applied rewrites 10.3%

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

8. Applied rewrites 10.3%

   \[\leadsto -\frac{\left(-\frac{\left(-\frac{\frac{r}{\pi} \cdot -0.06944444444444445}{s}\right) - \frac{0.16666666666666666}{\pi}}{s}\right) - \frac{\frac{0.25}{r}}{\pi}}{s} \]
9. Add Preprocessing

Alternative 8: 10.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ -\frac{\left(-\frac{\frac{0.06944444444444445}{s} \cdot \frac{r}{\pi} - \frac{0.16666666666666666}{\pi}}{s}\right) - \frac{0.25}{\pi \cdot r}}{s} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in s around -inf

   \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{\frac{-1}{16} \cdot \frac{r}{\mathsf{PI}\left(\right)} + \frac{-1}{144} \cdot \frac{r}{\mathsf{PI}\left(\right)}}{s} - \frac{1}{6} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{s} - \frac{1}{4} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s}} \]

4. Applied rewrites 10.3%

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

6. Applied rewrites 10.3%

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

7. Taylor expanded in s around 0

   \[\leadsto -\frac{\left(-\frac{\frac{5}{72} \cdot \frac{r}{s \cdot \mathsf{PI}\left(\right)} - \frac{\frac{1}{6}}{\pi}}{s}\right) - \frac{\frac{1}{4}}{\pi \cdot r}}{s} \]

9. Applied rewrites 10.3%

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

11. Applied rewrites 10.3%

    \[\leadsto -\frac{\left(-\frac{\frac{0.06944444444444445}{s} \cdot \frac{r}{\pi} - \frac{0.16666666666666666}{\pi}}{s}\right) - \frac{0.25}{\pi \cdot r}}{s} \]
12. Add Preprocessing

Alternative 9: 10.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ -\frac{\left(-\frac{\frac{0.06944444444444445 \cdot r}{s \cdot \pi} - \frac{0.16666666666666666}{\pi}}{s}\right) - \frac{0.25}{\pi \cdot r}}{s} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in s around -inf

   \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{\frac{-1}{16} \cdot \frac{r}{\mathsf{PI}\left(\right)} + \frac{-1}{144} \cdot \frac{r}{\mathsf{PI}\left(\right)}}{s} - \frac{1}{6} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{s} - \frac{1}{4} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s}} \]

4. Applied rewrites 10.3%

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

6. Applied rewrites 10.3%

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

7. Taylor expanded in s around 0

   \[\leadsto -\frac{\left(-\frac{\frac{5}{72} \cdot \frac{r}{s \cdot \mathsf{PI}\left(\right)} - \frac{\frac{1}{6}}{\pi}}{s}\right) - \frac{\frac{1}{4}}{\pi \cdot r}}{s} \]

9. Applied rewrites 10.3%

   \[\leadsto -\frac{\left(-\frac{\frac{0.06944444444444445 \cdot r}{s \cdot \pi} - \frac{0.16666666666666666}{\pi}}{s}\right) - \frac{0.25}{\pi \cdot r}}{s} \]
10. Add Preprocessing

Alternative 10: 9.4% accurate, 3.3× speedup

Math

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

FPCore

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

C

float code(float s, float r) {
	return ((0.25f / (r * ((float) M_PI))) - (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(0.16666666666666666) / Float32(Float32(pi) * s))) / s)
end

MATLAB

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

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in s around inf

   \[\leadsto \color{blue}{\frac{\left(\frac{1}{144} \cdot \frac{r}{{s}^{2} \cdot \mathsf{PI}\left(\right)} + \left(\frac{1}{16} \cdot \frac{r}{{s}^{2} \cdot \mathsf{PI}\left(\right)} + \frac{1}{4} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}\right)\right) - \frac{\frac{1}{6}}{s \cdot \mathsf{PI}\left(\right)}}{s}} \]

4. Applied rewrites 10.3%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.006944444444444444, \frac{r}{\pi \cdot \left(s \cdot s\right)}, \mathsf{fma}\left(\frac{r}{\pi \cdot \left(s \cdot s\right)}, 0.0625, \frac{0.25 \cdot 1}{\pi \cdot r}\right)\right) - \frac{0.16666666666666666}{\pi \cdot s}}{s}} \]

5. Taylor expanded in s around inf

   \[\leadsto \frac{\frac{\frac{1}{4}}{r \cdot \mathsf{PI}\left(\right)} - \frac{\frac{1}{6}}{\pi \cdot s}}{s} \]

7. Applied rewrites 9.4%

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

Alternative 11: 9.2% accurate, 6.0× speedup

Math

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

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(0.25) / 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%

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

2. Taylor expanded in s around 0

   \[\leadsto \color{blue}{\frac{1}{8} \cdot \frac{e^{-1 \cdot \frac{r}{s}}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}} + \frac{\frac{3}{4} \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

4. Applied rewrites 99.6%

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

5. Taylor expanded in s around inf

   \[\leadsto \color{blue}{\frac{\frac{1}{4}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}} \]

7. Applied rewrites 9.2%

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

9. Applied rewrites 9.2%

   \[\leadsto \frac{\frac{0.25}{r}}{\color{blue}{s \cdot \pi}} \]
10. Add Preprocessing

Alternative 12: 9.2% accurate, 6.4× 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%

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

2. Taylor expanded in s around 0

   \[\leadsto \color{blue}{\frac{1}{8} \cdot \frac{e^{-1 \cdot \frac{r}{s}}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}} + \frac{\frac{3}{4} \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

4. Applied rewrites 99.6%

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

5. Taylor expanded in s around inf

   \[\leadsto \color{blue}{\frac{\frac{1}{4}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}} \]

7. Applied rewrites 9.2%

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

Alternative 13: 9.2% accurate, 6.4× speedup

Math

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

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(0.25) / Float32(Float32(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%

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

2. Taylor expanded in s around inf

   \[\leadsto \color{blue}{\frac{\frac{1}{4}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}} \]

4. Applied rewrites 9.2%

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

Reproduce

herbie shell --seed 2025132 
(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))))