Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.5%

Time: 5.6s

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: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

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

4. Applied rewrites 99.5%

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

6. Applied rewrites 99.5%

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

8. Applied rewrites 99.5%

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

10. Applied rewrites 99.4%

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

Alternative 2: 99.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

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

6. Applied rewrites 99.5%

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

8. Applied rewrites 99.5%

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

10. Applied rewrites 99.5%

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

Alternative 3: 99.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

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

6. Applied rewrites 99.5%

   \[\leadsto \color{blue}{0.125 \cdot \frac{\frac{e^{-\frac{r}{s}} + e^{\frac{r \cdot -0.3333333333333333}{s}}}{r \cdot \pi}}{s}} \]
7. Add Preprocessing

Alternative 4: 99.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

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

4. Applied rewrites 99.5%

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

6. Applied rewrites 99.5%

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

8. Applied rewrites 99.5%

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

10. Applied rewrites 99.5%

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

Alternative 5: 99.4% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

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

4. Applied rewrites 99.5%

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

6. Applied rewrites 99.5%

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

8. Applied rewrites 99.5%

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

9. Taylor expanded in s around 0

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

11. Applied rewrites 99.5%

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

Alternative 6: 10.3% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

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

5. Taylor expanded in s around -inf

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

7. Applied rewrites 10.3%

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

9. Applied rewrites 10.3%

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

Alternative 7: 10.3% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

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

5. Taylor expanded in s around -inf

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

7. Applied rewrites 10.3%

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

9. Applied rewrites 10.3%

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

Alternative 8: 10.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

float code(float s, float r) {
	return ((fmaf((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(fma(Float32(r / Float32(pi)), Float32(Float32(0.06944444444444445) / s), Float32(Float32(-0.16666666666666666) / Float32(pi))) / s) - Float32(Float32(-0.25) / Float32(r * Float32(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 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%

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

5. Taylor expanded in s around -inf

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

7. Applied rewrites 10.3%

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

9. Applied rewrites 10.3%

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

Alternative 9: 9.3% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

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

4. Applied rewrites 9.3%

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

6. Applied rewrites 9.3%

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

Alternative 10: 9.3% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \frac{1}{s} \cdot \left(\frac{0.25}{r \cdot \pi} - \frac{0.16666666666666666}{s \cdot \pi}\right) \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (* (/ 1.0 s) (- (/ 0.25 (* r PI)) (/ 0.16666666666666666 (* s PI)))))

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = (single(1.0) / s) * ((single(0.25) / (r * single(pi))) - (single(0.16666666666666666) / (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

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

4. Applied rewrites 9.3%

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

6. Applied rewrites 9.3%

   \[\leadsto \frac{1}{s} \cdot \color{blue}{\left(\frac{0.25}{r \cdot \pi} - \frac{0.16666666666666666}{s \cdot \pi}\right)} \]
7. Add Preprocessing

Alternative 11: 9.3% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

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

4. Applied rewrites 9.3%

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

6. Applied rewrites 9.3%

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

Alternative 12: 9.2% accurate, 6.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = 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

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

4. Applied rewrites 9.2%

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

6. Applied rewrites 9.2%

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

Alternative 13: 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 inf

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

4. Applied rewrites 9.2%

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

Reproduce

herbie shell --seed 2025161 
(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))))