Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.6%

Time: 3.7s

Alternatives: 6

Speedup: 1.1×

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, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

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

4. Taylor expanded in s around 0

   \[\leadsto \mathsf{fma}\left(\frac{3}{4}, \frac{e^{\frac{-r}{3 \cdot s}}}{\left(\left(\pi \cdot 6\right) \cdot s\right) \cdot r}, \color{blue}{\frac{1}{8} \cdot \frac{e^{-1 \cdot \frac{r}{s}}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}}\right) \]
5. Step-by-step derivation

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. mul-1-neg N/A

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

   4. distribute-frac-neg N/A

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

   5. lower-/.f32 N/A

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

   6. lift-neg.f32 N/A

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

   7. lift-/.f32 N/A

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

   8. lift-exp.f32 N/A

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

   9. *-commutative N/A

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

   10. *-commutative N/A

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

   11. lift-*.f32 N/A

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

   12. lift-PI.f32 N/A

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

   13. lift-*.f32 99.6

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

6. Applied rewrites 99.6%

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

Alternative 2: 99.5% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

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

   1. lower-/.f32 N/A

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

4. Applied rewrites 99.5%

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

Alternative 3: 99.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

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

   1. lower-/.f32 N/A

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

4. Applied rewrites 99.5%

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

6. Applied rewrites 99.5%

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

Alternative 4: 38.7% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = single(0.25) / log((exp(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

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

   1. lower-/.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lift-PI.f32 8.9

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

4. Applied rewrites 8.9%

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

   1. lift-*.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. add-log-exp N/A

      \[\leadsto \frac{\frac{1}{4}}{\left(r \cdot s\right) \cdot \log \left(e^{\mathsf{PI}\left(\right)}\right)} \]

   8. log-pow-rev N/A

      \[\leadsto \frac{\frac{1}{4}}{\log \left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{\left(r \cdot s\right)}\right)} \]

   9. lower-log.f32 N/A

      \[\leadsto \frac{\frac{1}{4}}{\log \left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{\left(r \cdot s\right)}\right)} \]

   10. lower-pow.f32 N/A

       \[\leadsto \frac{\frac{1}{4}}{\log \left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{\left(r \cdot s\right)}\right)} \]

   11. lower-exp.f32 N/A

       \[\leadsto \frac{\frac{1}{4}}{\log \left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{\left(r \cdot s\right)}\right)} \]

   12. lift-PI.f32 N/A

       \[\leadsto \frac{\frac{1}{4}}{\log \left({\left(e^{\pi}\right)}^{\left(r \cdot s\right)}\right)} \]

   13. *-commutative N/A

       \[\leadsto \frac{\frac{1}{4}}{\log \left({\left(e^{\pi}\right)}^{\left(s \cdot r\right)}\right)} \]

   14. lower-*.f32 9.9

       \[\leadsto \frac{0.25}{\log \left({\left(e^{\pi}\right)}^{\left(s \cdot r\right)}\right)} \]

6. Applied rewrites 9.9%

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

   1. lift-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4}}{\log \left({\left(e^{\pi}\right)}^{\left(s \cdot r\right)}\right)} \]

   2. lift-pow.f32 N/A

      \[\leadsto \frac{\frac{1}{4}}{\log \left({\left(e^{\pi}\right)}^{\left(s \cdot r\right)}\right)} \]

   3. lift-PI.f32 N/A

      \[\leadsto \frac{\frac{1}{4}}{\log \left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{\left(s \cdot r\right)}\right)} \]

   4. lift-exp.f32 N/A

      \[\leadsto \frac{\frac{1}{4}}{\log \left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{\left(s \cdot r\right)}\right)} \]

   5. pow-exp N/A

      \[\leadsto \frac{\frac{1}{4}}{\log \left(e^{\mathsf{PI}\left(\right) \cdot \left(s \cdot r\right)}\right)} \]

   6. associate-*l* N/A

      \[\leadsto \frac{\frac{1}{4}}{\log \left(e^{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}\right)} \]

   7. *-commutative N/A

      \[\leadsto \frac{\frac{1}{4}}{\log \left(e^{\left(s \cdot \mathsf{PI}\left(\right)\right) \cdot r}\right)} \]

   8. *-commutative N/A

      \[\leadsto \frac{\frac{1}{4}}{\log \left(e^{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}\right)} \]

   9. exp-prod N/A

      \[\leadsto \frac{\frac{1}{4}}{\log \left({\left(e^{r}\right)}^{\left(s \cdot \mathsf{PI}\left(\right)\right)}\right)} \]

   10. lower-pow.f32 N/A

       \[\leadsto \frac{\frac{1}{4}}{\log \left({\left(e^{r}\right)}^{\left(s \cdot \mathsf{PI}\left(\right)\right)}\right)} \]

   11. lower-exp.f32 N/A

       \[\leadsto \frac{\frac{1}{4}}{\log \left({\left(e^{r}\right)}^{\left(s \cdot \mathsf{PI}\left(\right)\right)}\right)} \]

   12. *-commutative N/A

       \[\leadsto \frac{\frac{1}{4}}{\log \left({\left(e^{r}\right)}^{\left(\mathsf{PI}\left(\right) \cdot s\right)}\right)} \]

   13. lift-*.f32 N/A

       \[\leadsto \frac{\frac{1}{4}}{\log \left({\left(e^{r}\right)}^{\left(\mathsf{PI}\left(\right) \cdot s\right)}\right)} \]

   14. lift-PI.f32 38.7

       \[\leadsto \frac{0.25}{\log \left({\left(e^{r}\right)}^{\left(\pi \cdot s\right)}\right)} \]

8. Applied rewrites 38.7%

   \[\leadsto \frac{0.25}{\log \left({\left(e^{r}\right)}^{\left(\pi \cdot s\right)}\right)} \]
9. Add Preprocessing

Alternative 5: 9.9% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = single(0.25) / log(exp(((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 inf

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

   1. lower-/.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lift-PI.f32 8.9

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

4. Applied rewrites 8.9%

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

   1. lift-*.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. add-log-exp N/A

      \[\leadsto \frac{\frac{1}{4}}{\left(r \cdot s\right) \cdot \log \left(e^{\mathsf{PI}\left(\right)}\right)} \]

   8. log-pow-rev N/A

      \[\leadsto \frac{\frac{1}{4}}{\log \left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{\left(r \cdot s\right)}\right)} \]

   9. lower-log.f32 N/A

      \[\leadsto \frac{\frac{1}{4}}{\log \left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{\left(r \cdot s\right)}\right)} \]

   10. lower-pow.f32 N/A

       \[\leadsto \frac{\frac{1}{4}}{\log \left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{\left(r \cdot s\right)}\right)} \]

   11. lower-exp.f32 N/A

       \[\leadsto \frac{\frac{1}{4}}{\log \left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{\left(r \cdot s\right)}\right)} \]

   12. lift-PI.f32 N/A

       \[\leadsto \frac{\frac{1}{4}}{\log \left({\left(e^{\pi}\right)}^{\left(r \cdot s\right)}\right)} \]

   13. *-commutative N/A

       \[\leadsto \frac{\frac{1}{4}}{\log \left({\left(e^{\pi}\right)}^{\left(s \cdot r\right)}\right)} \]

   14. lower-*.f32 9.9

       \[\leadsto \frac{0.25}{\log \left({\left(e^{\pi}\right)}^{\left(s \cdot r\right)}\right)} \]

6. Applied rewrites 9.9%

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

   1. lift-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4}}{\log \left({\left(e^{\pi}\right)}^{\left(s \cdot r\right)}\right)} \]

   2. lift-pow.f32 N/A

      \[\leadsto \frac{\frac{1}{4}}{\log \left({\left(e^{\pi}\right)}^{\left(s \cdot r\right)}\right)} \]

   3. lift-PI.f32 N/A

      \[\leadsto \frac{\frac{1}{4}}{\log \left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{\left(s \cdot r\right)}\right)} \]

   4. lift-exp.f32 N/A

      \[\leadsto \frac{\frac{1}{4}}{\log \left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{\left(s \cdot r\right)}\right)} \]

   5. pow-exp N/A

      \[\leadsto \frac{\frac{1}{4}}{\log \left(e^{\mathsf{PI}\left(\right) \cdot \left(s \cdot r\right)}\right)} \]

   6. associate-*l* N/A

      \[\leadsto \frac{\frac{1}{4}}{\log \left(e^{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r}\right)} \]

   7. *-commutative N/A

      \[\leadsto \frac{\frac{1}{4}}{\log \left(e^{\left(s \cdot \mathsf{PI}\left(\right)\right) \cdot r}\right)} \]

   8. *-commutative N/A

      \[\leadsto \frac{\frac{1}{4}}{\log \left(e^{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}\right)} \]

   9. lower-exp.f32 N/A

      \[\leadsto \frac{\frac{1}{4}}{\log \left(e^{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}\right)} \]

   10. *-commutative N/A

       \[\leadsto \frac{\frac{1}{4}}{\log \left(e^{r \cdot \left(\mathsf{PI}\left(\right) \cdot s\right)}\right)} \]

   11. lift-*.f32 N/A

       \[\leadsto \frac{\frac{1}{4}}{\log \left(e^{r \cdot \left(\mathsf{PI}\left(\right) \cdot s\right)}\right)} \]

   12. lift-PI.f32 N/A

       \[\leadsto \frac{\frac{1}{4}}{\log \left(e^{r \cdot \left(\pi \cdot s\right)}\right)} \]

   13. *-commutative N/A

       \[\leadsto \frac{\frac{1}{4}}{\log \left(e^{\left(\pi \cdot s\right) \cdot r}\right)} \]

   14. lift-*.f32 9.9

       \[\leadsto \frac{0.25}{\log \left(e^{\left(\pi \cdot s\right) \cdot r}\right)} \]

8. Applied rewrites 9.9%

   \[\leadsto \frac{0.25}{\log \left(e^{\left(\pi \cdot s\right) \cdot r}\right)} \]
9. Add Preprocessing

Alternative 6: 8.9% accurate, 6.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

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

   1. lower-/.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lift-PI.f32 8.9

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

4. Applied rewrites 8.9%

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

   1. lift-*.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. lower-*.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

   10. lift-PI.f32 8.9

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

6. Applied rewrites 8.9%

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

Reproduce

herbie shell --seed 2025128 
(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))))