Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.6%

Time: 11.9s

Alternatives: 10

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

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = ((exp((r / (single(-3.0) * s))) * single(0.75)) / ((single(6.0) * r) * (single(pi) * s))) + ((exp((-r / s)) * single(0.125)) / ((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. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. associate-*l* N/A

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

   6. associate-*r* N/A

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

   7. lower-*.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. lower-*.f32 99.6

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

4. Applied rewrites 99.6%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. times-frac N/A

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

   5. *-commutative N/A

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

   6. lift-*.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. associate-*l* N/A

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

   9. lift-*.f32 N/A

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

   10. associate-/r* N/A

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

   11. metadata-eval N/A

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

   12. times-frac N/A

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

   13. *-commutative N/A

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

   14. lift-*.f32 N/A

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

   15. lower-/.f32 N/A

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

6. Applied rewrites 99.6%

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

   1. lift-/.f32 N/A

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

   2. frac-2neg N/A

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

   3. lift-neg.f32 N/A

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

   4. remove-double-neg N/A

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

   5. lower-/.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. distribute-lft-neg-in N/A

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

   8. lower-*.f32 N/A

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

   9. metadata-eval 99.6

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

8. Applied rewrites 99.6%

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

9. Final simplification 99.6%

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

Alternative 2: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = ((exp(((r / s) * single(-0.3333333333333333))) / r) + (exp((-r / s)) / r)) * (single(0.125) / (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. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f32 N/A

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

4. Applied rewrites 99.6%

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

5. Final simplification 99.6%

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

Alternative 3: 10.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.125}{\pi \cdot r} + \mathsf{fma}\left(0.006944444444444444, \frac{r}{\left(s \cdot s\right) \cdot \pi}, \frac{-0.041666666666666664}{\pi \cdot s}\right)}{s} + \frac{e^{\frac{-r}{s}} \cdot 0.125}{\left(\pi \cdot s\right) \cdot r} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/
   (+
    (/ 0.125 (* PI r))
    (fma
     0.006944444444444444
     (/ r (* (* s s) PI))
     (/ -0.041666666666666664 (* PI s))))
   s)
  (/ (* (exp (/ (- r) s)) 0.125) (* (* PI s) r))))

C

float code(float s, float r) {
	return (((0.125f / (((float) M_PI) * r)) + fmaf(0.006944444444444444f, (r / ((s * s) * ((float) M_PI))), (-0.041666666666666664f / (((float) M_PI) * s)))) / s) + ((expf((-r / s)) * 0.125f) / ((((float) M_PI) * s) * r));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(Float32(0.125) / Float32(Float32(pi) * r)) + fma(Float32(0.006944444444444444), Float32(r / Float32(Float32(s * s) * Float32(pi))), Float32(Float32(-0.041666666666666664) / Float32(Float32(pi) * s)))) / s) + Float32(Float32(exp(Float32(Float32(-r) / s)) * Float32(0.125)) / Float32(Float32(Float32(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. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. associate-*l* N/A

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

   6. associate-*r* N/A

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

   7. lower-*.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. lower-*.f32 99.6

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

4. Applied rewrites 99.6%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. times-frac N/A

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

   5. *-commutative N/A

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

   6. lift-*.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. associate-*l* N/A

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

   9. lift-*.f32 N/A

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

   10. associate-/r* N/A

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

   11. metadata-eval N/A

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

   12. times-frac N/A

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

   13. *-commutative N/A

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

   14. lift-*.f32 N/A

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

   15. lower-/.f32 N/A

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

6. Applied rewrites 99.6%

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

7. Taylor expanded in s around -inf

   \[\leadsto \frac{\frac{1}{8} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} + \color{blue}{-1 \cdot \frac{-1 \cdot \frac{\frac{1}{144} \cdot \frac{r}{s \cdot \mathsf{PI}\left(\right)} - \frac{1}{24} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{s} - \frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s}} \]
8. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \frac{\frac{1}{8} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} + \color{blue}{\frac{-1 \cdot \left(-1 \cdot \frac{\frac{1}{144} \cdot \frac{r}{s \cdot \mathsf{PI}\left(\right)} - \frac{1}{24} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{s} - \frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}\right)}{s}} \]

   2. mul-1-neg N/A

      \[\leadsto \frac{\frac{1}{8} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} + \frac{\color{blue}{\mathsf{neg}\left(\left(-1 \cdot \frac{\frac{1}{144} \cdot \frac{r}{s \cdot \mathsf{PI}\left(\right)} - \frac{1}{24} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{s} - \frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}\right)\right)}}{s} \]

   3. sub-neg N/A

      \[\leadsto \frac{\frac{1}{8} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} + \frac{\mathsf{neg}\left(\color{blue}{\left(-1 \cdot \frac{\frac{1}{144} \cdot \frac{r}{s \cdot \mathsf{PI}\left(\right)} - \frac{1}{24} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{s} + \left(\mathsf{neg}\left(\frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}\right)\right)\right)}\right)}{s} \]

   4. mul-1-neg N/A

      \[\leadsto \frac{\frac{1}{8} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} + \frac{\mathsf{neg}\left(\left(\color{blue}{\left(\mathsf{neg}\left(\frac{\frac{1}{144} \cdot \frac{r}{s \cdot \mathsf{PI}\left(\right)} - \frac{1}{24} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{s}\right)\right)} + \left(\mathsf{neg}\left(\frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}\right)\right)\right)\right)}{s} \]

   5. distribute-neg-out N/A

      \[\leadsto \frac{\frac{1}{8} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} + \frac{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\left(\frac{\frac{1}{144} \cdot \frac{r}{s \cdot \mathsf{PI}\left(\right)} - \frac{1}{24} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{s} + \frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}\right)\right)\right)}\right)}{s} \]

   6. remove-double-neg N/A

      \[\leadsto \frac{\frac{1}{8} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} + \frac{\color{blue}{\frac{\frac{1}{144} \cdot \frac{r}{s \cdot \mathsf{PI}\left(\right)} - \frac{1}{24} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{s} + \frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}}{s} \]

   7. lower-/.f32 N/A

      \[\leadsto \frac{\frac{1}{8} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} + \color{blue}{\frac{\frac{\frac{1}{144} \cdot \frac{r}{s \cdot \mathsf{PI}\left(\right)} - \frac{1}{24} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{s} + \frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s}} \]

9. Applied rewrites 11.2%

   \[\leadsto \frac{0.125 \cdot e^{\frac{-r}{s}}}{\left(\pi \cdot s\right) \cdot r} + \color{blue}{\frac{\mathsf{fma}\left(0.006944444444444444, \frac{r}{\left(s \cdot s\right) \cdot \pi}, \frac{-0.041666666666666664}{\pi \cdot s}\right) + \frac{0.125}{\pi \cdot r}}{s}} \]

10. Final simplification 11.2%

    \[\leadsto \frac{\frac{0.125}{\pi \cdot r} + \mathsf{fma}\left(0.006944444444444444, \frac{r}{\left(s \cdot s\right) \cdot \pi}, \frac{-0.041666666666666664}{\pi \cdot s}\right)}{s} + \frac{e^{\frac{-r}{s}} \cdot 0.125}{\left(\pi \cdot s\right) \cdot r} \]
11. Add Preprocessing

Alternative 4: 10.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(0.006944444444444444, \frac{r}{\left(s \cdot s\right) \cdot \pi}, \frac{0.125}{\pi \cdot r}\right) - \frac{0.041666666666666664}{\pi \cdot s}}{s} + \frac{0.125}{\left(\pi \cdot s\right) \cdot r} \cdot e^{\frac{-r}{s}} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/
   (-
    (fma 0.006944444444444444 (/ r (* (* s s) PI)) (/ 0.125 (* PI r)))
    (/ 0.041666666666666664 (* PI s)))
   s)
  (* (/ 0.125 (* (* PI s) r)) (exp (/ (- r) s)))))

C

float code(float s, float r) {
	return ((fmaf(0.006944444444444444f, (r / ((s * s) * ((float) M_PI))), (0.125f / (((float) M_PI) * r))) - (0.041666666666666664f / (((float) M_PI) * s))) / s) + ((0.125f / ((((float) M_PI) * s) * r)) * expf((-r / s)));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(fma(Float32(0.006944444444444444), Float32(r / Float32(Float32(s * s) * Float32(pi))), Float32(Float32(0.125) / Float32(Float32(pi) * r))) - Float32(Float32(0.041666666666666664) / Float32(Float32(pi) * s))) / s) + Float32(Float32(Float32(0.125) / Float32(Float32(Float32(pi) * s) * r)) * exp(Float32(Float32(-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. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. lift-*.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. associate-*l* N/A

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

   8. lift-*.f32 N/A

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

   9. associate-*l* N/A

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

   10. associate-/r* N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. associate-/r* N/A

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

   14. associate-*l* N/A

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

   15. lift-*.f32 N/A

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

   16. associate-*l* N/A

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

   17. lift-*.f32 N/A

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

   18. lift-*.f32 N/A

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

4. Applied rewrites 96.9%

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

5. Taylor expanded in s around inf

   \[\leadsto e^{\frac{\mathsf{neg}\left(r\right)}{s}} \cdot \frac{\frac{1}{8}}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} + \color{blue}{\frac{\left(\frac{1}{144} \cdot \frac{r}{{s}^{2} \cdot \mathsf{PI}\left(\right)} + \frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}\right) - \frac{\frac{1}{24}}{s \cdot \mathsf{PI}\left(\right)}}{s}} \]

7. Applied rewrites 11.1%

   \[\leadsto e^{\frac{-r}{s}} \cdot \frac{0.125}{\left(\pi \cdot s\right) \cdot r} + \color{blue}{\frac{\mathsf{fma}\left(0.006944444444444444, \frac{r}{\left(s \cdot s\right) \cdot \pi}, \frac{0.125}{\pi \cdot r}\right) - \frac{0.041666666666666664}{\pi \cdot s}}{s}} \]

8. Final simplification 11.1%

   \[\leadsto \frac{\mathsf{fma}\left(0.006944444444444444, \frac{r}{\left(s \cdot s\right) \cdot \pi}, \frac{0.125}{\pi \cdot r}\right) - \frac{0.041666666666666664}{\pi \cdot s}}{s} + \frac{0.125}{\left(\pi \cdot s\right) \cdot r} \cdot e^{\frac{-r}{s}} \]
9. Add Preprocessing

Alternative 5: 10.1% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Julia

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

3. Taylor expanded in s around inf

   \[\leadsto \color{blue}{\frac{\left(\frac{-1}{48} \cdot \frac{{r}^{2}}{{s}^{3} \cdot \mathsf{PI}\left(\right)} + \left(\frac{-1}{1296} \cdot \frac{{r}^{2}}{{s}^{3} \cdot \mathsf{PI}\left(\right)} + \left(\frac{1}{144} \cdot \frac{r}{{s}^{2} \cdot \mathsf{PI}\left(\right)} + \left(\frac{1}{16} \cdot \frac{r}{{s}^{2} \cdot \mathsf{PI}\left(\right)} + \frac{1}{4} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}\right)\right)\right)\right) - \frac{\frac{1}{6}}{s \cdot \mathsf{PI}\left(\right)}}{s}} \]

5. Applied rewrites 8.8%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.021604938271604937, \frac{r}{\left(\left(s \cdot s\right) \cdot s\right) \cdot \pi} \cdot r, \mathsf{fma}\left(0.06944444444444445, \frac{r}{\left(s \cdot s\right) \cdot \pi}, \frac{0.25}{\pi \cdot r}\right)\right) - \frac{0.16666666666666666}{\pi \cdot s}}{s}} \]

6. Taylor expanded in r around 0

   \[\leadsto \frac{\frac{r \cdot \left(\frac{5}{72} \cdot \frac{r}{{s}^{2} \cdot \mathsf{PI}\left(\right)} - \frac{1}{6} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}\right) + \frac{1}{4} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{r}}{s} \]

8. Applied rewrites 10.6%

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

Alternative 6: 10.1% accurate, 4.0× speedup

Math

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

FPCore

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

C

float code(float s, float r) {
	return (fmaf(0.06944444444444445f, (r / ((s * 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(fma(Float32(0.06944444444444445), Float32(r / Float32(Float32(s * s) * Float32(pi))), Float32(Float32(-0.16666666666666666) / Float32(Float32(pi) * s))) - Float32(Float32(-0.25) / Float32(Float32(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. Add Preprocessing

3. Taylor expanded in s around inf

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

5. Applied rewrites 10.6%

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

Alternative 7: 9.1% accurate, 5.9× speedup

Math

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

FPCore

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

C

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

Julia

function code(s, r)
	return Float32(fma(Float32(s / Float32(Float32(pi) * r)), Float32(0.25), Float32(Float32(-0.16666666666666666) / Float32(pi))) / Float32(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. Add Preprocessing

3. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lower-PI.f32 9.1

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

5. Applied rewrites 9.1%

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

6. Taylor expanded in r around 0

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

   1. lower-/.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

   4. lower-/.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f32 N/A

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

   8. lower-PI.f32 N/A

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

   9. associate-*r/ N/A

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

   10. metadata-eval N/A

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

   11. lower-/.f32 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f32 N/A

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

   14. lower-PI.f32 9.4

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

8. Applied rewrites 9.4%

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

9. Taylor expanded in s around 0

   \[\leadsto \frac{\frac{1}{4} \cdot \frac{s}{r \cdot \mathsf{PI}\left(\right)} - \frac{1}{6} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{\color{blue}{{s}^{2}}} \]
10. Step-by-step derivation

11. Applied rewrites 9.4%

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

Alternative 8: 9.1% accurate, 6.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.25) / (single(pi) * r)) - (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. Add Preprocessing

3. Taylor expanded in s around inf

   \[\leadsto \color{blue}{\frac{\left(\frac{-1}{48} \cdot \frac{{r}^{2}}{{s}^{3} \cdot \mathsf{PI}\left(\right)} + \left(\frac{-1}{1296} \cdot \frac{{r}^{2}}{{s}^{3} \cdot \mathsf{PI}\left(\right)} + \left(\frac{1}{144} \cdot \frac{r}{{s}^{2} \cdot \mathsf{PI}\left(\right)} + \left(\frac{1}{16} \cdot \frac{r}{{s}^{2} \cdot \mathsf{PI}\left(\right)} + \frac{1}{4} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}\right)\right)\right)\right) - \frac{\frac{1}{6}}{s \cdot \mathsf{PI}\left(\right)}}{s}} \]

5. Applied rewrites 8.8%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.021604938271604937, \frac{r}{\left(\left(s \cdot s\right) \cdot s\right) \cdot \pi} \cdot r, \mathsf{fma}\left(0.06944444444444445, \frac{r}{\left(s \cdot s\right) \cdot \pi}, \frac{0.25}{\pi \cdot r}\right)\right) - \frac{0.16666666666666666}{\pi \cdot s}}{s}} \]

6. Taylor expanded in s around inf

   \[\leadsto \frac{\frac{\frac{1}{4}}{r \cdot \mathsf{PI}\left(\right)} - \frac{\frac{1}{6}}{\mathsf{PI}\left(\right) \cdot s}}{s} \]
7. Step-by-step derivation

8. Applied rewrites 9.4%

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

Alternative 9: 9.0% accurate, 6.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

3. 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. Step-by-step derivation

   1. div-sub N/A

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

   2. sub-neg N/A

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

   3. associate-/l* N/A

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

   4. associate-/l/ N/A

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

   5. associate-*l* N/A

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

   6. unpow2 N/A

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

   7. +-commutative N/A

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

   8. remove-double-neg N/A

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

   9. sub-neg N/A

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

   10. lower--.f32 N/A

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

5. Applied rewrites 9.3%

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

Alternative 10: 9.1% accurate, 13.5× 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. Add Preprocessing

3. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lower-PI.f32 9.1

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

5. Applied rewrites 9.1%

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

7. Applied rewrites 9.1%

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

Reproduce

herbie shell --seed 2024240 
(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))))