Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.6%

Time: 14.8s

Alternatives: 17

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, 0.6× speedup

Math

\[\begin{array}{l} \\ \frac{e^{\frac{r}{-s}} \cdot 0.125}{\pi \cdot \left(r \cdot s\right)} + \frac{0.75 \cdot e^{r \cdot \frac{-0.3333333333333333}{s}}}{s \cdot \left({\left(\sqrt[3]{\pi}\right)}^{3} \cdot \left(r \cdot 6\right)\right)} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/ (* (exp (/ r (- s))) 0.125) (* PI (* r s)))
  (/
   (* 0.75 (exp (* r (/ -0.3333333333333333 s))))
   (* s (* (pow (cbrt PI) 3.0) (* r 6.0))))))

C

float code(float s, float r) {
	return ((expf((r / -s)) * 0.125f) / (((float) M_PI) * (r * s))) + ((0.75f * expf((r * (-0.3333333333333333f / s)))) / (s * (powf(cbrtf(((float) M_PI)), 3.0f) * (r * 6.0f))));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(exp(Float32(r / Float32(-s))) * Float32(0.125)) / Float32(Float32(pi) * Float32(r * s))) + Float32(Float32(Float32(0.75) * exp(Float32(r * Float32(Float32(-0.3333333333333333) / s)))) / Float32(s * Float32((cbrt(Float32(pi)) ^ Float32(3.0)) * Float32(r * Float32(6.0))))))
end

Derivation

1. Initial program 99.5%

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

   1. lift-/.f32 N/A

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

   2. frac-2neg N/A

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

   3. div-inv N/A

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

   4. lift-neg.f32 N/A

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

   5. remove-double-neg N/A

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

   6. lower-*.f32 N/A

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

   7. lift-*.f32 N/A

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

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

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

   9. associate-/r* N/A

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. lower-/.f32 N/A

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

   14. metadata-eval 99.5

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

4. Applied rewrites 99.5%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lift-*.f32 N/A

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

   5. *-commutative N/A

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

   6. times-frac N/A

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

   7. lift-*.f32 N/A

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

   8. lift-*.f32 N/A

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

   9. associate-*l* N/A

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

   10. *-commutative N/A

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

   11. lift-*.f32 N/A

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

   12. associate-/r* N/A

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

   13. metadata-eval N/A

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

   14. times-frac N/A

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

   15. lift-*.f32 N/A

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

   16. associate-*r* N/A

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

   17. lift-*.f32 N/A

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

   18. *-commutative N/A

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

   19. lift-*.f32 N/A

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

6. Applied rewrites 99.6%

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

7. Taylor expanded in s around 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-PI.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 99.6

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

9. Applied rewrites 99.6%

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

11. Applied rewrites 99.6%

    \[\leadsto \frac{e^{\frac{r}{-s}} \cdot 0.125}{\pi \cdot \left(r \cdot s\right)} + \frac{0.75 \cdot e^{r \cdot \frac{-0.3333333333333333}{s}}}{s \cdot \left({\left(\sqrt[3]{\pi}\right)}^{3} \cdot \left(\color{blue}{r} \cdot 6\right)\right)} \]
12. Add Preprocessing

Alternative 2: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

float code(float s, float r) {
	return ((expf((r / -s)) * 0.125f) / (((float) M_PI) * (r * s))) + ((0.75f * expf((r * (-0.3333333333333333f / s)))) / (s * (((float) M_PI) * (r * 6.0f))));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(exp(Float32(r / Float32(-s))) * Float32(0.125)) / Float32(Float32(pi) * Float32(r * s))) + Float32(Float32(Float32(0.75) * exp(Float32(r * Float32(Float32(-0.3333333333333333) / s)))) / Float32(s * Float32(Float32(pi) * Float32(r * Float32(6.0))))))
end

MATLAB

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

Derivation

1. Initial program 99.5%

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

   1. lift-/.f32 N/A

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

   2. frac-2neg N/A

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

   3. div-inv N/A

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

   4. lift-neg.f32 N/A

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

   5. remove-double-neg N/A

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

   6. lower-*.f32 N/A

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

   7. lift-*.f32 N/A

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

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

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

   9. associate-/r* N/A

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. lower-/.f32 N/A

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

   14. metadata-eval 99.5

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

4. Applied rewrites 99.5%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lift-*.f32 N/A

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

   5. *-commutative N/A

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

   6. times-frac N/A

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

   7. lift-*.f32 N/A

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

   8. lift-*.f32 N/A

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

   9. associate-*l* N/A

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

   10. *-commutative N/A

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

   11. lift-*.f32 N/A

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

   12. associate-/r* N/A

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

   13. metadata-eval N/A

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

   14. times-frac N/A

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

   15. lift-*.f32 N/A

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

   16. associate-*r* N/A

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

   17. lift-*.f32 N/A

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

   18. *-commutative N/A

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

   19. lift-*.f32 N/A

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

6. Applied rewrites 99.6%

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

7. Taylor expanded in s around 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-PI.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 99.6

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

9. Applied rewrites 99.6%

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

Alternative 3: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \left(r \cdot s\right)\\ \mathsf{fma}\left(0.16666666666666666, \frac{0.75 \cdot e^{\frac{r}{s \cdot -3}}}{t\_0}, \frac{e^{\frac{r}{-s}}}{t\_0 \cdot 8}\right) \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (let* ((t_0 (* PI (* r s))))
   (fma
    0.16666666666666666
    (/ (* 0.75 (exp (/ r (* s -3.0)))) t_0)
    (/ (exp (/ r (- s))) (* t_0 8.0)))))

C

float code(float s, float r) {
	float t_0 = ((float) M_PI) * (r * s);
	return fmaf(0.16666666666666666f, ((0.75f * expf((r / (s * -3.0f)))) / t_0), (expf((r / -s)) / (t_0 * 8.0f)));
}

Julia

function code(s, r)
	t_0 = Float32(Float32(pi) * Float32(r * s))
	return fma(Float32(0.16666666666666666), Float32(Float32(Float32(0.75) * exp(Float32(r / Float32(s * Float32(-3.0))))) / t_0), Float32(exp(Float32(r / Float32(-s))) / Float32(t_0 * Float32(8.0))))
end

Derivation

1. Initial program 99.5%

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

   1. lift-/.f32 N/A

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

   2. frac-2neg N/A

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

   3. div-inv N/A

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

   4. lift-neg.f32 N/A

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

   5. remove-double-neg N/A

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

   6. lower-*.f32 N/A

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

   7. lift-*.f32 N/A

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

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

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

   9. associate-/r* N/A

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. lower-/.f32 N/A

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

   14. metadata-eval 99.5

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

4. Applied rewrites 99.5%

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

6. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.16666666666666666, \frac{0.75 \cdot e^{\frac{r}{s \cdot -3}}}{\pi \cdot \left(r \cdot s\right)}, \frac{e^{\frac{r}{-s}}}{\left(\pi \cdot \left(r \cdot s\right)\right) \cdot 8}\right)} \]
7. Add Preprocessing

Alternative 4: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in r around 0

   \[\leadsto \color{blue}{\frac{\frac{1}{4}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}} \]
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. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 9.0

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

5. Applied rewrites 9.0%

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

6. Taylor expanded in s around 0

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

   1. lower-/.f32 N/A

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

8. Applied rewrites 99.5%

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

9. Final simplification 99.5%

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

Alternative 5: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in s around 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. Step-by-step derivation

   1. distribute-lft-out N/A

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

   2. *-commutative N/A

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

   3. associate-/l* N/A

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

   4. lower-*.f32 N/A

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

5. Applied rewrites 99.4%

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

6. Final simplification 99.4%

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

Alternative 6: 99.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

4. Applied rewrites 99.4%

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

Alternative 7: 10.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \frac{e^{\frac{r}{-s}} \cdot 0.25}{r \cdot \left(s \cdot \left(\pi \cdot 2\right)\right)} + \frac{\frac{0.125}{r \cdot \pi} + \mathsf{fma}\left(\frac{r}{s \cdot \left(s \cdot \pi\right)}, 0.006944444444444444, \frac{-0.041666666666666664}{s \cdot \pi}\right)}{s} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/ (* (exp (/ r (- s))) 0.25) (* r (* s (* PI 2.0))))
  (/
   (+
    (/ 0.125 (* r PI))
    (fma
     (/ r (* s (* s PI)))
     0.006944444444444444
     (/ -0.041666666666666664 (* s PI))))
   s)))

C

float code(float s, float r) {
	return ((expf((r / -s)) * 0.25f) / (r * (s * (((float) M_PI) * 2.0f)))) + (((0.125f / (r * ((float) M_PI))) + fmaf((r / (s * (s * ((float) M_PI)))), 0.006944444444444444f, (-0.041666666666666664f / (s * ((float) M_PI))))) / s);
}

Julia

function code(s, r)
	return Float32(Float32(Float32(exp(Float32(r / Float32(-s))) * Float32(0.25)) / Float32(r * Float32(s * Float32(Float32(pi) * Float32(2.0))))) + Float32(Float32(Float32(Float32(0.125) / Float32(r * Float32(pi))) + fma(Float32(r / Float32(s * Float32(s * Float32(pi)))), Float32(0.006944444444444444), Float32(Float32(-0.041666666666666664) / Float32(s * Float32(pi))))) / s))
end

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in r around 0

   \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \color{blue}{\frac{\frac{1}{8}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}} \]
4. Step-by-step derivation

   1. lower-/.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \color{blue}{\frac{\frac{1}{8}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}} \]

   2. lower-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{1}{8}}{\color{blue}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}} \]

   3. lower-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{1}{8}}{r \cdot \color{blue}{\left(s \cdot \mathsf{PI}\left(\right)\right)}} \]

   4. lower-PI.f32 9.4

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.125}{r \cdot \left(s \cdot \color{blue}{\pi}\right)} \]

5. Applied rewrites 9.4%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \color{blue}{\frac{0.125}{r \cdot \left(s \cdot \pi\right)}} \]

6. Taylor expanded in s around inf

   \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \color{blue}{\frac{\left(\frac{1}{144} \cdot \frac{r}{{s}^{2} \cdot \mathsf{PI}\left(\right)} + \frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}\right) - \frac{\frac{1}{24}}{s \cdot \mathsf{PI}\left(\right)}}{s}} \]

8. Applied rewrites 10.1%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \color{blue}{\frac{\frac{0.125}{r \cdot \pi} + \mathsf{fma}\left(\frac{r}{s \cdot \left(s \cdot \pi\right)}, 0.006944444444444444, \frac{-0.041666666666666664}{s \cdot \pi}\right)}{s}} \]

9. Final simplification 10.1%

   \[\leadsto \frac{e^{\frac{r}{-s}} \cdot 0.25}{r \cdot \left(s \cdot \left(\pi \cdot 2\right)\right)} + \frac{\frac{0.125}{r \cdot \pi} + \mathsf{fma}\left(\frac{r}{s \cdot \left(s \cdot \pi\right)}, 0.006944444444444444, \frac{-0.041666666666666664}{s \cdot \pi}\right)}{s} \]
10. Add Preprocessing

Alternative 8: 10.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \frac{e^{\frac{r}{-s}} \cdot 0.25}{r \cdot \left(s \cdot \left(\pi \cdot 2\right)\right)} + \frac{\frac{0.125}{r \cdot \pi} + \mathsf{fma}\left(r, \frac{0.006944444444444444}{s \cdot \left(s \cdot \pi\right)}, \frac{-0.041666666666666664}{s \cdot \pi}\right)}{s} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/ (* (exp (/ r (- s))) 0.25) (* r (* s (* PI 2.0))))
  (/
   (+
    (/ 0.125 (* r PI))
    (fma
     r
     (/ 0.006944444444444444 (* s (* s PI)))
     (/ -0.041666666666666664 (* s PI))))
   s)))

C

float code(float s, float r) {
	return ((expf((r / -s)) * 0.25f) / (r * (s * (((float) M_PI) * 2.0f)))) + (((0.125f / (r * ((float) M_PI))) + fmaf(r, (0.006944444444444444f / (s * (s * ((float) M_PI)))), (-0.041666666666666664f / (s * ((float) M_PI))))) / s);
}

Julia

function code(s, r)
	return Float32(Float32(Float32(exp(Float32(r / Float32(-s))) * Float32(0.25)) / Float32(r * Float32(s * Float32(Float32(pi) * Float32(2.0))))) + Float32(Float32(Float32(Float32(0.125) / Float32(r * Float32(pi))) + fma(r, Float32(Float32(0.006944444444444444) / Float32(s * Float32(s * Float32(pi)))), Float32(Float32(-0.041666666666666664) / Float32(s * Float32(pi))))) / s))
end

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in s around inf

   \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \color{blue}{\frac{\left(\frac{1}{144} \cdot \frac{r}{{s}^{2} \cdot \mathsf{PI}\left(\right)} + \frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}\right) - \frac{\frac{1}{24}}{s \cdot \mathsf{PI}\left(\right)}}{s}} \]

5. Applied rewrites 10.1%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \color{blue}{\frac{\frac{0.125}{r \cdot \pi} + \mathsf{fma}\left(r, \frac{0.006944444444444444}{s \cdot \left(s \cdot \pi\right)}, \frac{-0.041666666666666664}{s \cdot \pi}\right)}{s}} \]

6. Final simplification 10.1%

   \[\leadsto \frac{e^{\frac{r}{-s}} \cdot 0.25}{r \cdot \left(s \cdot \left(\pi \cdot 2\right)\right)} + \frac{\frac{0.125}{r \cdot \pi} + \mathsf{fma}\left(r, \frac{0.006944444444444444}{s \cdot \left(s \cdot \pi\right)}, \frac{-0.041666666666666664}{s \cdot \pi}\right)}{s} \]
7. Add Preprocessing

Alternative 9: 10.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

function code(s, r)
	return fma(Float32(Float32(0.125) / Float32(Float32(pi) * Float32(r * s))), exp(Float32(r / Float32(-s))), Float32(Float32(Float32(Float32(0.125) / Float32(r * Float32(pi))) + fma(Float32(r / Float32(s * Float32(s * Float32(pi)))), Float32(0.006944444444444444), Float32(Float32(-0.041666666666666664) / Float32(s * Float32(pi))))) / s))
end

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in s around -inf

   \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{\frac{-1}{144} \cdot \frac{r}{\mathsf{PI}\left(\right)} + \frac{1}{1296} \cdot \frac{{r}^{2}}{s \cdot \mathsf{PI}\left(\right)}}{s} - \frac{1}{24} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{s} - \frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s}} \]

5. Applied rewrites 9.8%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \color{blue}{\left(\frac{\frac{-0.041666666666666664}{\pi} - \frac{\mathsf{fma}\left(r, \frac{r}{s \cdot \pi} \cdot 0.0007716049382716049, \frac{r}{\pi} \cdot -0.006944444444444444\right)}{s}}{s \cdot s} + \frac{0.125}{r \cdot \left(s \cdot \pi\right)}\right)} \]

7. Applied rewrites 9.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0.125}{\pi \cdot \left(r \cdot s\right)}, e^{\frac{r}{-s}}, \frac{0.125}{\pi \cdot \left(r \cdot s\right)} + \frac{\mathsf{fma}\left(s, -0.041666666666666664, \left(-\pi\right) \cdot \mathsf{fma}\left(r, \frac{-0.006944444444444444}{\pi}, \frac{r \cdot \left(r \cdot 0.0007716049382716049\right)}{s \cdot \pi}\right)\right)}{\left(s \cdot s\right) \cdot \left(s \cdot \pi\right)}\right)} \]

8. Taylor expanded in s around inf

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

10. Applied rewrites 10.1%

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

Alternative 10: 10.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.125}{\pi \cdot \left(r \cdot s\right)}\\ \mathsf{fma}\left(t\_0, e^{\frac{r}{-s}}, t\_0 + \frac{\mathsf{fma}\left(s, -0.041666666666666664, r \cdot 0.006944444444444444\right)}{\left(s \cdot \pi\right) \cdot \left(s \cdot s\right)}\right) \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (let* ((t_0 (/ 0.125 (* PI (* r s)))))
   (fma
    t_0
    (exp (/ r (- s)))
    (+
     t_0
     (/
      (fma s -0.041666666666666664 (* r 0.006944444444444444))
      (* (* s PI) (* s s)))))))

C

float code(float s, float r) {
	float t_0 = 0.125f / (((float) M_PI) * (r * s));
	return fmaf(t_0, expf((r / -s)), (t_0 + (fmaf(s, -0.041666666666666664f, (r * 0.006944444444444444f)) / ((s * ((float) M_PI)) * (s * s)))));
}

Julia

function code(s, r)
	t_0 = Float32(Float32(0.125) / Float32(Float32(pi) * Float32(r * s)))
	return fma(t_0, exp(Float32(r / Float32(-s))), Float32(t_0 + Float32(fma(s, Float32(-0.041666666666666664), Float32(r * Float32(0.006944444444444444))) / Float32(Float32(s * Float32(pi)) * Float32(s * s)))))
end

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in s around -inf

   \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{\frac{-1}{144} \cdot \frac{r}{\mathsf{PI}\left(\right)} + \frac{1}{1296} \cdot \frac{{r}^{2}}{s \cdot \mathsf{PI}\left(\right)}}{s} - \frac{1}{24} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{s} - \frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s}} \]

5. Applied rewrites 9.8%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \color{blue}{\left(\frac{\frac{-0.041666666666666664}{\pi} - \frac{\mathsf{fma}\left(r, \frac{r}{s \cdot \pi} \cdot 0.0007716049382716049, \frac{r}{\pi} \cdot -0.006944444444444444\right)}{s}}{s \cdot s} + \frac{0.125}{r \cdot \left(s \cdot \pi\right)}\right)} \]

7. Applied rewrites 9.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0.125}{\pi \cdot \left(r \cdot s\right)}, e^{\frac{r}{-s}}, \frac{0.125}{\pi \cdot \left(r \cdot s\right)} + \frac{\mathsf{fma}\left(s, -0.041666666666666664, \left(-\pi\right) \cdot \mathsf{fma}\left(r, \frac{-0.006944444444444444}{\pi}, \frac{r \cdot \left(r \cdot 0.0007716049382716049\right)}{s \cdot \pi}\right)\right)}{\left(s \cdot s\right) \cdot \left(s \cdot \pi\right)}\right)} \]

8. Taylor expanded in r around 0

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

10. Applied rewrites 10.1%

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

11. Final simplification 10.1%

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

Alternative 11: 10.0% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.5%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
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 9.8%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.06944444444444445, \frac{r}{s \cdot \pi}, \frac{-0.16666666666666666}{\pi}\right)}{s \cdot s} + \frac{0.25}{r \cdot \left(s \cdot \pi\right)}} \]
6. Step-by-step derivation

7. Applied rewrites 9.8%

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

8. Final simplification 9.8%

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

Alternative 12: 10.1% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.5%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
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 9.8%

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

Alternative 13: 9.1% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \frac{1}{r} \cdot \frac{\frac{0.25}{s}}{\pi} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = (single(1.0) / r) * ((single(0.25) / s) / single(pi));
end

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in r around 0

   \[\leadsto \color{blue}{\frac{\frac{1}{4}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}} \]
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. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 9.0

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

5. Applied rewrites 9.0%

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

7. Applied rewrites 9.0%

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

9. Applied rewrites 9.0%

   \[\leadsto \frac{1}{r} \cdot \color{blue}{\frac{\frac{0.25}{s}}{\pi}} \]
10. Add Preprocessing

Alternative 14: 9.1% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{s \cdot \pi} \cdot \frac{0.25}{r} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = (single(1.0) / (s * single(pi))) * (single(0.25) / r);
end

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in r around 0

   \[\leadsto \color{blue}{\frac{\frac{1}{4}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}} \]
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. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 9.0

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

5. Applied rewrites 9.0%

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

7. Applied rewrites 9.0%

   \[\leadsto \frac{1}{s \cdot \pi} \cdot \color{blue}{\frac{0.25}{r}} \]
8. Add Preprocessing

Alternative 15: 9.1% accurate, 10.6× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in r around 0

   \[\leadsto \color{blue}{\frac{\frac{1}{4}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}} \]
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. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 9.0

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

5. Applied rewrites 9.0%

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

7. Applied rewrites 9.0%

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

Alternative 16: 9.1% accurate, 13.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in r around 0

   \[\leadsto \color{blue}{\frac{\frac{1}{4}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}} \]
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. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 9.0

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

5. Applied rewrites 9.0%

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

7. Applied rewrites 9.0%

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

8. Final simplification 9.0%

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

Alternative 17: 9.1% accurate, 13.5× 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 99.5%

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

3. Taylor expanded in r around 0

   \[\leadsto \color{blue}{\frac{\frac{1}{4}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}} \]
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. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 9.0

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

5. Applied rewrites 9.0%

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

Reproduce

herbie shell --seed 2024223 
(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))))