Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.5% → 99.5%

Time: 5.7s

Alternatives: 18

Speedup: 1.1×

Specification

\[\left(0 \leq s \land s \leq 256\right) \land \left(10^{-6} < r \land r < 1000000\right)\]

Math

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

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/ (* 0.25 (exp (/ (- r) s))) (* (* (* 2.0 PI) s) r))
  (/ (* 0.75 (exp (/ (- r) (* 3.0 s)))) (* (* (* 6.0 PI) s) r))))

C

float code(float s, float r) {
	return ((0.25f * expf((-r / s))) / (((2.0f * ((float) M_PI)) * s) * r)) + ((0.75f * expf((-r / (3.0f * s)))) / (((6.0f * ((float) M_PI)) * s) * r));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.25) * exp(Float32(Float32(-r) / s))) / Float32(Float32(Float32(Float32(2.0) * Float32(pi)) * s) * r)) + Float32(Float32(Float32(0.75) * exp(Float32(Float32(-r) / Float32(Float32(3.0) * s)))) / Float32(Float32(Float32(Float32(6.0) * Float32(pi)) * s) * r)))
end

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.25) * exp((-r / s))) / (((single(2.0) * single(pi)) * s) * r)) + ((single(0.75) * exp((-r / (single(3.0) * s)))) / (((single(6.0) * single(pi)) * s) * r));
end

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/ (* 0.25 (exp (/ (- r) s))) (* (* (* 2.0 PI) s) r))
  (/ (* 0.75 (exp (/ (- r) (* 3.0 s)))) (* (* (* 6.0 PI) s) r))))

C

float code(float s, float r) {
	return ((0.25f * expf((-r / s))) / (((2.0f * ((float) M_PI)) * s) * r)) + ((0.75f * expf((-r / (3.0f * s)))) / (((6.0f * ((float) M_PI)) * s) * r));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.25) * exp(Float32(Float32(-r) / s))) / Float32(Float32(Float32(Float32(2.0) * Float32(pi)) * s) * r)) + Float32(Float32(Float32(0.75) * exp(Float32(Float32(-r) / Float32(Float32(3.0) * s)))) / Float32(Float32(Float32(Float32(6.0) * Float32(pi)) * s) * r)))
end

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.25) * exp((-r / s))) / (((single(2.0) * single(pi)) * s) * r)) + ((single(0.75) * exp((-r / (single(3.0) * s)))) / (((single(6.0) * single(pi)) * s) * r));
end

Alternative 1: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lift-*.f32 N/A

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

   5. *-commutative N/A

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

   6. times-frac N/A

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

   7. lift-*.f32 N/A

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

   8. lift-*.f32 N/A

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

   9. associate-*l* N/A

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

   10. associate-/r* N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. associate-/r* N/A

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

   14. associate-*l* N/A

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

   15. lift-*.f32 N/A

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

   16. lift-*.f32 N/A

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

   17. *-commutative N/A

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

   18. associate-*r/ N/A

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

   19. lower-/.f32 N/A

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

3. Applied rewrites 99.5%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 99.5

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

   4. lift-/.f32 N/A

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

   5. lift-neg.f32 N/A

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

   6. distribute-frac-neg N/A

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

   7. distribute-neg-frac2 N/A

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

   8. lower-/.f32 N/A

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

   9. lift-*.f32 N/A

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

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

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

   11. lower-*.f32 N/A

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

   12. metadata-eval 99.5

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

5. Applied rewrites 99.5%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-/l* N/A

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

   4. lift-/.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f32 N/A

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

   8. frac-times N/A

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

   9. *-commutative N/A

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

   10. lift-*.f32 N/A

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

   11. associate-*l* N/A

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

   12. lift-*.f32 N/A

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

   13. rem-square-sqrt N/A

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

   14. lift-sqrt.f32 N/A

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

   15. lift-sqrt.f32 N/A

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

   16. associate-*l* N/A

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

   17. lift-*.f32 N/A

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

   18. lift-*.f32 N/A

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

7. Applied rewrites 99.5%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. lift-*.f32 N/A

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

   7. lower-*.f32 99.5

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

   8. lift-*.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f32 99.5

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

9. Applied rewrites 99.5%

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

Alternative 2: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lift-*.f32 N/A

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

   5. *-commutative N/A

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

   6. times-frac N/A

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

   7. lift-*.f32 N/A

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

   8. lift-*.f32 N/A

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

   9. associate-*l* N/A

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

   10. associate-/r* N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. associate-/r* N/A

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

   14. associate-*l* N/A

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

   15. lift-*.f32 N/A

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

   16. lift-*.f32 N/A

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

   17. *-commutative N/A

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

   18. associate-*r/ N/A

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

   19. lower-/.f32 N/A

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

3. Applied rewrites 99.5%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 99.5

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

   4. lift-/.f32 N/A

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

   5. lift-neg.f32 N/A

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

   6. distribute-frac-neg N/A

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

   7. distribute-neg-frac2 N/A

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

   8. lower-/.f32 N/A

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

   9. lift-*.f32 N/A

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

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

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

   11. lower-*.f32 N/A

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

   12. metadata-eval 99.5

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

5. Applied rewrites 99.5%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-/l* N/A

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

   4. lift-/.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f32 N/A

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

   8. frac-times N/A

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

   9. *-commutative N/A

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

   10. lift-*.f32 N/A

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

   11. associate-*l* N/A

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

   12. lift-*.f32 N/A

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

   13. rem-square-sqrt N/A

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

   14. lift-sqrt.f32 N/A

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

   15. lift-sqrt.f32 N/A

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

   16. associate-*l* N/A

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

   17. lift-*.f32 N/A

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

   18. lift-*.f32 N/A

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

7. Applied rewrites 99.5%

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

Alternative 3: 99.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.125) / ((r * single(pi)) * exp((r / s)))) / s) + ((exp((single(-0.3333333333333333) * (r / s))) * single(0.125)) / (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. Taylor expanded in s around 0

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

   1. lower-/.f32 N/A

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

4. Applied rewrites 99.4%

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

6. Applied rewrites 99.4%

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

Alternative 4: 99.4% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = (((exp((-r / s)) + 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. Taylor expanded in s around 0

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

   1. lower-/.f32 N/A

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

4. Applied rewrites 99.4%

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

6. Applied rewrites 99.4%

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

Alternative 5: 99.4% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = single(0.125) / (s * ((single(pi) * r) / (exp(((r / s) * single(-0.3333333333333333))) + exp((-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. Taylor expanded in s around 0

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

   1. lower-/.f32 N/A

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

4. Applied rewrites 99.4%

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

   1. lift-/.f32 N/A

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

   2. div-flip N/A

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

   3. lower-/.f32 N/A

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

   4. lower-/.f32 99.4

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

   5. lift-fma.f32 N/A

      \[\leadsto \frac{1}{\frac{s}{\frac{1}{8} \cdot \frac{e^{-1 \cdot \frac{r}{s}}}{r \cdot \pi} + \color{blue}{\frac{1}{8} \cdot \frac{e^{\frac{-1}{3} \cdot \frac{r}{s}}}{r \cdot \pi}}}} \]

   6. lift-*.f32 N/A

      \[\leadsto \frac{1}{\frac{s}{\frac{1}{8} \cdot \frac{e^{-1 \cdot \frac{r}{s}}}{r \cdot \pi} + \frac{1}{8} \cdot \color{blue}{\frac{e^{\frac{-1}{3} \cdot \frac{r}{s}}}{r \cdot \pi}}}} \]

   7. distribute-lft-out N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

6. Applied rewrites 99.4%

   \[\leadsto \frac{1}{\color{blue}{\frac{s}{\frac{e^{\frac{-r}{s}} + e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r \cdot \pi} \cdot 0.125}}} \]
7. Step-by-step derivation

   1. lift-/.f32 N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{s}{\frac{e^{\frac{-r}{s}} + e^{\frac{-1}{3} \cdot \frac{r}{s}}}{r \cdot \pi} \cdot \frac{1}{8}}}} \]

   2. lift-/.f32 N/A

      \[\leadsto \frac{1}{\frac{s}{\color{blue}{\frac{e^{\frac{-r}{s}} + e^{\frac{-1}{3} \cdot \frac{r}{s}}}{r \cdot \pi} \cdot \frac{1}{8}}}} \]

   3. lift-*.f32 N/A

      \[\leadsto \frac{1}{\frac{s}{\frac{e^{\frac{-r}{s}} + e^{\frac{-1}{3} \cdot \frac{r}{s}}}{r \cdot \pi} \cdot \color{blue}{\frac{1}{8}}}} \]

   4. associate-/r* N/A

      \[\leadsto \frac{1}{\frac{\frac{s}{\frac{e^{\frac{-r}{s}} + e^{\frac{-1}{3} \cdot \frac{r}{s}}}{r \cdot \pi}}}{\color{blue}{\frac{1}{8}}}} \]

   5. div-flip-rev N/A

      \[\leadsto \frac{\frac{1}{8}}{\color{blue}{\frac{s}{\frac{e^{\frac{-r}{s}} + e^{\frac{-1}{3} \cdot \frac{r}{s}}}{r \cdot \pi}}}} \]

   6. lower-/.f32 N/A

      \[\leadsto \frac{\frac{1}{8}}{\color{blue}{\frac{s}{\frac{e^{\frac{-r}{s}} + e^{\frac{-1}{3} \cdot \frac{r}{s}}}{r \cdot \pi}}}} \]

   7. mult-flip N/A

      \[\leadsto \frac{\frac{1}{8}}{s \cdot \color{blue}{\frac{1}{\frac{e^{\frac{-r}{s}} + e^{\frac{-1}{3} \cdot \frac{r}{s}}}{r \cdot \pi}}}} \]

   8. lift-/.f32 N/A

      \[\leadsto \frac{\frac{1}{8}}{s \cdot \frac{1}{\frac{e^{\frac{-r}{s}} + e^{\frac{-1}{3} \cdot \frac{r}{s}}}{\color{blue}{r \cdot \pi}}}} \]

   9. div-flip N/A

      \[\leadsto \frac{\frac{1}{8}}{s \cdot \frac{r \cdot \pi}{\color{blue}{e^{\frac{-r}{s}} + e^{\frac{-1}{3} \cdot \frac{r}{s}}}}} \]

8. Applied rewrites 99.4%

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

Alternative 6: 99.4% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = ((exp(((r / s) * single(-0.3333333333333333))) + exp((-r / s))) / (single(pi) * r)) * (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. Taylor expanded in s around 0

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

   1. lower-/.f32 N/A

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

4. Applied rewrites 99.4%

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

   1. lift-/.f32 N/A

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

   2. div-flip N/A

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

   3. lower-/.f32 N/A

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

   4. lower-/.f32 99.4

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

   5. lift-fma.f32 N/A

      \[\leadsto \frac{1}{\frac{s}{\frac{1}{8} \cdot \frac{e^{-1 \cdot \frac{r}{s}}}{r \cdot \pi} + \color{blue}{\frac{1}{8} \cdot \frac{e^{\frac{-1}{3} \cdot \frac{r}{s}}}{r \cdot \pi}}}} \]

   6. lift-*.f32 N/A

      \[\leadsto \frac{1}{\frac{s}{\frac{1}{8} \cdot \frac{e^{-1 \cdot \frac{r}{s}}}{r \cdot \pi} + \frac{1}{8} \cdot \color{blue}{\frac{e^{\frac{-1}{3} \cdot \frac{r}{s}}}{r \cdot \pi}}}} \]

   7. distribute-lft-out N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

6. Applied rewrites 99.4%

   \[\leadsto \frac{1}{\color{blue}{\frac{s}{\frac{e^{\frac{-r}{s}} + e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r \cdot \pi} \cdot 0.125}}} \]
7. Step-by-step derivation

   1. lift-/.f32 N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{s}{\frac{e^{\frac{-r}{s}} + e^{\frac{-1}{3} \cdot \frac{r}{s}}}{r \cdot \pi} \cdot \frac{1}{8}}}} \]

   2. lift-/.f32 N/A

      \[\leadsto \frac{1}{\frac{s}{\color{blue}{\frac{e^{\frac{-r}{s}} + e^{\frac{-1}{3} \cdot \frac{r}{s}}}{r \cdot \pi} \cdot \frac{1}{8}}}} \]

   3. div-flip-rev N/A

      \[\leadsto \frac{\frac{e^{\frac{-r}{s}} + e^{\frac{-1}{3} \cdot \frac{r}{s}}}{r \cdot \pi} \cdot \frac{1}{8}}{\color{blue}{s}} \]

   4. lift-*.f32 N/A

      \[\leadsto \frac{\frac{e^{\frac{-r}{s}} + e^{\frac{-1}{3} \cdot \frac{r}{s}}}{r \cdot \pi} \cdot \frac{1}{8}}{s} \]

   5. associate-/l* N/A

      \[\leadsto \frac{e^{\frac{-r}{s}} + e^{\frac{-1}{3} \cdot \frac{r}{s}}}{r \cdot \pi} \cdot \color{blue}{\frac{\frac{1}{8}}{s}} \]

   6. lower-*.f32 N/A

      \[\leadsto \frac{e^{\frac{-r}{s}} + e^{\frac{-1}{3} \cdot \frac{r}{s}}}{r \cdot \pi} \cdot \color{blue}{\frac{\frac{1}{8}}{s}} \]

8. Applied rewrites 99.4%

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

Alternative 7: 99.4% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = ((exp(((r / s) * single(-0.3333333333333333))) + exp((-r / s))) * single(0.125)) / (single(pi) * (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. Taylor expanded in s around 0

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

   1. lower-/.f32 N/A

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

4. Applied rewrites 99.4%

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

   1. lift-/.f32 N/A

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

   2. div-flip N/A

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

   3. lower-/.f32 N/A

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

   4. lower-/.f32 99.4

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

   5. lift-fma.f32 N/A

      \[\leadsto \frac{1}{\frac{s}{\frac{1}{8} \cdot \frac{e^{-1 \cdot \frac{r}{s}}}{r \cdot \pi} + \color{blue}{\frac{1}{8} \cdot \frac{e^{\frac{-1}{3} \cdot \frac{r}{s}}}{r \cdot \pi}}}} \]

   6. lift-*.f32 N/A

      \[\leadsto \frac{1}{\frac{s}{\frac{1}{8} \cdot \frac{e^{-1 \cdot \frac{r}{s}}}{r \cdot \pi} + \frac{1}{8} \cdot \color{blue}{\frac{e^{\frac{-1}{3} \cdot \frac{r}{s}}}{r \cdot \pi}}}} \]

   7. distribute-lft-out N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

6. Applied rewrites 99.4%

   \[\leadsto \frac{1}{\color{blue}{\frac{s}{\frac{e^{\frac{-r}{s}} + e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r \cdot \pi} \cdot 0.125}}} \]
7. Step-by-step derivation

   1. lift-/.f32 N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{s}{\frac{e^{\frac{-r}{s}} + e^{\frac{-1}{3} \cdot \frac{r}{s}}}{r \cdot \pi} \cdot \frac{1}{8}}}} \]

   2. lift-/.f32 N/A

      \[\leadsto \frac{1}{\frac{s}{\color{blue}{\frac{e^{\frac{-r}{s}} + e^{\frac{-1}{3} \cdot \frac{r}{s}}}{r \cdot \pi} \cdot \frac{1}{8}}}} \]

   3. div-flip-rev N/A

      \[\leadsto \frac{\frac{e^{\frac{-r}{s}} + e^{\frac{-1}{3} \cdot \frac{r}{s}}}{r \cdot \pi} \cdot \frac{1}{8}}{\color{blue}{s}} \]

   4. lift-*.f32 N/A

      \[\leadsto \frac{\frac{e^{\frac{-r}{s}} + e^{\frac{-1}{3} \cdot \frac{r}{s}}}{r \cdot \pi} \cdot \frac{1}{8}}{s} \]

   5. lift-/.f32 N/A

      \[\leadsto \frac{\frac{e^{\frac{-r}{s}} + e^{\frac{-1}{3} \cdot \frac{r}{s}}}{r \cdot \pi} \cdot \frac{1}{8}}{s} \]

   6. associate-*l/ N/A

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

8. Applied rewrites 99.4%

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

Alternative 8: 48.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 29.56999969482422:\\ \;\;\;\;\frac{1}{r \cdot \mathsf{fma}\left(4, s \cdot \pi, r \cdot \mathsf{fma}\left(-8, r \cdot \mathsf{fma}\left(-0.2222222222222222, \frac{\pi}{s}, 0.1388888888888889 \cdot \frac{\pi}{s}\right), 2.6666666666666665 \cdot \pi\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{\log \left({\left(e^{\pi}\right)}^{r}\right) \cdot s}\\ \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (if (<= r 29.56999969482422)
   (/
    1.0
    (*
     r
     (fma
      4.0
      (* s PI)
      (*
       r
       (fma
        -8.0
        (*
         r
         (fma -0.2222222222222222 (/ PI s) (* 0.1388888888888889 (/ PI s))))
        (* 2.6666666666666665 PI))))))
   (/ 0.25 (* (log (pow (exp PI) r)) s))))

C

float code(float s, float r) {
	float tmp;
	if (r <= 29.56999969482422f) {
		tmp = 1.0f / (r * fmaf(4.0f, (s * ((float) M_PI)), (r * fmaf(-8.0f, (r * fmaf(-0.2222222222222222f, (((float) M_PI) / s), (0.1388888888888889f * (((float) M_PI) / s)))), (2.6666666666666665f * ((float) M_PI))))));
	} else {
		tmp = 0.25f / (logf(powf(expf(((float) M_PI)), r)) * s);
	}
	return tmp;
}

Julia

function code(s, r)
	tmp = Float32(0.0)
	if (r <= Float32(29.56999969482422))
		tmp = Float32(Float32(1.0) / Float32(r * fma(Float32(4.0), Float32(s * Float32(pi)), Float32(r * fma(Float32(-8.0), Float32(r * fma(Float32(-0.2222222222222222), Float32(Float32(pi) / s), Float32(Float32(0.1388888888888889) * Float32(Float32(pi) / s)))), Float32(Float32(2.6666666666666665) * Float32(pi)))))));
	else
		tmp = Float32(Float32(0.25) / Float32(log((exp(Float32(pi)) ^ r)) * s));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if r < 29.5699997

   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. Taylor expanded in s around 0

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

      1. lower-/.f32 N/A

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

   4. Applied rewrites 99.4%

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

      1. lift-/.f32 N/A

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

      2. div-flip N/A

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

      3. lower-/.f32 N/A

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

      4. lower-/.f32 99.4

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

      5. lift-fma.f32 N/A

         \[\leadsto \frac{1}{\frac{s}{\frac{1}{8} \cdot \frac{e^{-1 \cdot \frac{r}{s}}}{r \cdot \pi} + \color{blue}{\frac{1}{8} \cdot \frac{e^{\frac{-1}{3} \cdot \frac{r}{s}}}{r \cdot \pi}}}} \]

      6. lift-*.f32 N/A

         \[\leadsto \frac{1}{\frac{s}{\frac{1}{8} \cdot \frac{e^{-1 \cdot \frac{r}{s}}}{r \cdot \pi} + \frac{1}{8} \cdot \color{blue}{\frac{e^{\frac{-1}{3} \cdot \frac{r}{s}}}{r \cdot \pi}}}} \]

      7. distribute-lft-out N/A

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

      8. *-commutative N/A

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

      9. lower-*.f32 N/A

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

   6. Applied rewrites 99.4%

      \[\leadsto \frac{1}{\color{blue}{\frac{s}{\frac{e^{\frac{-r}{s}} + e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r \cdot \pi} \cdot 0.125}}} \]

   7. Taylor expanded in r around 0

      \[\leadsto \frac{1}{r \cdot \color{blue}{\left(4 \cdot \left(s \cdot \mathsf{PI}\left(\right)\right) + r \cdot \left(-8 \cdot \left(r \cdot \left(\frac{-2}{9} \cdot \frac{\mathsf{PI}\left(\right)}{s} + \frac{5}{36} \cdot \frac{\mathsf{PI}\left(\right)}{s}\right)\right) + \frac{8}{3} \cdot \mathsf{PI}\left(\right)\right)\right)}} \]
   8. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \frac{1}{r \cdot \left(4 \cdot \left(s \cdot \mathsf{PI}\left(\right)\right) + \color{blue}{r \cdot \left(-8 \cdot \left(r \cdot \left(\frac{-2}{9} \cdot \frac{\mathsf{PI}\left(\right)}{s} + \frac{5}{36} \cdot \frac{\mathsf{PI}\left(\right)}{s}\right)\right) + \frac{8}{3} \cdot \mathsf{PI}\left(\right)\right)}\right)} \]

      2. lower-fma.f32 N/A

         \[\leadsto \frac{1}{r \cdot \mathsf{fma}\left(4, s \cdot \color{blue}{\mathsf{PI}\left(\right)}, r \cdot \left(-8 \cdot \left(r \cdot \left(\frac{-2}{9} \cdot \frac{\mathsf{PI}\left(\right)}{s} + \frac{5}{36} \cdot \frac{\mathsf{PI}\left(\right)}{s}\right)\right) + \frac{8}{3} \cdot \mathsf{PI}\left(\right)\right)\right)} \]

      3. lower-*.f32 N/A

         \[\leadsto \frac{1}{r \cdot \mathsf{fma}\left(4, s \cdot \mathsf{PI}\left(\right), r \cdot \left(-8 \cdot \left(r \cdot \left(\frac{-2}{9} \cdot \frac{\mathsf{PI}\left(\right)}{s} + \frac{5}{36} \cdot \frac{\mathsf{PI}\left(\right)}{s}\right)\right) + \frac{8}{3} \cdot \mathsf{PI}\left(\right)\right)\right)} \]

      4. lower-PI.f32 N/A

         \[\leadsto \frac{1}{r \cdot \mathsf{fma}\left(4, s \cdot \pi, r \cdot \left(-8 \cdot \left(r \cdot \left(\frac{-2}{9} \cdot \frac{\mathsf{PI}\left(\right)}{s} + \frac{5}{36} \cdot \frac{\mathsf{PI}\left(\right)}{s}\right)\right) + \frac{8}{3} \cdot \mathsf{PI}\left(\right)\right)\right)} \]

      5. lower-*.f32 N/A

         \[\leadsto \frac{1}{r \cdot \mathsf{fma}\left(4, s \cdot \pi, r \cdot \left(-8 \cdot \left(r \cdot \left(\frac{-2}{9} \cdot \frac{\mathsf{PI}\left(\right)}{s} + \frac{5}{36} \cdot \frac{\mathsf{PI}\left(\right)}{s}\right)\right) + \frac{8}{3} \cdot \mathsf{PI}\left(\right)\right)\right)} \]

      6. lower-fma.f32 N/A

         \[\leadsto \frac{1}{r \cdot \mathsf{fma}\left(4, s \cdot \pi, r \cdot \mathsf{fma}\left(-8, r \cdot \left(\frac{-2}{9} \cdot \frac{\mathsf{PI}\left(\right)}{s} + \frac{5}{36} \cdot \frac{\mathsf{PI}\left(\right)}{s}\right), \frac{8}{3} \cdot \mathsf{PI}\left(\right)\right)\right)} \]

   9. Applied rewrites 25.8%

      \[\leadsto \frac{1}{r \cdot \color{blue}{\mathsf{fma}\left(4, s \cdot \pi, r \cdot \mathsf{fma}\left(-8, r \cdot \mathsf{fma}\left(-0.2222222222222222, \frac{\pi}{s}, 0.1388888888888889 \cdot \frac{\pi}{s}\right), 2.6666666666666665 \cdot \pi\right)\right)}} \]

   if 29.5699997 < r

   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. Taylor expanded in s around inf

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

      1. lower-/.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-PI.f32 9.0

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

   4. Applied rewrites 9.0%

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

      1. lift-*.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f32 N/A

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

      6. lower-*.f32 9.0

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

   6. Applied rewrites 9.0%

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

      1. lift-*.f32 N/A

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

      2. lift-PI.f32 N/A

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

      3. add-log-exp N/A

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

      4. log-pow-rev N/A

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

      5. lower-log.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. lift-PI.f32 N/A

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

      8. lower-exp.f32 43.0

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

   8. Applied rewrites 43.0%

      \[\leadsto \frac{0.25}{\log \left({\left(e^{\pi}\right)}^{r}\right) \cdot s} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 46.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 29.56999969482422:\\ \;\;\;\;\frac{1}{s \cdot \mathsf{fma}\left(-2, \frac{r \cdot \left(\pi \cdot \mathsf{fma}\left(-1, r, -0.3333333333333333 \cdot r\right)\right)}{s}, 4 \cdot \left(r \cdot \pi\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{\log \left({\left(e^{\pi}\right)}^{r}\right) \cdot s}\\ \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (if (<= r 29.56999969482422)
   (/
    1.0
    (*
     s
     (fma
      -2.0
      (/ (* r (* PI (fma -1.0 r (* -0.3333333333333333 r)))) s)
      (* 4.0 (* r PI)))))
   (/ 0.25 (* (log (pow (exp PI) r)) s))))

C

float code(float s, float r) {
	float tmp;
	if (r <= 29.56999969482422f) {
		tmp = 1.0f / (s * fmaf(-2.0f, ((r * (((float) M_PI) * fmaf(-1.0f, r, (-0.3333333333333333f * r)))) / s), (4.0f * (r * ((float) M_PI)))));
	} else {
		tmp = 0.25f / (logf(powf(expf(((float) M_PI)), r)) * s);
	}
	return tmp;
}

Julia

function code(s, r)
	tmp = Float32(0.0)
	if (r <= Float32(29.56999969482422))
		tmp = Float32(Float32(1.0) / Float32(s * fma(Float32(-2.0), Float32(Float32(r * Float32(Float32(pi) * fma(Float32(-1.0), r, Float32(Float32(-0.3333333333333333) * r)))) / s), Float32(Float32(4.0) * Float32(r * Float32(pi))))));
	else
		tmp = Float32(Float32(0.25) / Float32(log((exp(Float32(pi)) ^ r)) * s));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if r < 29.5699997

   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. Taylor expanded in s around 0

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

      1. lower-/.f32 N/A

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

   4. Applied rewrites 99.4%

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

      1. lift-/.f32 N/A

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

      2. div-flip N/A

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

      3. lower-/.f32 N/A

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

      4. lower-/.f32 99.4

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

      5. lift-fma.f32 N/A

         \[\leadsto \frac{1}{\frac{s}{\frac{1}{8} \cdot \frac{e^{-1 \cdot \frac{r}{s}}}{r \cdot \pi} + \color{blue}{\frac{1}{8} \cdot \frac{e^{\frac{-1}{3} \cdot \frac{r}{s}}}{r \cdot \pi}}}} \]

      6. lift-*.f32 N/A

         \[\leadsto \frac{1}{\frac{s}{\frac{1}{8} \cdot \frac{e^{-1 \cdot \frac{r}{s}}}{r \cdot \pi} + \frac{1}{8} \cdot \color{blue}{\frac{e^{\frac{-1}{3} \cdot \frac{r}{s}}}{r \cdot \pi}}}} \]

      7. distribute-lft-out N/A

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

      8. *-commutative N/A

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

      9. lower-*.f32 N/A

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

   6. Applied rewrites 99.4%

      \[\leadsto \frac{1}{\color{blue}{\frac{s}{\frac{e^{\frac{-r}{s}} + e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r \cdot \pi} \cdot 0.125}}} \]

   7. Taylor expanded in s around inf

      \[\leadsto \frac{1}{s \cdot \color{blue}{\left(-2 \cdot \frac{r \cdot \left(\mathsf{PI}\left(\right) \cdot \left(-1 \cdot r + \frac{-1}{3} \cdot r\right)\right)}{s} + 4 \cdot \left(r \cdot \mathsf{PI}\left(\right)\right)\right)}} \]
   8. Step-by-step derivation

      1. lower-*.f32 N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-/.f32 N/A

         \[\leadsto \frac{1}{s \cdot \mathsf{fma}\left(-2, \frac{r \cdot \left(\mathsf{PI}\left(\right) \cdot \left(-1 \cdot r + \frac{-1}{3} \cdot r\right)\right)}{s}, 4 \cdot \left(r \cdot \mathsf{PI}\left(\right)\right)\right)} \]

      4. lower-*.f32 N/A

         \[\leadsto \frac{1}{s \cdot \mathsf{fma}\left(-2, \frac{r \cdot \left(\mathsf{PI}\left(\right) \cdot \left(-1 \cdot r + \frac{-1}{3} \cdot r\right)\right)}{s}, 4 \cdot \left(r \cdot \mathsf{PI}\left(\right)\right)\right)} \]

      5. lower-*.f32 N/A

         \[\leadsto \frac{1}{s \cdot \mathsf{fma}\left(-2, \frac{r \cdot \left(\mathsf{PI}\left(\right) \cdot \left(-1 \cdot r + \frac{-1}{3} \cdot r\right)\right)}{s}, 4 \cdot \left(r \cdot \mathsf{PI}\left(\right)\right)\right)} \]

      6. lower-PI.f32 N/A

         \[\leadsto \frac{1}{s \cdot \mathsf{fma}\left(-2, \frac{r \cdot \left(\pi \cdot \left(-1 \cdot r + \frac{-1}{3} \cdot r\right)\right)}{s}, 4 \cdot \left(r \cdot \mathsf{PI}\left(\right)\right)\right)} \]

      7. lower-fma.f32 N/A

         \[\leadsto \frac{1}{s \cdot \mathsf{fma}\left(-2, \frac{r \cdot \left(\pi \cdot \mathsf{fma}\left(-1, r, \frac{-1}{3} \cdot r\right)\right)}{s}, 4 \cdot \left(r \cdot \mathsf{PI}\left(\right)\right)\right)} \]

      8. lower-*.f32 N/A

         \[\leadsto \frac{1}{s \cdot \mathsf{fma}\left(-2, \frac{r \cdot \left(\pi \cdot \mathsf{fma}\left(-1, r, \frac{-1}{3} \cdot r\right)\right)}{s}, 4 \cdot \left(r \cdot \mathsf{PI}\left(\right)\right)\right)} \]

      9. lower-*.f32 N/A

         \[\leadsto \frac{1}{s \cdot \mathsf{fma}\left(-2, \frac{r \cdot \left(\pi \cdot \mathsf{fma}\left(-1, r, \frac{-1}{3} \cdot r\right)\right)}{s}, 4 \cdot \left(r \cdot \mathsf{PI}\left(\right)\right)\right)} \]

      10. lower-*.f32 N/A

          \[\leadsto \frac{1}{s \cdot \mathsf{fma}\left(-2, \frac{r \cdot \left(\pi \cdot \mathsf{fma}\left(-1, r, \frac{-1}{3} \cdot r\right)\right)}{s}, 4 \cdot \left(r \cdot \mathsf{PI}\left(\right)\right)\right)} \]

      11. lower-PI.f32 19.5

          \[\leadsto \frac{1}{s \cdot \mathsf{fma}\left(-2, \frac{r \cdot \left(\pi \cdot \mathsf{fma}\left(-1, r, -0.3333333333333333 \cdot r\right)\right)}{s}, 4 \cdot \left(r \cdot \pi\right)\right)} \]

   9. Applied rewrites 19.5%

      \[\leadsto \frac{1}{s \cdot \color{blue}{\mathsf{fma}\left(-2, \frac{r \cdot \left(\pi \cdot \mathsf{fma}\left(-1, r, -0.3333333333333333 \cdot r\right)\right)}{s}, 4 \cdot \left(r \cdot \pi\right)\right)}} \]

   if 29.5699997 < r

   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. Taylor expanded in s around inf

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

      1. lower-/.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-PI.f32 9.0

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

   4. Applied rewrites 9.0%

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

      1. lift-*.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f32 N/A

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

      6. lower-*.f32 9.0

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

   6. Applied rewrites 9.0%

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

      1. lift-*.f32 N/A

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

      2. lift-PI.f32 N/A

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

      3. add-log-exp N/A

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

      4. log-pow-rev N/A

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

      5. lower-log.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. lift-PI.f32 N/A

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

      8. lower-exp.f32 43.0

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

   8. Applied rewrites 43.0%

      \[\leadsto \frac{0.25}{\log \left({\left(e^{\pi}\right)}^{r}\right) \cdot s} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 46.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 29.56999969482422:\\ \;\;\;\;\frac{1}{s \cdot \mathsf{fma}\left(-2, \frac{r \cdot \left(\pi \cdot \mathsf{fma}\left(-1, r, -0.3333333333333333 \cdot r\right)\right)}{s}, 4 \cdot \left(r \cdot \pi\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{\log \left({\left(e^{r \cdot \pi}\right)}^{s}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (if (<= r 29.56999969482422)
   (/
    1.0
    (*
     s
     (fma
      -2.0
      (/ (* r (* PI (fma -1.0 r (* -0.3333333333333333 r)))) s)
      (* 4.0 (* r PI)))))
   (/ 0.25 (log (pow (exp (* r PI)) s)))))

C

float code(float s, float r) {
	float tmp;
	if (r <= 29.56999969482422f) {
		tmp = 1.0f / (s * fmaf(-2.0f, ((r * (((float) M_PI) * fmaf(-1.0f, r, (-0.3333333333333333f * r)))) / s), (4.0f * (r * ((float) M_PI)))));
	} else {
		tmp = 0.25f / logf(powf(expf((r * ((float) M_PI))), s));
	}
	return tmp;
}

Julia

function code(s, r)
	tmp = Float32(0.0)
	if (r <= Float32(29.56999969482422))
		tmp = Float32(Float32(1.0) / Float32(s * fma(Float32(-2.0), Float32(Float32(r * Float32(Float32(pi) * fma(Float32(-1.0), r, Float32(Float32(-0.3333333333333333) * r)))) / s), Float32(Float32(4.0) * Float32(r * Float32(pi))))));
	else
		tmp = Float32(Float32(0.25) / log((exp(Float32(r * Float32(pi))) ^ s)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if r < 29.5699997

   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. Taylor expanded in s around 0

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

      1. lower-/.f32 N/A

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

   4. Applied rewrites 99.4%

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

      1. lift-/.f32 N/A

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

      2. div-flip N/A

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

      3. lower-/.f32 N/A

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

      4. lower-/.f32 99.4

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

      5. lift-fma.f32 N/A

         \[\leadsto \frac{1}{\frac{s}{\frac{1}{8} \cdot \frac{e^{-1 \cdot \frac{r}{s}}}{r \cdot \pi} + \color{blue}{\frac{1}{8} \cdot \frac{e^{\frac{-1}{3} \cdot \frac{r}{s}}}{r \cdot \pi}}}} \]

      6. lift-*.f32 N/A

         \[\leadsto \frac{1}{\frac{s}{\frac{1}{8} \cdot \frac{e^{-1 \cdot \frac{r}{s}}}{r \cdot \pi} + \frac{1}{8} \cdot \color{blue}{\frac{e^{\frac{-1}{3} \cdot \frac{r}{s}}}{r \cdot \pi}}}} \]

      7. distribute-lft-out N/A

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

      8. *-commutative N/A

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

      9. lower-*.f32 N/A

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

   6. Applied rewrites 99.4%

      \[\leadsto \frac{1}{\color{blue}{\frac{s}{\frac{e^{\frac{-r}{s}} + e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r \cdot \pi} \cdot 0.125}}} \]

   7. Taylor expanded in s around inf

      \[\leadsto \frac{1}{s \cdot \color{blue}{\left(-2 \cdot \frac{r \cdot \left(\mathsf{PI}\left(\right) \cdot \left(-1 \cdot r + \frac{-1}{3} \cdot r\right)\right)}{s} + 4 \cdot \left(r \cdot \mathsf{PI}\left(\right)\right)\right)}} \]
   8. Step-by-step derivation

      1. lower-*.f32 N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-/.f32 N/A

         \[\leadsto \frac{1}{s \cdot \mathsf{fma}\left(-2, \frac{r \cdot \left(\mathsf{PI}\left(\right) \cdot \left(-1 \cdot r + \frac{-1}{3} \cdot r\right)\right)}{s}, 4 \cdot \left(r \cdot \mathsf{PI}\left(\right)\right)\right)} \]

      4. lower-*.f32 N/A

         \[\leadsto \frac{1}{s \cdot \mathsf{fma}\left(-2, \frac{r \cdot \left(\mathsf{PI}\left(\right) \cdot \left(-1 \cdot r + \frac{-1}{3} \cdot r\right)\right)}{s}, 4 \cdot \left(r \cdot \mathsf{PI}\left(\right)\right)\right)} \]

      5. lower-*.f32 N/A

         \[\leadsto \frac{1}{s \cdot \mathsf{fma}\left(-2, \frac{r \cdot \left(\mathsf{PI}\left(\right) \cdot \left(-1 \cdot r + \frac{-1}{3} \cdot r\right)\right)}{s}, 4 \cdot \left(r \cdot \mathsf{PI}\left(\right)\right)\right)} \]

      6. lower-PI.f32 N/A

         \[\leadsto \frac{1}{s \cdot \mathsf{fma}\left(-2, \frac{r \cdot \left(\pi \cdot \left(-1 \cdot r + \frac{-1}{3} \cdot r\right)\right)}{s}, 4 \cdot \left(r \cdot \mathsf{PI}\left(\right)\right)\right)} \]

      7. lower-fma.f32 N/A

         \[\leadsto \frac{1}{s \cdot \mathsf{fma}\left(-2, \frac{r \cdot \left(\pi \cdot \mathsf{fma}\left(-1, r, \frac{-1}{3} \cdot r\right)\right)}{s}, 4 \cdot \left(r \cdot \mathsf{PI}\left(\right)\right)\right)} \]

      8. lower-*.f32 N/A

         \[\leadsto \frac{1}{s \cdot \mathsf{fma}\left(-2, \frac{r \cdot \left(\pi \cdot \mathsf{fma}\left(-1, r, \frac{-1}{3} \cdot r\right)\right)}{s}, 4 \cdot \left(r \cdot \mathsf{PI}\left(\right)\right)\right)} \]

      9. lower-*.f32 N/A

         \[\leadsto \frac{1}{s \cdot \mathsf{fma}\left(-2, \frac{r \cdot \left(\pi \cdot \mathsf{fma}\left(-1, r, \frac{-1}{3} \cdot r\right)\right)}{s}, 4 \cdot \left(r \cdot \mathsf{PI}\left(\right)\right)\right)} \]

      10. lower-*.f32 N/A

          \[\leadsto \frac{1}{s \cdot \mathsf{fma}\left(-2, \frac{r \cdot \left(\pi \cdot \mathsf{fma}\left(-1, r, \frac{-1}{3} \cdot r\right)\right)}{s}, 4 \cdot \left(r \cdot \mathsf{PI}\left(\right)\right)\right)} \]

      11. lower-PI.f32 19.5

          \[\leadsto \frac{1}{s \cdot \mathsf{fma}\left(-2, \frac{r \cdot \left(\pi \cdot \mathsf{fma}\left(-1, r, -0.3333333333333333 \cdot r\right)\right)}{s}, 4 \cdot \left(r \cdot \pi\right)\right)} \]

   9. Applied rewrites 19.5%

      \[\leadsto \frac{1}{s \cdot \color{blue}{\mathsf{fma}\left(-2, \frac{r \cdot \left(\pi \cdot \mathsf{fma}\left(-1, r, -0.3333333333333333 \cdot r\right)\right)}{s}, 4 \cdot \left(r \cdot \pi\right)\right)}} \]

   if 29.5699997 < r

   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. Taylor expanded in s around inf

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

      1. lower-/.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-PI.f32 9.0

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

   4. Applied rewrites 9.0%

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

      1. lift-*.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f32 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f32 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. lift-PI.f32 N/A

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

      11. add-log-exp N/A

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

      12. log-pow-rev N/A

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

      13. log-pow-rev N/A

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

      14. lower-log.f32 N/A

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

      15. lower-pow.f32 N/A

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

      16. lift-PI.f32 N/A

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

      17. pow-exp N/A

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

      18. *-commutative N/A

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

      19. lift-*.f32 N/A

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

      20. lower-exp.f32 41.7

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

   6. Applied rewrites 41.7%

      \[\leadsto \frac{0.25}{\log \left({\left(e^{r \cdot \pi}\right)}^{s}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 19.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \frac{1}{s \cdot \mathsf{fma}\left(-2, \frac{r \cdot \left(\pi \cdot \mathsf{fma}\left(-1, r, -0.3333333333333333 \cdot r\right)\right)}{s}, 4 \cdot \left(r \cdot \pi\right)\right)} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (/
  1.0
  (*
   s
   (fma
    -2.0
    (/ (* r (* PI (fma -1.0 r (* -0.3333333333333333 r)))) s)
    (* 4.0 (* r PI))))))

C

float code(float s, float r) {
	return 1.0f / (s * fmaf(-2.0f, ((r * (((float) M_PI) * fmaf(-1.0f, r, (-0.3333333333333333f * r)))) / s), (4.0f * (r * ((float) M_PI)))));
}

Julia

function code(s, r)
	return Float32(Float32(1.0) / Float32(s * fma(Float32(-2.0), Float32(Float32(r * Float32(Float32(pi) * fma(Float32(-1.0), r, Float32(Float32(-0.3333333333333333) * r)))) / s), Float32(Float32(4.0) * Float32(r * 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. Taylor expanded in s around 0

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

   1. lower-/.f32 N/A

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

4. Applied rewrites 99.4%

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

   1. lift-/.f32 N/A

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

   2. div-flip N/A

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

   3. lower-/.f32 N/A

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

   4. lower-/.f32 99.4

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

   5. lift-fma.f32 N/A

      \[\leadsto \frac{1}{\frac{s}{\frac{1}{8} \cdot \frac{e^{-1 \cdot \frac{r}{s}}}{r \cdot \pi} + \color{blue}{\frac{1}{8} \cdot \frac{e^{\frac{-1}{3} \cdot \frac{r}{s}}}{r \cdot \pi}}}} \]

   6. lift-*.f32 N/A

      \[\leadsto \frac{1}{\frac{s}{\frac{1}{8} \cdot \frac{e^{-1 \cdot \frac{r}{s}}}{r \cdot \pi} + \frac{1}{8} \cdot \color{blue}{\frac{e^{\frac{-1}{3} \cdot \frac{r}{s}}}{r \cdot \pi}}}} \]

   7. distribute-lft-out N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

6. Applied rewrites 99.4%

   \[\leadsto \frac{1}{\color{blue}{\frac{s}{\frac{e^{\frac{-r}{s}} + e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r \cdot \pi} \cdot 0.125}}} \]

7. Taylor expanded in s around inf

   \[\leadsto \frac{1}{s \cdot \color{blue}{\left(-2 \cdot \frac{r \cdot \left(\mathsf{PI}\left(\right) \cdot \left(-1 \cdot r + \frac{-1}{3} \cdot r\right)\right)}{s} + 4 \cdot \left(r \cdot \mathsf{PI}\left(\right)\right)\right)}} \]
8. Step-by-step derivation

   1. lower-*.f32 N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-/.f32 N/A

      \[\leadsto \frac{1}{s \cdot \mathsf{fma}\left(-2, \frac{r \cdot \left(\mathsf{PI}\left(\right) \cdot \left(-1 \cdot r + \frac{-1}{3} \cdot r\right)\right)}{s}, 4 \cdot \left(r \cdot \mathsf{PI}\left(\right)\right)\right)} \]

   4. lower-*.f32 N/A

      \[\leadsto \frac{1}{s \cdot \mathsf{fma}\left(-2, \frac{r \cdot \left(\mathsf{PI}\left(\right) \cdot \left(-1 \cdot r + \frac{-1}{3} \cdot r\right)\right)}{s}, 4 \cdot \left(r \cdot \mathsf{PI}\left(\right)\right)\right)} \]

   5. lower-*.f32 N/A

      \[\leadsto \frac{1}{s \cdot \mathsf{fma}\left(-2, \frac{r \cdot \left(\mathsf{PI}\left(\right) \cdot \left(-1 \cdot r + \frac{-1}{3} \cdot r\right)\right)}{s}, 4 \cdot \left(r \cdot \mathsf{PI}\left(\right)\right)\right)} \]

   6. lower-PI.f32 N/A

      \[\leadsto \frac{1}{s \cdot \mathsf{fma}\left(-2, \frac{r \cdot \left(\pi \cdot \left(-1 \cdot r + \frac{-1}{3} \cdot r\right)\right)}{s}, 4 \cdot \left(r \cdot \mathsf{PI}\left(\right)\right)\right)} \]

   7. lower-fma.f32 N/A

      \[\leadsto \frac{1}{s \cdot \mathsf{fma}\left(-2, \frac{r \cdot \left(\pi \cdot \mathsf{fma}\left(-1, r, \frac{-1}{3} \cdot r\right)\right)}{s}, 4 \cdot \left(r \cdot \mathsf{PI}\left(\right)\right)\right)} \]

   8. lower-*.f32 N/A

      \[\leadsto \frac{1}{s \cdot \mathsf{fma}\left(-2, \frac{r \cdot \left(\pi \cdot \mathsf{fma}\left(-1, r, \frac{-1}{3} \cdot r\right)\right)}{s}, 4 \cdot \left(r \cdot \mathsf{PI}\left(\right)\right)\right)} \]

   9. lower-*.f32 N/A

      \[\leadsto \frac{1}{s \cdot \mathsf{fma}\left(-2, \frac{r \cdot \left(\pi \cdot \mathsf{fma}\left(-1, r, \frac{-1}{3} \cdot r\right)\right)}{s}, 4 \cdot \left(r \cdot \mathsf{PI}\left(\right)\right)\right)} \]

   10. lower-*.f32 N/A

       \[\leadsto \frac{1}{s \cdot \mathsf{fma}\left(-2, \frac{r \cdot \left(\pi \cdot \mathsf{fma}\left(-1, r, \frac{-1}{3} \cdot r\right)\right)}{s}, 4 \cdot \left(r \cdot \mathsf{PI}\left(\right)\right)\right)} \]

   11. lower-PI.f32 19.5

       \[\leadsto \frac{1}{s \cdot \mathsf{fma}\left(-2, \frac{r \cdot \left(\pi \cdot \mathsf{fma}\left(-1, r, -0.3333333333333333 \cdot r\right)\right)}{s}, 4 \cdot \left(r \cdot \pi\right)\right)} \]

9. Applied rewrites 19.5%

   \[\leadsto \frac{1}{s \cdot \color{blue}{\mathsf{fma}\left(-2, \frac{r \cdot \left(\pi \cdot \mathsf{fma}\left(-1, r, -0.3333333333333333 \cdot r\right)\right)}{s}, 4 \cdot \left(r \cdot \pi\right)\right)}} \]
10. Add Preprocessing

Alternative 12: 12.4% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \frac{1}{r \cdot \mathsf{fma}\left(2.6666666666666665, r \cdot \pi, 4 \cdot \left(s \cdot \pi\right)\right)} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (/ 1.0 (* r (fma 2.6666666666666665 (* r PI) (* 4.0 (* s PI))))))

C

float code(float s, float r) {
	return 1.0f / (r * fmaf(2.6666666666666665f, (r * ((float) M_PI)), (4.0f * (s * ((float) M_PI)))));
}

Julia

function code(s, r)
	return Float32(Float32(1.0) / Float32(r * fma(Float32(2.6666666666666665), Float32(r * Float32(pi)), Float32(Float32(4.0) * 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. Taylor expanded in s around 0

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

   1. lower-/.f32 N/A

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

4. Applied rewrites 99.4%

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

   1. lift-/.f32 N/A

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

   2. div-flip N/A

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

   3. lower-/.f32 N/A

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

   4. lower-/.f32 99.4

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

   5. lift-fma.f32 N/A

      \[\leadsto \frac{1}{\frac{s}{\frac{1}{8} \cdot \frac{e^{-1 \cdot \frac{r}{s}}}{r \cdot \pi} + \color{blue}{\frac{1}{8} \cdot \frac{e^{\frac{-1}{3} \cdot \frac{r}{s}}}{r \cdot \pi}}}} \]

   6. lift-*.f32 N/A

      \[\leadsto \frac{1}{\frac{s}{\frac{1}{8} \cdot \frac{e^{-1 \cdot \frac{r}{s}}}{r \cdot \pi} + \frac{1}{8} \cdot \color{blue}{\frac{e^{\frac{-1}{3} \cdot \frac{r}{s}}}{r \cdot \pi}}}} \]

   7. distribute-lft-out N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

6. Applied rewrites 99.4%

   \[\leadsto \frac{1}{\color{blue}{\frac{s}{\frac{e^{\frac{-r}{s}} + e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r \cdot \pi} \cdot 0.125}}} \]

7. Taylor expanded in r around 0

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

   1. lower-*.f32 N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower-PI.f32 12.4

      \[\leadsto \frac{1}{r \cdot \mathsf{fma}\left(2.6666666666666665, r \cdot \pi, 4 \cdot \left(s \cdot \pi\right)\right)} \]

9. Applied rewrites 12.4%

   \[\leadsto \frac{1}{r \cdot \color{blue}{\mathsf{fma}\left(2.6666666666666665, r \cdot \pi, 4 \cdot \left(s \cdot \pi\right)\right)}} \]
10. Add Preprocessing

Alternative 13: 9.0% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = (single(0.25) / sqrt(single(pi))) / ((sqrt(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. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 9.0

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

4. Applied rewrites 9.0%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 9.0

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

6. Applied rewrites 9.0%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. rem-square-sqrt N/A

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

   4. lift-sqrt.f32 N/A

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

   5. lift-sqrt.f32 N/A

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

   6. associate-*l* N/A

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. associate-/r* N/A

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

   10. lower-/.f32 N/A

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

   11. lower-/.f32 9.0

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

   12. lift-*.f32 N/A

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

   13. *-commutative N/A

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

   14. lift-*.f32 N/A

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

   15. associate-*r* N/A

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

   16. lower-*.f32 N/A

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

   17. lower-*.f32 9.0

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

8. Applied rewrites 9.0%

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

Alternative 14: 9.0% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = single(0.25) / ((sqrt(single(pi)) * r) * (sqrt(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. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 9.0

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

4. Applied rewrites 9.0%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 9.0

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

6. Applied rewrites 9.0%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. rem-square-sqrt N/A

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

   4. lift-sqrt.f32 N/A

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

   5. lift-sqrt.f32 N/A

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

   6. associate-*l* N/A

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

   7. *-commutative N/A

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

   8. lift-*.f32 N/A

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

   9. associate-*l* N/A

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

   10. associate-*r* N/A

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

   11. lower-*.f32 N/A

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

   12. lower-*.f32 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f32 9.0

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

8. Applied rewrites 9.0%

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

Alternative 15: 9.0% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = (single(0.25) / (single(pi) * r)) * (single(1.0) / 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. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 9.0

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

4. Applied rewrites 9.0%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. lift-*.f32 N/A

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

   6. lower-*.f32 9.0

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

6. Applied rewrites 9.0%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-/r* N/A

      \[\leadsto \frac{\frac{\frac{1}{4}}{r \cdot \pi}}{\color{blue}{s}} \]

   4. mult-flip N/A

      \[\leadsto \frac{\frac{1}{4}}{r \cdot \pi} \cdot \color{blue}{\frac{1}{s}} \]

   5. lower-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4}}{r \cdot \pi} \cdot \color{blue}{\frac{1}{s}} \]

   6. lower-/.f32 N/A

      \[\leadsto \frac{\frac{1}{4}}{r \cdot \pi} \cdot \frac{\color{blue}{1}}{s} \]

   7. lift-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4}}{r \cdot \pi} \cdot \frac{1}{s} \]

   8. *-commutative N/A

      \[\leadsto \frac{\frac{1}{4}}{\pi \cdot r} \cdot \frac{1}{s} \]

   9. lower-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4}}{\pi \cdot r} \cdot \frac{1}{s} \]

   10. lower-/.f32 9.0

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

8. Applied rewrites 9.0%

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

Alternative 16: 9.0% accurate, 6.0× speedup

Math

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

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(Float32(0.25) / 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. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 9.0

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

4. Applied rewrites 9.0%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 9.0

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

6. Applied rewrites 9.0%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-/r* N/A

      \[\leadsto \frac{\frac{\frac{1}{4}}{\pi}}{\color{blue}{r \cdot s}} \]

   5. mult-flip-rev N/A

      \[\leadsto \frac{\frac{1}{4} \cdot \frac{1}{\pi}}{\color{blue}{r} \cdot s} \]

   6. lift-/.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot \frac{1}{\pi}}{r \cdot s} \]

   7. lift-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot \frac{1}{\pi}}{\color{blue}{r} \cdot s} \]

   8. lower-/.f32 9.0

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

   9. lift-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot \frac{1}{\pi}}{\color{blue}{r} \cdot s} \]

   10. lift-/.f32 N/A

       \[\leadsto \frac{\frac{1}{4} \cdot \frac{1}{\pi}}{r \cdot s} \]

   11. mult-flip-rev N/A

       \[\leadsto \frac{\frac{\frac{1}{4}}{\pi}}{\color{blue}{r} \cdot s} \]

   12. lower-/.f32 9.0

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

8. Applied rewrites 9.0%

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

Alternative 17: 9.0% accurate, 6.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

2. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 9.0

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

4. Applied rewrites 9.0%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 9.0

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

6. Applied rewrites 9.0%

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

Alternative 18: 9.0% accurate, 6.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

2. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 9.0

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

4. Applied rewrites 9.0%

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

Reproduce

herbie shell --seed 2025156 
(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))))