Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.6%

Time: 15.6s

Alternatives: 18

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{0.125 \cdot e^{0 - \frac{r}{s}}}{r \cdot \left(s \cdot \pi\right)} + \frac{{e}^{\left(\frac{r \cdot -0.3333333333333333}{s}\right)} \cdot -0.75}{\left(r \cdot s\right) \cdot \left(\pi \cdot -6\right)} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/ (* 0.125 (exp (- 0.0 (/ r s)))) (* r (* s PI)))
  (/
   (* (pow E (/ (* r -0.3333333333333333) s)) -0.75)
   (* (* r s) (* PI -6.0)))))

C

float code(float s, float r) {
	return ((0.125f * expf((0.0f - (r / s)))) / (r * (s * ((float) M_PI)))) + ((powf(((float) M_E), ((r * -0.3333333333333333f) / s)) * -0.75f) / ((r * s) * (((float) M_PI) * -6.0f)));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.125) * exp(Float32(Float32(0.0) - Float32(r / s)))) / Float32(r * Float32(s * Float32(pi)))) + Float32(Float32((Float32(exp(1)) ^ Float32(Float32(r * Float32(-0.3333333333333333)) / s)) * Float32(-0.75)) / Float32(Float32(r * s) * Float32(Float32(pi) * Float32(-6.0)))))
end

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.125) * exp((single(0.0) - (r / s)))) / (r * (s * single(pi)))) + (((single(2.71828182845904523536) ^ ((r * single(-0.3333333333333333)) / s)) * single(-0.75)) / ((r * s) * (single(pi) * single(-6.0))));
end

Derivation

1. Initial program 99.4%

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

   1. frac-2neg N/A

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

   2. /-lowering-/.f32 N/A

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

   3. *-commutative N/A

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

   4. distribute-rgt-neg-in N/A

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

   5. *-lowering-*.f32 N/A

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

   6. exp-lowering-exp.f32 N/A

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

   7. remove-double-neg N/A

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

   8. frac-2neg N/A

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

   9. /-lowering-/.f32 N/A

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

   10. *-commutative N/A

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

   11. distribute-rgt-neg-in N/A

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

   12. *-lowering-*.f32 N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. associate-*l* N/A

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

   16. *-commutative N/A

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

   17. distribute-rgt-neg-in N/A

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

   18. *-lowering-*.f32 N/A

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

4. Applied egg-rr 99.4%

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

   1. clear-num N/A

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

   2. div-inv N/A

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

   3. clear-num N/A

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

   4. exp-prod N/A

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

   5. pow-lowering-pow.f32 N/A

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

   6. exp-1-e N/A

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

   7. E-lowering-E.f32 N/A

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

   8. /-lowering-/.f32 N/A

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

   9. *-lowering-*.f32 99.5

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

6. Applied egg-rr 99.5%

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

   1. associate-/r* N/A

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

   2. div-inv N/A

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

   3. metadata-eval N/A

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

   4. associate-*l/ N/A

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

   5. /-lowering-/.f32 N/A

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

   6. *-lowering-*.f32 99.6

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

8. Applied egg-rr 99.6%

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

9. Taylor expanded in r around inf

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

    1. associate-*r/ N/A

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

    2. /-lowering-/.f32 N/A

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

    3. *-lowering-*.f32 N/A

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

    4. exp-lowering-exp.f32 N/A

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

    5. mul-1-neg N/A

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

    6. neg-sub0 N/A

       \[\leadsto \frac{\frac{1}{8} \cdot e^{\color{blue}{0 - \frac{r}{s}}}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} + \frac{{\mathsf{E}\left(\right)}^{\left(\frac{r \cdot \frac{-1}{3}}{s}\right)} \cdot \frac{-3}{4}}{\left(r \cdot s\right) \cdot \left(\mathsf{PI}\left(\right) \cdot -6\right)} \]

    7. --lowering--.f32 N/A

       \[\leadsto \frac{\frac{1}{8} \cdot e^{\color{blue}{0 - \frac{r}{s}}}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} + \frac{{\mathsf{E}\left(\right)}^{\left(\frac{r \cdot \frac{-1}{3}}{s}\right)} \cdot \frac{-3}{4}}{\left(r \cdot s\right) \cdot \left(\mathsf{PI}\left(\right) \cdot -6\right)} \]

    8. /-lowering-/.f32 N/A

       \[\leadsto \frac{\frac{1}{8} \cdot e^{0 - \color{blue}{\frac{r}{s}}}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} + \frac{{\mathsf{E}\left(\right)}^{\left(\frac{r \cdot \frac{-1}{3}}{s}\right)} \cdot \frac{-3}{4}}{\left(r \cdot s\right) \cdot \left(\mathsf{PI}\left(\right) \cdot -6\right)} \]

    9. *-lowering-*.f32 N/A

       \[\leadsto \frac{\frac{1}{8} \cdot e^{0 - \frac{r}{s}}}{\color{blue}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}} + \frac{{\mathsf{E}\left(\right)}^{\left(\frac{r \cdot \frac{-1}{3}}{s}\right)} \cdot \frac{-3}{4}}{\left(r \cdot s\right) \cdot \left(\mathsf{PI}\left(\right) \cdot -6\right)} \]

    10. *-lowering-*.f32 N/A

        \[\leadsto \frac{\frac{1}{8} \cdot e^{0 - \frac{r}{s}}}{r \cdot \color{blue}{\left(s \cdot \mathsf{PI}\left(\right)\right)}} + \frac{{\mathsf{E}\left(\right)}^{\left(\frac{r \cdot \frac{-1}{3}}{s}\right)} \cdot \frac{-3}{4}}{\left(r \cdot s\right) \cdot \left(\mathsf{PI}\left(\right) \cdot -6\right)} \]

    11. PI-lowering-PI.f32 99.6

        \[\leadsto \frac{0.125 \cdot e^{0 - \frac{r}{s}}}{r \cdot \left(s \cdot \color{blue}{\pi}\right)} + \frac{{e}^{\left(\frac{r \cdot -0.3333333333333333}{s}\right)} \cdot -0.75}{\left(r \cdot s\right) \cdot \left(\pi \cdot -6\right)} \]

11. Simplified 99.6%

    \[\leadsto \color{blue}{\frac{0.125 \cdot e^{0 - \frac{r}{s}}}{r \cdot \left(s \cdot \pi\right)}} + \frac{{e}^{\left(\frac{r \cdot -0.3333333333333333}{s}\right)} \cdot -0.75}{\left(r \cdot s\right) \cdot \left(\pi \cdot -6\right)} \]
12. Add Preprocessing

Alternative 2: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{0.125 \cdot e^{0 - \frac{r}{s}}}{r \cdot \left(s \cdot \pi\right)} + \frac{-0.75 \cdot {e}^{\left(\frac{r}{s \cdot -3}\right)}}{\left(r \cdot s\right) \cdot \left(\pi \cdot -6\right)} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/ (* 0.125 (exp (- 0.0 (/ r s)))) (* r (* s PI)))
  (/ (* -0.75 (pow E (/ r (* s -3.0)))) (* (* r s) (* PI -6.0)))))

C

float code(float s, float r) {
	return ((0.125f * expf((0.0f - (r / s)))) / (r * (s * ((float) M_PI)))) + ((-0.75f * powf(((float) M_E), (r / (s * -3.0f)))) / ((r * s) * (((float) M_PI) * -6.0f)));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.125) * exp(Float32(Float32(0.0) - Float32(r / s)))) / Float32(r * Float32(s * Float32(pi)))) + Float32(Float32(Float32(-0.75) * (Float32(exp(1)) ^ Float32(r / Float32(s * Float32(-3.0))))) / Float32(Float32(r * s) * Float32(Float32(pi) * Float32(-6.0)))))
end

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.125) * exp((single(0.0) - (r / s)))) / (r * (s * single(pi)))) + ((single(-0.75) * (single(2.71828182845904523536) ^ (r / (s * single(-3.0))))) / ((r * s) * (single(pi) * single(-6.0))));
end

Derivation

1. Initial program 99.4%

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

   1. frac-2neg N/A

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

   2. /-lowering-/.f32 N/A

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

   3. *-commutative N/A

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

   4. distribute-rgt-neg-in N/A

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

   5. *-lowering-*.f32 N/A

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

   6. exp-lowering-exp.f32 N/A

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

   7. remove-double-neg N/A

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

   8. frac-2neg N/A

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

   9. /-lowering-/.f32 N/A

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

   10. *-commutative N/A

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

   11. distribute-rgt-neg-in N/A

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

   12. *-lowering-*.f32 N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. associate-*l* N/A

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

   16. *-commutative N/A

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

   17. distribute-rgt-neg-in N/A

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

   18. *-lowering-*.f32 N/A

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

4. Applied egg-rr 99.4%

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

   1. clear-num N/A

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

   2. div-inv N/A

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

   3. clear-num N/A

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

   4. exp-prod N/A

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

   5. pow-lowering-pow.f32 N/A

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

   6. exp-1-e N/A

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

   7. E-lowering-E.f32 N/A

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

   8. /-lowering-/.f32 N/A

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

   9. *-lowering-*.f32 99.5

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

6. Applied egg-rr 99.5%

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

7. Taylor expanded in r around inf

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

   1. associate-*r/ N/A

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

   2. /-lowering-/.f32 N/A

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

   3. *-lowering-*.f32 N/A

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

   4. exp-lowering-exp.f32 N/A

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

   5. neg-mul-1 N/A

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

   6. neg-sub0 N/A

      \[\leadsto \frac{\frac{1}{8} \cdot e^{\color{blue}{0 - \frac{r}{s}}}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} + \frac{{\mathsf{E}\left(\right)}^{\left(\frac{r}{s \cdot -3}\right)} \cdot \frac{-3}{4}}{\left(r \cdot s\right) \cdot \left(\mathsf{PI}\left(\right) \cdot -6\right)} \]

   7. --lowering--.f32 N/A

      \[\leadsto \frac{\frac{1}{8} \cdot e^{\color{blue}{0 - \frac{r}{s}}}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} + \frac{{\mathsf{E}\left(\right)}^{\left(\frac{r}{s \cdot -3}\right)} \cdot \frac{-3}{4}}{\left(r \cdot s\right) \cdot \left(\mathsf{PI}\left(\right) \cdot -6\right)} \]

   8. /-lowering-/.f32 N/A

      \[\leadsto \frac{\frac{1}{8} \cdot e^{0 - \color{blue}{\frac{r}{s}}}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} + \frac{{\mathsf{E}\left(\right)}^{\left(\frac{r}{s \cdot -3}\right)} \cdot \frac{-3}{4}}{\left(r \cdot s\right) \cdot \left(\mathsf{PI}\left(\right) \cdot -6\right)} \]

   9. *-lowering-*.f32 N/A

      \[\leadsto \frac{\frac{1}{8} \cdot e^{0 - \frac{r}{s}}}{\color{blue}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}} + \frac{{\mathsf{E}\left(\right)}^{\left(\frac{r}{s \cdot -3}\right)} \cdot \frac{-3}{4}}{\left(r \cdot s\right) \cdot \left(\mathsf{PI}\left(\right) \cdot -6\right)} \]

   10. *-lowering-*.f32 N/A

       \[\leadsto \frac{\frac{1}{8} \cdot e^{0 - \frac{r}{s}}}{r \cdot \color{blue}{\left(s \cdot \mathsf{PI}\left(\right)\right)}} + \frac{{\mathsf{E}\left(\right)}^{\left(\frac{r}{s \cdot -3}\right)} \cdot \frac{-3}{4}}{\left(r \cdot s\right) \cdot \left(\mathsf{PI}\left(\right) \cdot -6\right)} \]

   11. PI-lowering-PI.f32 99.5

       \[\leadsto \frac{0.125 \cdot e^{0 - \frac{r}{s}}}{r \cdot \left(s \cdot \color{blue}{\pi}\right)} + \frac{{e}^{\left(\frac{r}{s \cdot -3}\right)} \cdot -0.75}{\left(r \cdot s\right) \cdot \left(\pi \cdot -6\right)} \]

9. Simplified 99.5%

   \[\leadsto \color{blue}{\frac{0.125 \cdot e^{0 - \frac{r}{s}}}{r \cdot \left(s \cdot \pi\right)}} + \frac{{e}^{\left(\frac{r}{s \cdot -3}\right)} \cdot -0.75}{\left(r \cdot s\right) \cdot \left(\pi \cdot -6\right)} \]

10. Final simplification 99.5%

    \[\leadsto \frac{0.125 \cdot e^{0 - \frac{r}{s}}}{r \cdot \left(s \cdot \pi\right)} + \frac{-0.75 \cdot {e}^{\left(\frac{r}{s \cdot -3}\right)}}{\left(r \cdot s\right) \cdot \left(\pi \cdot -6\right)} \]
11. Add Preprocessing

Alternative 3: 99.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{0.125}{s \cdot \pi} \cdot \left(\frac{{e}^{\left(\frac{r}{s \cdot -3}\right)}}{r} + \frac{e^{0 - \frac{r}{s}}}{r}\right) \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (*
  (/ 0.125 (* s PI))
  (+ (/ (pow E (/ r (* s -3.0))) r) (/ (exp (- 0.0 (/ r s))) r))))

C

float code(float s, float r) {
	return (0.125f / (s * ((float) M_PI))) * ((powf(((float) M_E), (r / (s * -3.0f))) / r) + (expf((0.0f - (r / s))) / r));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(0.125) / Float32(s * Float32(pi))) * Float32(Float32((Float32(exp(1)) ^ Float32(r / Float32(s * Float32(-3.0)))) / r) + Float32(exp(Float32(Float32(0.0) - Float32(r / s))) / r)))
end

MATLAB

function tmp = code(s, r)
	tmp = (single(0.125) / (s * single(pi))) * (((single(2.71828182845904523536) ^ (r / (s * single(-3.0)))) / r) + (exp((single(0.0) - (r / s))) / r));
end

Derivation

1. Initial program 99.4%

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

   1. +-commutative N/A

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

   2. times-frac N/A

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

   3. times-frac N/A

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

   4. associate-*l* N/A

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

   5. associate-/r* N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. associate-/r* N/A

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

   9. associate-*l* N/A

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

   10. distribute-lft-out N/A

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

   11. *-lowering-*.f32 N/A

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

4. Applied egg-rr 99.4%

   \[\leadsto \color{blue}{\frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{s \cdot -3}}}{r} + \frac{e^{0 - \frac{r}{s}}}{r}\right)} \]
5. Step-by-step derivation

   1. clear-num N/A

      \[\leadsto \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \cdot \left(\frac{e^{\color{blue}{\frac{1}{\frac{s \cdot -3}{r}}}}}{r} + \frac{e^{0 - \frac{r}{s}}}{r}\right) \]

   2. div-inv N/A

      \[\leadsto \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \cdot \left(\frac{e^{\color{blue}{1 \cdot \frac{1}{\frac{s \cdot -3}{r}}}}}{r} + \frac{e^{0 - \frac{r}{s}}}{r}\right) \]

   3. clear-num N/A

      \[\leadsto \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \cdot \left(\frac{e^{1 \cdot \color{blue}{\frac{r}{s \cdot -3}}}}{r} + \frac{e^{0 - \frac{r}{s}}}{r}\right) \]

   4. exp-prod N/A

      \[\leadsto \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \cdot \left(\frac{\color{blue}{{\left(e^{1}\right)}^{\left(\frac{r}{s \cdot -3}\right)}}}{r} + \frac{e^{0 - \frac{r}{s}}}{r}\right) \]

   5. pow-lowering-pow.f32 N/A

      \[\leadsto \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \cdot \left(\frac{\color{blue}{{\left(e^{1}\right)}^{\left(\frac{r}{s \cdot -3}\right)}}}{r} + \frac{e^{0 - \frac{r}{s}}}{r}\right) \]

   6. exp-1-e N/A

      \[\leadsto \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \cdot \left(\frac{{\color{blue}{\mathsf{E}\left(\right)}}^{\left(\frac{r}{s \cdot -3}\right)}}{r} + \frac{e^{0 - \frac{r}{s}}}{r}\right) \]

   7. E-lowering-E.f32 N/A

      \[\leadsto \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \cdot \left(\frac{{\color{blue}{\mathsf{E}\left(\right)}}^{\left(\frac{r}{s \cdot -3}\right)}}{r} + \frac{e^{0 - \frac{r}{s}}}{r}\right) \]

   8. /-lowering-/.f32 N/A

      \[\leadsto \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \cdot \left(\frac{{\mathsf{E}\left(\right)}^{\color{blue}{\left(\frac{r}{s \cdot -3}\right)}}}{r} + \frac{e^{0 - \frac{r}{s}}}{r}\right) \]

   9. *-lowering-*.f32 99.5

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{{e}^{\left(\frac{r}{\color{blue}{s \cdot -3}}\right)}}{r} + \frac{e^{0 - \frac{r}{s}}}{r}\right) \]

6. Applied egg-rr 99.5%

   \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{\color{blue}{{e}^{\left(\frac{r}{s \cdot -3}\right)}}}{r} + \frac{e^{0 - \frac{r}{s}}}{r}\right) \]
7. Add Preprocessing

Alternative 4: 99.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{0.125 \cdot \left(e^{0 - \frac{r}{s}} + {e}^{\left(\frac{r}{s \cdot -3}\right)}\right)}{r \cdot \left(s \cdot \pi\right)} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (/
  (* 0.125 (+ (exp (- 0.0 (/ r s))) (pow E (/ r (* s -3.0)))))
  (* r (* s PI))))

C

float code(float s, float r) {
	return (0.125f * (expf((0.0f - (r / s))) + powf(((float) M_E), (r / (s * -3.0f))))) / (r * (s * ((float) M_PI)));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(0.125) * Float32(exp(Float32(Float32(0.0) - Float32(r / s))) + (Float32(exp(1)) ^ Float32(r / Float32(s * Float32(-3.0)))))) / Float32(r * Float32(s * Float32(pi))))
end

MATLAB

function tmp = code(s, r)
	tmp = (single(0.125) * (exp((single(0.0) - (r / s))) + (single(2.71828182845904523536) ^ (r / (s * single(-3.0)))))) / (r * (s * single(pi)));
end

Derivation

1. Initial program 99.4%

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

   1. +-commutative N/A

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

   2. times-frac N/A

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

   3. times-frac N/A

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

   4. associate-*l* N/A

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

   5. associate-/r* N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. associate-/r* N/A

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

   9. associate-*l* N/A

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

   10. distribute-lft-out N/A

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

   11. *-lowering-*.f32 N/A

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

4. Applied egg-rr 99.4%

   \[\leadsto \color{blue}{\frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{s \cdot -3}}}{r} + \frac{e^{0 - \frac{r}{s}}}{r}\right)} \]

5. Taylor expanded in r around inf

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

   1. associate-*r/ N/A

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

   2. metadata-eval N/A

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

   3. associate-*r* N/A

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

   4. neg-mul-1 N/A

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

   5. distribute-lft-out N/A

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

   6. /-lowering-/.f32 N/A

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

7. Simplified 99.4%

   \[\leadsto \color{blue}{\frac{0.125 \cdot \left(e^{0 - \frac{r}{s}} + e^{\frac{r}{s} \cdot -0.3333333333333333}\right)}{r \cdot \left(s \cdot \pi\right)}} \]
8. Step-by-step derivation

   1. metadata-eval N/A

      \[\leadsto \frac{\frac{1}{8} \cdot \left(e^{0 - \frac{r}{s}} + e^{\frac{r}{s} \cdot \color{blue}{\frac{1}{-3}}}\right)}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} \]

   2. div-inv N/A

      \[\leadsto \frac{\frac{1}{8} \cdot \left(e^{0 - \frac{r}{s}} + e^{\color{blue}{\frac{\frac{r}{s}}{-3}}}\right)}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} \]

   3. associate-/r* N/A

      \[\leadsto \frac{\frac{1}{8} \cdot \left(e^{0 - \frac{r}{s}} + e^{\color{blue}{\frac{r}{s \cdot -3}}}\right)}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} \]

   4. clear-num N/A

      \[\leadsto \frac{\frac{1}{8} \cdot \left(e^{0 - \frac{r}{s}} + e^{\color{blue}{\frac{1}{\frac{s \cdot -3}{r}}}}\right)}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} \]

   5. div-inv N/A

      \[\leadsto \frac{\frac{1}{8} \cdot \left(e^{0 - \frac{r}{s}} + e^{\color{blue}{1 \cdot \frac{1}{\frac{s \cdot -3}{r}}}}\right)}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} \]

   6. log-E N/A

      \[\leadsto \frac{\frac{1}{8} \cdot \left(e^{0 - \frac{r}{s}} + e^{\color{blue}{\log \mathsf{E}\left(\right)} \cdot \frac{1}{\frac{s \cdot -3}{r}}}\right)}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} \]

   7. clear-num N/A

      \[\leadsto \frac{\frac{1}{8} \cdot \left(e^{0 - \frac{r}{s}} + e^{\log \mathsf{E}\left(\right) \cdot \color{blue}{\frac{r}{s \cdot -3}}}\right)}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} \]

   8. pow-to-exp N/A

      \[\leadsto \frac{\frac{1}{8} \cdot \left(e^{0 - \frac{r}{s}} + \color{blue}{{\mathsf{E}\left(\right)}^{\left(\frac{r}{s \cdot -3}\right)}}\right)}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} \]

   9. pow-lowering-pow.f32 N/A

      \[\leadsto \frac{\frac{1}{8} \cdot \left(e^{0 - \frac{r}{s}} + \color{blue}{{\mathsf{E}\left(\right)}^{\left(\frac{r}{s \cdot -3}\right)}}\right)}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} \]

   10. E-lowering-E.f32 N/A

       \[\leadsto \frac{\frac{1}{8} \cdot \left(e^{0 - \frac{r}{s}} + {\color{blue}{\mathsf{E}\left(\right)}}^{\left(\frac{r}{s \cdot -3}\right)}\right)}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} \]

   11. /-lowering-/.f32 N/A

       \[\leadsto \frac{\frac{1}{8} \cdot \left(e^{0 - \frac{r}{s}} + {\mathsf{E}\left(\right)}^{\color{blue}{\left(\frac{r}{s \cdot -3}\right)}}\right)}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} \]

   12. *-lowering-*.f32 99.4

       \[\leadsto \frac{0.125 \cdot \left(e^{0 - \frac{r}{s}} + {e}^{\left(\frac{r}{\color{blue}{s \cdot -3}}\right)}\right)}{r \cdot \left(s \cdot \pi\right)} \]

9. Applied egg-rr 99.4%

   \[\leadsto \frac{0.125 \cdot \left(e^{0 - \frac{r}{s}} + \color{blue}{{e}^{\left(\frac{r}{s \cdot -3}\right)}}\right)}{r \cdot \left(s \cdot \pi\right)} \]
10. Add Preprocessing

Alternative 5: 99.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{0.125 \cdot \left(e^{0 - \frac{r}{s}} + e^{\frac{r \cdot -0.3333333333333333}{s}}\right)}{r \cdot \left(s \cdot \pi\right)} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (/
  (* 0.125 (+ (exp (- 0.0 (/ r s))) (exp (/ (* r -0.3333333333333333) s))))
  (* r (* s PI))))

C

float code(float s, float r) {
	return (0.125f * (expf((0.0f - (r / s))) + expf(((r * -0.3333333333333333f) / s)))) / (r * (s * ((float) M_PI)));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(0.125) * Float32(exp(Float32(Float32(0.0) - Float32(r / s))) + exp(Float32(Float32(r * Float32(-0.3333333333333333)) / s)))) / Float32(r * Float32(s * Float32(pi))))
end

MATLAB

function tmp = code(s, r)
	tmp = (single(0.125) * (exp((single(0.0) - (r / s))) + exp(((r * single(-0.3333333333333333)) / s)))) / (r * (s * single(pi)));
end

Derivation

1. Initial program 99.4%

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

   1. +-commutative N/A

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

   2. times-frac N/A

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

   3. times-frac N/A

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

   4. associate-*l* N/A

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

   5. associate-/r* N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. associate-/r* N/A

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

   9. associate-*l* N/A

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

   10. distribute-lft-out N/A

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

   11. *-lowering-*.f32 N/A

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

4. Applied egg-rr 99.4%

   \[\leadsto \color{blue}{\frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{s \cdot -3}}}{r} + \frac{e^{0 - \frac{r}{s}}}{r}\right)} \]

5. Taylor expanded in r around inf

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

   1. associate-*r/ N/A

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

   2. metadata-eval N/A

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

   3. associate-*r* N/A

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

   4. neg-mul-1 N/A

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

   5. distribute-lft-out N/A

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

   6. /-lowering-/.f32 N/A

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

7. Simplified 99.4%

   \[\leadsto \color{blue}{\frac{0.125 \cdot \left(e^{0 - \frac{r}{s}} + e^{\frac{r}{s} \cdot -0.3333333333333333}\right)}{r \cdot \left(s \cdot \pi\right)}} \]
8. Step-by-step derivation

   1. associate-*l/ N/A

      \[\leadsto \frac{\frac{1}{8} \cdot \left(e^{0 - \frac{r}{s}} + e^{\color{blue}{\frac{r \cdot \frac{-1}{3}}{s}}}\right)}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} \]

   2. /-lowering-/.f32 N/A

      \[\leadsto \frac{\frac{1}{8} \cdot \left(e^{0 - \frac{r}{s}} + e^{\color{blue}{\frac{r \cdot \frac{-1}{3}}{s}}}\right)}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} \]

   3. *-lowering-*.f32 99.4

      \[\leadsto \frac{0.125 \cdot \left(e^{0 - \frac{r}{s}} + e^{\frac{\color{blue}{r \cdot -0.3333333333333333}}{s}}\right)}{r \cdot \left(s \cdot \pi\right)} \]

9. Applied egg-rr 99.4%

   \[\leadsto \frac{0.125 \cdot \left(e^{0 - \frac{r}{s}} + e^{\color{blue}{\frac{r \cdot -0.3333333333333333}{s}}}\right)}{r \cdot \left(s \cdot \pi\right)} \]
10. Add Preprocessing

Alternative 6: 99.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{0.125 \cdot \left(e^{0 - \frac{r}{s}} + e^{\frac{r}{s} \cdot -0.3333333333333333}\right)}{r \cdot \left(s \cdot \pi\right)} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (/
  (* 0.125 (+ (exp (- 0.0 (/ r s))) (exp (* (/ r s) -0.3333333333333333))))
  (* r (* s PI))))

C

float code(float s, float r) {
	return (0.125f * (expf((0.0f - (r / s))) + expf(((r / s) * -0.3333333333333333f)))) / (r * (s * ((float) M_PI)));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(0.125) * Float32(exp(Float32(Float32(0.0) - Float32(r / s))) + exp(Float32(Float32(r / s) * Float32(-0.3333333333333333))))) / Float32(r * Float32(s * Float32(pi))))
end

MATLAB

function tmp = code(s, r)
	tmp = (single(0.125) * (exp((single(0.0) - (r / s))) + exp(((r / s) * single(-0.3333333333333333))))) / (r * (s * single(pi)));
end

Derivation

1. Initial program 99.4%

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

   1. +-commutative N/A

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

   2. times-frac N/A

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

   3. times-frac N/A

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

   4. associate-*l* N/A

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

   5. associate-/r* N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. associate-/r* N/A

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

   9. associate-*l* N/A

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

   10. distribute-lft-out N/A

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

   11. *-lowering-*.f32 N/A

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

4. Applied egg-rr 99.4%

   \[\leadsto \color{blue}{\frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{s \cdot -3}}}{r} + \frac{e^{0 - \frac{r}{s}}}{r}\right)} \]

5. Taylor expanded in r around inf

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

   1. associate-*r/ N/A

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

   2. metadata-eval N/A

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

   3. associate-*r* N/A

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

   4. neg-mul-1 N/A

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

   5. distribute-lft-out N/A

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

   6. /-lowering-/.f32 N/A

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

7. Simplified 99.4%

   \[\leadsto \color{blue}{\frac{0.125 \cdot \left(e^{0 - \frac{r}{s}} + e^{\frac{r}{s} \cdot -0.3333333333333333}\right)}{r \cdot \left(s \cdot \pi\right)}} \]
8. Step-by-step derivation

   1. exp-lowering-exp.f32 N/A

      \[\leadsto \frac{\frac{1}{8} \cdot \left(\color{blue}{e^{0 - \frac{r}{s}}} + e^{\frac{r}{s} \cdot \frac{-1}{3}}\right)}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} \]

   2. sub0-neg N/A

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

   3. distribute-frac-neg2 N/A

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

   4. /-lowering-/.f32 N/A

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

   5. neg-lowering-neg.f32 99.4

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

9. Applied egg-rr 99.4%

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

10. Final simplification 99.4%

    \[\leadsto \frac{0.125 \cdot \left(e^{0 - \frac{r}{s}} + e^{\frac{r}{s} \cdot -0.3333333333333333}\right)}{r \cdot \left(s \cdot \pi\right)} \]
11. Add Preprocessing

Alternative 7: 97.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \left(e^{0 - \frac{r}{s}} + e^{\frac{r}{s \cdot -3}}\right) \cdot \frac{0.125}{\pi \cdot \left(r \cdot s\right)} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (* (+ (exp (- 0.0 (/ r s))) (exp (/ r (* s -3.0)))) (/ 0.125 (* PI (* r s)))))

C

float code(float s, float r) {
	return (expf((0.0f - (r / s))) + expf((r / (s * -3.0f)))) * (0.125f / (((float) M_PI) * (r * s)));
}

Julia

function code(s, r)
	return Float32(Float32(exp(Float32(Float32(0.0) - Float32(r / s))) + exp(Float32(r / Float32(s * Float32(-3.0))))) * Float32(Float32(0.125) / Float32(Float32(pi) * Float32(r * s))))
end

MATLAB

function tmp = code(s, r)
	tmp = (exp((single(0.0) - (r / s))) + exp((r / (s * single(-3.0))))) * (single(0.125) / (single(pi) * (r * s)));
end

Derivation

1. Initial program 99.4%

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

   1. +-commutative N/A

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

   2. times-frac N/A

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

   3. times-frac N/A

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

   4. associate-*l* N/A

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

   5. associate-/r* N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. associate-/r* N/A

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

   9. associate-*l* N/A

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

   10. distribute-lft-out N/A

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

   11. *-lowering-*.f32 N/A

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

4. Applied egg-rr 99.4%

   \[\leadsto \color{blue}{\frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{s \cdot -3}}}{r} + \frac{e^{0 - \frac{r}{s}}}{r}\right)} \]

5. Taylor expanded in r around inf

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

   1. associate-*r/ N/A

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

   2. metadata-eval N/A

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

   3. associate-*r* N/A

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

   4. neg-mul-1 N/A

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

   5. distribute-lft-out N/A

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

   6. /-lowering-/.f32 N/A

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

7. Simplified 99.4%

   \[\leadsto \color{blue}{\frac{0.125 \cdot \left(e^{0 - \frac{r}{s}} + e^{\frac{r}{s} \cdot -0.3333333333333333}\right)}{r \cdot \left(s \cdot \pi\right)}} \]
8. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\left(e^{0 - \frac{r}{s}} + e^{\frac{r}{s} \cdot \frac{-1}{3}}\right) \cdot \frac{1}{8}}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} \]

   2. associate-*r* N/A

      \[\leadsto \frac{\left(e^{0 - \frac{r}{s}} + e^{\frac{r}{s} \cdot \frac{-1}{3}}\right) \cdot \frac{1}{8}}{\color{blue}{\left(r \cdot s\right) \cdot \mathsf{PI}\left(\right)}} \]

   3. associate-/l* N/A

      \[\leadsto \color{blue}{\left(e^{0 - \frac{r}{s}} + e^{\frac{r}{s} \cdot \frac{-1}{3}}\right) \cdot \frac{\frac{1}{8}}{\left(r \cdot s\right) \cdot \mathsf{PI}\left(\right)}} \]

   4. *-lowering-*.f32 N/A

      \[\leadsto \color{blue}{\left(e^{0 - \frac{r}{s}} + e^{\frac{r}{s} \cdot \frac{-1}{3}}\right) \cdot \frac{\frac{1}{8}}{\left(r \cdot s\right) \cdot \mathsf{PI}\left(\right)}} \]

9. Applied egg-rr 96.6%

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

10. Final simplification 96.6%

    \[\leadsto \left(e^{0 - \frac{r}{s}} + e^{\frac{r}{s \cdot -3}}\right) \cdot \frac{0.125}{\pi \cdot \left(r \cdot s\right)} \]
11. Add Preprocessing

Alternative 8: 10.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{0 - \frac{r}{s}}}{r} + \frac{\mathsf{fma}\left(r, \mathsf{fma}\left(r, \frac{0.05555555555555555}{s \cdot s}, \frac{-0.3333333333333333}{s}\right), 1\right)}{r}\right) \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (*
  (/ 0.125 (* s PI))
  (+
   (/ (exp (- 0.0 (/ r s))) r)
   (/
    (fma
     r
     (fma r (/ 0.05555555555555555 (* s s)) (/ -0.3333333333333333 s))
     1.0)
    r))))

C

float code(float s, float r) {
	return (0.125f / (s * ((float) M_PI))) * ((expf((0.0f - (r / s))) / r) + (fmaf(r, fmaf(r, (0.05555555555555555f / (s * s)), (-0.3333333333333333f / s)), 1.0f) / r));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(0.125) / Float32(s * Float32(pi))) * Float32(Float32(exp(Float32(Float32(0.0) - Float32(r / s))) / r) + Float32(fma(r, fma(r, Float32(Float32(0.05555555555555555) / Float32(s * s)), Float32(Float32(-0.3333333333333333) / s)), Float32(1.0)) / r)))
end

Derivation

1. Initial program 99.4%

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

   1. +-commutative N/A

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

   2. times-frac N/A

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

   3. times-frac N/A

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

   4. associate-*l* N/A

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

   5. associate-/r* N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. associate-/r* N/A

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

   9. associate-*l* N/A

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

   10. distribute-lft-out N/A

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

   11. *-lowering-*.f32 N/A

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

4. Applied egg-rr 99.4%

   \[\leadsto \color{blue}{\frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{s \cdot -3}}}{r} + \frac{e^{0 - \frac{r}{s}}}{r}\right)} \]

5. Taylor expanded in r around 0

   \[\leadsto \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \cdot \left(\frac{\color{blue}{1 + r \cdot \left(\frac{1}{18} \cdot \frac{r}{{s}^{2}} - \frac{1}{3} \cdot \frac{1}{s}\right)}}{r} + \frac{e^{0 - \frac{r}{s}}}{r}\right) \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \cdot \left(\frac{\color{blue}{r \cdot \left(\frac{1}{18} \cdot \frac{r}{{s}^{2}} - \frac{1}{3} \cdot \frac{1}{s}\right) + 1}}{r} + \frac{e^{0 - \frac{r}{s}}}{r}\right) \]

   2. accelerator-lowering-fma.f32 N/A

      \[\leadsto \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \cdot \left(\frac{\color{blue}{\mathsf{fma}\left(r, \frac{1}{18} \cdot \frac{r}{{s}^{2}} - \frac{1}{3} \cdot \frac{1}{s}, 1\right)}}{r} + \frac{e^{0 - \frac{r}{s}}}{r}\right) \]

   3. sub-neg N/A

      \[\leadsto \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \cdot \left(\frac{\mathsf{fma}\left(r, \color{blue}{\frac{1}{18} \cdot \frac{r}{{s}^{2}} + \left(\mathsf{neg}\left(\frac{1}{3} \cdot \frac{1}{s}\right)\right)}, 1\right)}{r} + \frac{e^{0 - \frac{r}{s}}}{r}\right) \]

   4. associate-*r/ N/A

      \[\leadsto \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \cdot \left(\frac{\mathsf{fma}\left(r, \color{blue}{\frac{\frac{1}{18} \cdot r}{{s}^{2}}} + \left(\mathsf{neg}\left(\frac{1}{3} \cdot \frac{1}{s}\right)\right), 1\right)}{r} + \frac{e^{0 - \frac{r}{s}}}{r}\right) \]

   5. *-commutative N/A

      \[\leadsto \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \cdot \left(\frac{\mathsf{fma}\left(r, \frac{\color{blue}{r \cdot \frac{1}{18}}}{{s}^{2}} + \left(\mathsf{neg}\left(\frac{1}{3} \cdot \frac{1}{s}\right)\right), 1\right)}{r} + \frac{e^{0 - \frac{r}{s}}}{r}\right) \]

   6. associate-*r/ N/A

      \[\leadsto \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \cdot \left(\frac{\mathsf{fma}\left(r, \color{blue}{r \cdot \frac{\frac{1}{18}}{{s}^{2}}} + \left(\mathsf{neg}\left(\frac{1}{3} \cdot \frac{1}{s}\right)\right), 1\right)}{r} + \frac{e^{0 - \frac{r}{s}}}{r}\right) \]

   7. metadata-eval N/A

      \[\leadsto \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \cdot \left(\frac{\mathsf{fma}\left(r, r \cdot \frac{\color{blue}{\frac{1}{18} \cdot 1}}{{s}^{2}} + \left(\mathsf{neg}\left(\frac{1}{3} \cdot \frac{1}{s}\right)\right), 1\right)}{r} + \frac{e^{0 - \frac{r}{s}}}{r}\right) \]

   8. associate-*r/ N/A

      \[\leadsto \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \cdot \left(\frac{\mathsf{fma}\left(r, r \cdot \color{blue}{\left(\frac{1}{18} \cdot \frac{1}{{s}^{2}}\right)} + \left(\mathsf{neg}\left(\frac{1}{3} \cdot \frac{1}{s}\right)\right), 1\right)}{r} + \frac{e^{0 - \frac{r}{s}}}{r}\right) \]

   9. accelerator-lowering-fma.f32 N/A

      \[\leadsto \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \cdot \left(\frac{\mathsf{fma}\left(r, \color{blue}{\mathsf{fma}\left(r, \frac{1}{18} \cdot \frac{1}{{s}^{2}}, \mathsf{neg}\left(\frac{1}{3} \cdot \frac{1}{s}\right)\right)}, 1\right)}{r} + \frac{e^{0 - \frac{r}{s}}}{r}\right) \]

   10. associate-*r/ N/A

       \[\leadsto \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \cdot \left(\frac{\mathsf{fma}\left(r, \mathsf{fma}\left(r, \color{blue}{\frac{\frac{1}{18} \cdot 1}{{s}^{2}}}, \mathsf{neg}\left(\frac{1}{3} \cdot \frac{1}{s}\right)\right), 1\right)}{r} + \frac{e^{0 - \frac{r}{s}}}{r}\right) \]

   11. metadata-eval N/A

       \[\leadsto \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \cdot \left(\frac{\mathsf{fma}\left(r, \mathsf{fma}\left(r, \frac{\color{blue}{\frac{1}{18}}}{{s}^{2}}, \mathsf{neg}\left(\frac{1}{3} \cdot \frac{1}{s}\right)\right), 1\right)}{r} + \frac{e^{0 - \frac{r}{s}}}{r}\right) \]

   12. /-lowering-/.f32 N/A

       \[\leadsto \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \cdot \left(\frac{\mathsf{fma}\left(r, \mathsf{fma}\left(r, \color{blue}{\frac{\frac{1}{18}}{{s}^{2}}}, \mathsf{neg}\left(\frac{1}{3} \cdot \frac{1}{s}\right)\right), 1\right)}{r} + \frac{e^{0 - \frac{r}{s}}}{r}\right) \]

   13. unpow2 N/A

       \[\leadsto \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \cdot \left(\frac{\mathsf{fma}\left(r, \mathsf{fma}\left(r, \frac{\frac{1}{18}}{\color{blue}{s \cdot s}}, \mathsf{neg}\left(\frac{1}{3} \cdot \frac{1}{s}\right)\right), 1\right)}{r} + \frac{e^{0 - \frac{r}{s}}}{r}\right) \]

   14. *-lowering-*.f32 N/A

       \[\leadsto \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \cdot \left(\frac{\mathsf{fma}\left(r, \mathsf{fma}\left(r, \frac{\frac{1}{18}}{\color{blue}{s \cdot s}}, \mathsf{neg}\left(\frac{1}{3} \cdot \frac{1}{s}\right)\right), 1\right)}{r} + \frac{e^{0 - \frac{r}{s}}}{r}\right) \]

   15. associate-*r/ N/A

       \[\leadsto \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \cdot \left(\frac{\mathsf{fma}\left(r, \mathsf{fma}\left(r, \frac{\frac{1}{18}}{s \cdot s}, \mathsf{neg}\left(\color{blue}{\frac{\frac{1}{3} \cdot 1}{s}}\right)\right), 1\right)}{r} + \frac{e^{0 - \frac{r}{s}}}{r}\right) \]

   16. metadata-eval N/A

       \[\leadsto \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \cdot \left(\frac{\mathsf{fma}\left(r, \mathsf{fma}\left(r, \frac{\frac{1}{18}}{s \cdot s}, \mathsf{neg}\left(\frac{\color{blue}{\frac{1}{3}}}{s}\right)\right), 1\right)}{r} + \frac{e^{0 - \frac{r}{s}}}{r}\right) \]

   17. distribute-neg-frac N/A

       \[\leadsto \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \cdot \left(\frac{\mathsf{fma}\left(r, \mathsf{fma}\left(r, \frac{\frac{1}{18}}{s \cdot s}, \color{blue}{\frac{\mathsf{neg}\left(\frac{1}{3}\right)}{s}}\right), 1\right)}{r} + \frac{e^{0 - \frac{r}{s}}}{r}\right) \]

   18. metadata-eval N/A

       \[\leadsto \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \cdot \left(\frac{\mathsf{fma}\left(r, \mathsf{fma}\left(r, \frac{\frac{1}{18}}{s \cdot s}, \frac{\color{blue}{\frac{-1}{3}}}{s}\right), 1\right)}{r} + \frac{e^{0 - \frac{r}{s}}}{r}\right) \]

   19. /-lowering-/.f32 7.7

       \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{\mathsf{fma}\left(r, \mathsf{fma}\left(r, \frac{0.05555555555555555}{s \cdot s}, \color{blue}{\frac{-0.3333333333333333}{s}}\right), 1\right)}{r} + \frac{e^{0 - \frac{r}{s}}}{r}\right) \]

7. Simplified 7.7%

   \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{\color{blue}{\mathsf{fma}\left(r, \mathsf{fma}\left(r, \frac{0.05555555555555555}{s \cdot s}, \frac{-0.3333333333333333}{s}\right), 1\right)}}{r} + \frac{e^{0 - \frac{r}{s}}}{r}\right) \]

8. Final simplification 7.7%

   \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{0 - \frac{r}{s}}}{r} + \frac{\mathsf{fma}\left(r, \mathsf{fma}\left(r, \frac{0.05555555555555555}{s \cdot s}, \frac{-0.3333333333333333}{s}\right), 1\right)}{r}\right) \]
9. Add Preprocessing

Alternative 9: 10.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \frac{0.125 \cdot \left(e^{0 - \frac{r}{s}} + \mathsf{fma}\left(r, \mathsf{fma}\left(r, \frac{0.05555555555555555}{s \cdot s}, \frac{-0.3333333333333333}{s}\right), 1\right)\right)}{r \cdot \left(s \cdot \pi\right)} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (/
  (*
   0.125
   (+
    (exp (- 0.0 (/ r s)))
    (fma
     r
     (fma r (/ 0.05555555555555555 (* s s)) (/ -0.3333333333333333 s))
     1.0)))
  (* r (* s PI))))

C

float code(float s, float r) {
	return (0.125f * (expf((0.0f - (r / s))) + fmaf(r, fmaf(r, (0.05555555555555555f / (s * s)), (-0.3333333333333333f / s)), 1.0f))) / (r * (s * ((float) M_PI)));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(0.125) * Float32(exp(Float32(Float32(0.0) - Float32(r / s))) + fma(r, fma(r, Float32(Float32(0.05555555555555555) / Float32(s * s)), Float32(Float32(-0.3333333333333333) / s)), Float32(1.0)))) / Float32(r * Float32(s * Float32(pi))))
end

Derivation

1. Initial program 99.4%

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

   1. +-commutative N/A

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

   2. times-frac N/A

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

   3. times-frac N/A

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

   4. associate-*l* N/A

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

   5. associate-/r* N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. associate-/r* N/A

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

   9. associate-*l* N/A

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

   10. distribute-lft-out N/A

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

   11. *-lowering-*.f32 N/A

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

4. Applied egg-rr 99.4%

   \[\leadsto \color{blue}{\frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{s \cdot -3}}}{r} + \frac{e^{0 - \frac{r}{s}}}{r}\right)} \]

5. Taylor expanded in r around inf

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

   1. associate-*r/ N/A

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

   2. metadata-eval N/A

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

   3. associate-*r* N/A

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

   4. neg-mul-1 N/A

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

   5. distribute-lft-out N/A

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

   6. /-lowering-/.f32 N/A

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

7. Simplified 99.4%

   \[\leadsto \color{blue}{\frac{0.125 \cdot \left(e^{0 - \frac{r}{s}} + e^{\frac{r}{s} \cdot -0.3333333333333333}\right)}{r \cdot \left(s \cdot \pi\right)}} \]

8. Taylor expanded in r around 0

   \[\leadsto \frac{\frac{1}{8} \cdot \left(e^{0 - \frac{r}{s}} + \color{blue}{\left(1 + r \cdot \left(\frac{1}{18} \cdot \frac{r}{{s}^{2}} - \frac{1}{3} \cdot \frac{1}{s}\right)\right)}\right)}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} \]
9. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\frac{1}{8} \cdot \left(e^{0 - \frac{r}{s}} + \color{blue}{\left(r \cdot \left(\frac{1}{18} \cdot \frac{r}{{s}^{2}} - \frac{1}{3} \cdot \frac{1}{s}\right) + 1\right)}\right)}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} \]

   2. accelerator-lowering-fma.f32 N/A

      \[\leadsto \frac{\frac{1}{8} \cdot \left(e^{0 - \frac{r}{s}} + \color{blue}{\mathsf{fma}\left(r, \frac{1}{18} \cdot \frac{r}{{s}^{2}} - \frac{1}{3} \cdot \frac{1}{s}, 1\right)}\right)}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} \]

   3. sub-neg N/A

      \[\leadsto \frac{\frac{1}{8} \cdot \left(e^{0 - \frac{r}{s}} + \mathsf{fma}\left(r, \color{blue}{\frac{1}{18} \cdot \frac{r}{{s}^{2}} + \left(\mathsf{neg}\left(\frac{1}{3} \cdot \frac{1}{s}\right)\right)}, 1\right)\right)}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} \]

   4. *-commutative N/A

      \[\leadsto \frac{\frac{1}{8} \cdot \left(e^{0 - \frac{r}{s}} + \mathsf{fma}\left(r, \color{blue}{\frac{r}{{s}^{2}} \cdot \frac{1}{18}} + \left(\mathsf{neg}\left(\frac{1}{3} \cdot \frac{1}{s}\right)\right), 1\right)\right)}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} \]

   5. associate-*l/ N/A

      \[\leadsto \frac{\frac{1}{8} \cdot \left(e^{0 - \frac{r}{s}} + \mathsf{fma}\left(r, \color{blue}{\frac{r \cdot \frac{1}{18}}{{s}^{2}}} + \left(\mathsf{neg}\left(\frac{1}{3} \cdot \frac{1}{s}\right)\right), 1\right)\right)}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} \]

   6. associate-*r/ N/A

      \[\leadsto \frac{\frac{1}{8} \cdot \left(e^{0 - \frac{r}{s}} + \mathsf{fma}\left(r, \color{blue}{r \cdot \frac{\frac{1}{18}}{{s}^{2}}} + \left(\mathsf{neg}\left(\frac{1}{3} \cdot \frac{1}{s}\right)\right), 1\right)\right)}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} \]

   7. metadata-eval N/A

      \[\leadsto \frac{\frac{1}{8} \cdot \left(e^{0 - \frac{r}{s}} + \mathsf{fma}\left(r, r \cdot \frac{\color{blue}{\frac{1}{18} \cdot 1}}{{s}^{2}} + \left(\mathsf{neg}\left(\frac{1}{3} \cdot \frac{1}{s}\right)\right), 1\right)\right)}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} \]

   8. associate-*r/ N/A

      \[\leadsto \frac{\frac{1}{8} \cdot \left(e^{0 - \frac{r}{s}} + \mathsf{fma}\left(r, r \cdot \color{blue}{\left(\frac{1}{18} \cdot \frac{1}{{s}^{2}}\right)} + \left(\mathsf{neg}\left(\frac{1}{3} \cdot \frac{1}{s}\right)\right), 1\right)\right)}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} \]

   9. accelerator-lowering-fma.f32 N/A

      \[\leadsto \frac{\frac{1}{8} \cdot \left(e^{0 - \frac{r}{s}} + \mathsf{fma}\left(r, \color{blue}{\mathsf{fma}\left(r, \frac{1}{18} \cdot \frac{1}{{s}^{2}}, \mathsf{neg}\left(\frac{1}{3} \cdot \frac{1}{s}\right)\right)}, 1\right)\right)}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} \]

   10. associate-*r/ N/A

       \[\leadsto \frac{\frac{1}{8} \cdot \left(e^{0 - \frac{r}{s}} + \mathsf{fma}\left(r, \mathsf{fma}\left(r, \color{blue}{\frac{\frac{1}{18} \cdot 1}{{s}^{2}}}, \mathsf{neg}\left(\frac{1}{3} \cdot \frac{1}{s}\right)\right), 1\right)\right)}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} \]

   11. metadata-eval N/A

       \[\leadsto \frac{\frac{1}{8} \cdot \left(e^{0 - \frac{r}{s}} + \mathsf{fma}\left(r, \mathsf{fma}\left(r, \frac{\color{blue}{\frac{1}{18}}}{{s}^{2}}, \mathsf{neg}\left(\frac{1}{3} \cdot \frac{1}{s}\right)\right), 1\right)\right)}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} \]

   12. /-lowering-/.f32 N/A

       \[\leadsto \frac{\frac{1}{8} \cdot \left(e^{0 - \frac{r}{s}} + \mathsf{fma}\left(r, \mathsf{fma}\left(r, \color{blue}{\frac{\frac{1}{18}}{{s}^{2}}}, \mathsf{neg}\left(\frac{1}{3} \cdot \frac{1}{s}\right)\right), 1\right)\right)}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} \]

   13. unpow2 N/A

       \[\leadsto \frac{\frac{1}{8} \cdot \left(e^{0 - \frac{r}{s}} + \mathsf{fma}\left(r, \mathsf{fma}\left(r, \frac{\frac{1}{18}}{\color{blue}{s \cdot s}}, \mathsf{neg}\left(\frac{1}{3} \cdot \frac{1}{s}\right)\right), 1\right)\right)}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} \]

   14. *-lowering-*.f32 N/A

       \[\leadsto \frac{\frac{1}{8} \cdot \left(e^{0 - \frac{r}{s}} + \mathsf{fma}\left(r, \mathsf{fma}\left(r, \frac{\frac{1}{18}}{\color{blue}{s \cdot s}}, \mathsf{neg}\left(\frac{1}{3} \cdot \frac{1}{s}\right)\right), 1\right)\right)}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} \]

   15. associate-*r/ N/A

       \[\leadsto \frac{\frac{1}{8} \cdot \left(e^{0 - \frac{r}{s}} + \mathsf{fma}\left(r, \mathsf{fma}\left(r, \frac{\frac{1}{18}}{s \cdot s}, \mathsf{neg}\left(\color{blue}{\frac{\frac{1}{3} \cdot 1}{s}}\right)\right), 1\right)\right)}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} \]

   16. metadata-eval N/A

       \[\leadsto \frac{\frac{1}{8} \cdot \left(e^{0 - \frac{r}{s}} + \mathsf{fma}\left(r, \mathsf{fma}\left(r, \frac{\frac{1}{18}}{s \cdot s}, \mathsf{neg}\left(\frac{\color{blue}{\frac{1}{3}}}{s}\right)\right), 1\right)\right)}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} \]

   17. distribute-neg-frac N/A

       \[\leadsto \frac{\frac{1}{8} \cdot \left(e^{0 - \frac{r}{s}} + \mathsf{fma}\left(r, \mathsf{fma}\left(r, \frac{\frac{1}{18}}{s \cdot s}, \color{blue}{\frac{\mathsf{neg}\left(\frac{1}{3}\right)}{s}}\right), 1\right)\right)}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} \]

   18. metadata-eval N/A

       \[\leadsto \frac{\frac{1}{8} \cdot \left(e^{0 - \frac{r}{s}} + \mathsf{fma}\left(r, \mathsf{fma}\left(r, \frac{\frac{1}{18}}{s \cdot s}, \frac{\color{blue}{\frac{-1}{3}}}{s}\right), 1\right)\right)}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} \]

   19. /-lowering-/.f32 7.7

       \[\leadsto \frac{0.125 \cdot \left(e^{0 - \frac{r}{s}} + \mathsf{fma}\left(r, \mathsf{fma}\left(r, \frac{0.05555555555555555}{s \cdot s}, \color{blue}{\frac{-0.3333333333333333}{s}}\right), 1\right)\right)}{r \cdot \left(s \cdot \pi\right)} \]

10. Simplified 7.7%

    \[\leadsto \frac{0.125 \cdot \left(e^{0 - \frac{r}{s}} + \color{blue}{\mathsf{fma}\left(r, \mathsf{fma}\left(r, \frac{0.05555555555555555}{s \cdot s}, \frac{-0.3333333333333333}{s}\right), 1\right)}\right)}{r \cdot \left(s \cdot \pi\right)} \]
11. Add Preprocessing

Alternative 10: 9.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{0 - \frac{r}{s}}}{r} + \frac{1}{r}\right) \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (* (/ 0.125 (* s PI)) (+ (/ (exp (- 0.0 (/ r s))) r) (/ 1.0 r))))

C

float code(float s, float r) {
	return (0.125f / (s * ((float) M_PI))) * ((expf((0.0f - (r / s))) / r) + (1.0f / r));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(0.125) / Float32(s * Float32(pi))) * Float32(Float32(exp(Float32(Float32(0.0) - Float32(r / s))) / r) + Float32(Float32(1.0) / r)))
end

MATLAB

function tmp = code(s, r)
	tmp = (single(0.125) / (s * single(pi))) * ((exp((single(0.0) - (r / s))) / r) + (single(1.0) / r));
end

Derivation

1. Initial program 99.4%

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

   1. +-commutative N/A

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

   2. times-frac N/A

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

   3. times-frac N/A

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

   4. associate-*l* N/A

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

   5. associate-/r* N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. associate-/r* N/A

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

   9. associate-*l* N/A

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

   10. distribute-lft-out N/A

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

   11. *-lowering-*.f32 N/A

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

4. Applied egg-rr 99.4%

   \[\leadsto \color{blue}{\frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{s \cdot -3}}}{r} + \frac{e^{0 - \frac{r}{s}}}{r}\right)} \]

5. Taylor expanded in r around 0

   \[\leadsto \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \cdot \left(\frac{\color{blue}{1}}{r} + \frac{e^{0 - \frac{r}{s}}}{r}\right) \]
6. Step-by-step derivation

7. Simplified 7.5%

   \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{\color{blue}{1}}{r} + \frac{e^{0 - \frac{r}{s}}}{r}\right) \]

8. Final simplification 7.5%

   \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{0 - \frac{r}{s}}}{r} + \frac{1}{r}\right) \]
9. Add Preprocessing

Alternative 11: 9.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \frac{0.125 \cdot \left(e^{0 - \frac{r}{s}} + 1\right)}{r \cdot \left(s \cdot \pi\right)} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (/ (* 0.125 (+ (exp (- 0.0 (/ r s))) 1.0)) (* r (* s PI))))

C

float code(float s, float r) {
	return (0.125f * (expf((0.0f - (r / s))) + 1.0f)) / (r * (s * ((float) M_PI)));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(0.125) * Float32(exp(Float32(Float32(0.0) - Float32(r / s))) + Float32(1.0))) / Float32(r * Float32(s * Float32(pi))))
end

MATLAB

function tmp = code(s, r)
	tmp = (single(0.125) * (exp((single(0.0) - (r / s))) + single(1.0))) / (r * (s * single(pi)));
end

Derivation

1. Initial program 99.4%

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

   1. +-commutative N/A

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

   2. times-frac N/A

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

   3. times-frac N/A

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

   4. associate-*l* N/A

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

   5. associate-/r* N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. associate-/r* N/A

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

   9. associate-*l* N/A

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

   10. distribute-lft-out N/A

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

   11. *-lowering-*.f32 N/A

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

4. Applied egg-rr 99.4%

   \[\leadsto \color{blue}{\frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{s \cdot -3}}}{r} + \frac{e^{0 - \frac{r}{s}}}{r}\right)} \]

5. Taylor expanded in r around inf

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

   1. associate-*r/ N/A

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

   2. metadata-eval N/A

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

   3. associate-*r* N/A

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

   4. neg-mul-1 N/A

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

   5. distribute-lft-out N/A

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

   6. /-lowering-/.f32 N/A

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

7. Simplified 99.4%

   \[\leadsto \color{blue}{\frac{0.125 \cdot \left(e^{0 - \frac{r}{s}} + e^{\frac{r}{s} \cdot -0.3333333333333333}\right)}{r \cdot \left(s \cdot \pi\right)}} \]

8. Taylor expanded in r around 0

   \[\leadsto \frac{\frac{1}{8} \cdot \left(e^{0 - \frac{r}{s}} + \color{blue}{1}\right)}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} \]
9. Step-by-step derivation

10. Simplified 7.5%

    \[\leadsto \frac{0.125 \cdot \left(e^{0 - \frac{r}{s}} + \color{blue}{1}\right)}{r \cdot \left(s \cdot \pi\right)} \]
11. Add Preprocessing

Alternative 12: 10.3% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in r around 0

   \[\leadsto \color{blue}{\frac{r \cdot \left(\frac{5}{72} \cdot \frac{r}{{s}^{3} \cdot \mathsf{PI}\left(\right)} - \frac{1}{6} \cdot \frac{1}{{s}^{2} \cdot \mathsf{PI}\left(\right)}\right) + \frac{1}{4} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}}{r}} \]

5. Simplified 7.4%

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

Alternative 13: 10.3% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in s around inf

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

5. Simplified 7.4%

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

Alternative 14: 10.3% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \frac{0.125}{s \cdot \pi} \cdot \left(\frac{2}{r} - \frac{\mathsf{fma}\left(-0.5555555555555556, \frac{r}{s}, 1.3333333333333333\right)}{s}\right) \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (*
  (/ 0.125 (* s PI))
  (- (/ 2.0 r) (/ (fma -0.5555555555555556 (/ r s) 1.3333333333333333) s))))

C

float code(float s, float r) {
	return (0.125f / (s * ((float) M_PI))) * ((2.0f / r) - (fmaf(-0.5555555555555556f, (r / s), 1.3333333333333333f) / s));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(0.125) / Float32(s * Float32(pi))) * Float32(Float32(Float32(2.0) / r) - Float32(fma(Float32(-0.5555555555555556), Float32(r / s), Float32(1.3333333333333333)) / s)))
end

Derivation

1. Initial program 99.4%

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

   1. +-commutative N/A

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

   2. times-frac N/A

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

   3. times-frac N/A

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

   4. associate-*l* N/A

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

   5. associate-/r* N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. associate-/r* N/A

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

   9. associate-*l* N/A

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

   10. distribute-lft-out N/A

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

   11. *-lowering-*.f32 N/A

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

4. Applied egg-rr 99.4%

   \[\leadsto \color{blue}{\frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{s \cdot -3}}}{r} + \frac{e^{0 - \frac{r}{s}}}{r}\right)} \]

5. Taylor expanded in s around -inf

   \[\leadsto \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \cdot \color{blue}{\left(-1 \cdot \frac{\frac{4}{3} + -1 \cdot \frac{\frac{1}{18} \cdot r + \frac{1}{2} \cdot r}{s}}{s} + 2 \cdot \frac{1}{r}\right)} \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \cdot \color{blue}{\left(2 \cdot \frac{1}{r} + -1 \cdot \frac{\frac{4}{3} + -1 \cdot \frac{\frac{1}{18} \cdot r + \frac{1}{2} \cdot r}{s}}{s}\right)} \]

   2. mul-1-neg N/A

      \[\leadsto \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \cdot \left(2 \cdot \frac{1}{r} + \color{blue}{\left(\mathsf{neg}\left(\frac{\frac{4}{3} + -1 \cdot \frac{\frac{1}{18} \cdot r + \frac{1}{2} \cdot r}{s}}{s}\right)\right)}\right) \]

   3. unsub-neg N/A

      \[\leadsto \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \cdot \color{blue}{\left(2 \cdot \frac{1}{r} - \frac{\frac{4}{3} + -1 \cdot \frac{\frac{1}{18} \cdot r + \frac{1}{2} \cdot r}{s}}{s}\right)} \]

   4. --lowering--.f32 N/A

      \[\leadsto \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \cdot \color{blue}{\left(2 \cdot \frac{1}{r} - \frac{\frac{4}{3} + -1 \cdot \frac{\frac{1}{18} \cdot r + \frac{1}{2} \cdot r}{s}}{s}\right)} \]

   5. associate-*r/ N/A

      \[\leadsto \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \cdot \left(\color{blue}{\frac{2 \cdot 1}{r}} - \frac{\frac{4}{3} + -1 \cdot \frac{\frac{1}{18} \cdot r + \frac{1}{2} \cdot r}{s}}{s}\right) \]

   6. metadata-eval N/A

      \[\leadsto \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \cdot \left(\frac{\color{blue}{2}}{r} - \frac{\frac{4}{3} + -1 \cdot \frac{\frac{1}{18} \cdot r + \frac{1}{2} \cdot r}{s}}{s}\right) \]

   7. /-lowering-/.f32 N/A

      \[\leadsto \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \cdot \left(\color{blue}{\frac{2}{r}} - \frac{\frac{4}{3} + -1 \cdot \frac{\frac{1}{18} \cdot r + \frac{1}{2} \cdot r}{s}}{s}\right) \]

   8. /-lowering-/.f32 N/A

      \[\leadsto \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \cdot \left(\frac{2}{r} - \color{blue}{\frac{\frac{4}{3} + -1 \cdot \frac{\frac{1}{18} \cdot r + \frac{1}{2} \cdot r}{s}}{s}}\right) \]

   9. +-commutative N/A

      \[\leadsto \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \cdot \left(\frac{2}{r} - \frac{\color{blue}{-1 \cdot \frac{\frac{1}{18} \cdot r + \frac{1}{2} \cdot r}{s} + \frac{4}{3}}}{s}\right) \]

   10. mul-1-neg N/A

       \[\leadsto \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \cdot \left(\frac{2}{r} - \frac{\color{blue}{\left(\mathsf{neg}\left(\frac{\frac{1}{18} \cdot r + \frac{1}{2} \cdot r}{s}\right)\right)} + \frac{4}{3}}{s}\right) \]

   11. distribute-neg-frac2 N/A

       \[\leadsto \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \cdot \left(\frac{2}{r} - \frac{\color{blue}{\frac{\frac{1}{18} \cdot r + \frac{1}{2} \cdot r}{\mathsf{neg}\left(s\right)}} + \frac{4}{3}}{s}\right) \]

   12. distribute-rgt-out N/A

       \[\leadsto \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \cdot \left(\frac{2}{r} - \frac{\frac{\color{blue}{r \cdot \left(\frac{1}{18} + \frac{1}{2}\right)}}{\mathsf{neg}\left(s\right)} + \frac{4}{3}}{s}\right) \]

   13. metadata-eval N/A

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

   14. *-commutative N/A

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

   15. neg-mul-1 N/A

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

   16. times-frac N/A

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

   17. accelerator-lowering-fma.f32 N/A

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

   18. metadata-eval N/A

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

   19. /-lowering-/.f32 7.4

       \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{2}{r} - \frac{\mathsf{fma}\left(-0.5555555555555556, \color{blue}{\frac{r}{s}}, 1.3333333333333333\right)}{s}\right) \]

7. Simplified 7.4%

   \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \color{blue}{\left(\frac{2}{r} - \frac{\mathsf{fma}\left(-0.5555555555555556, \frac{r}{s}, 1.3333333333333333\right)}{s}\right)} \]
8. Add Preprocessing

Alternative 15: 10.3% accurate, 4.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.4%

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

   1. +-commutative N/A

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

   2. times-frac N/A

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

   3. times-frac N/A

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

   4. associate-*l* N/A

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

   5. associate-/r* N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

   8. associate-/r* N/A

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

   9. associate-*l* N/A

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

   10. distribute-lft-out N/A

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

   11. *-lowering-*.f32 N/A

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

4. Applied egg-rr 99.4%

   \[\leadsto \color{blue}{\frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{s \cdot -3}}}{r} + \frac{e^{0 - \frac{r}{s}}}{r}\right)} \]

5. Taylor expanded in r around inf

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

   1. associate-*r/ N/A

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

   2. metadata-eval N/A

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

   3. associate-*r* N/A

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

   4. neg-mul-1 N/A

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

   5. distribute-lft-out N/A

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

   6. /-lowering-/.f32 N/A

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

7. Simplified 99.4%

   \[\leadsto \color{blue}{\frac{0.125 \cdot \left(e^{0 - \frac{r}{s}} + e^{\frac{r}{s} \cdot -0.3333333333333333}\right)}{r \cdot \left(s \cdot \pi\right)}} \]

8. Taylor expanded in r around 0

   \[\leadsto \frac{\color{blue}{\frac{1}{4} + r \cdot \left(\frac{5}{72} \cdot \frac{r}{{s}^{2}} - \frac{1}{6} \cdot \frac{1}{s}\right)}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} \]
9. Step-by-step derivation

   1. +-commutative N/A

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

   2. accelerator-lowering-fma.f32 N/A

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

   3. sub-neg N/A

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

   4. associate-*r/ N/A

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

   5. *-commutative N/A

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

   6. associate-*r/ N/A

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

   7. metadata-eval N/A

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

   8. associate-*r/ N/A

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

   9. accelerator-lowering-fma.f32 N/A

      \[\leadsto \frac{\mathsf{fma}\left(r, \color{blue}{\mathsf{fma}\left(r, \frac{5}{72} \cdot \frac{1}{{s}^{2}}, \mathsf{neg}\left(\frac{1}{6} \cdot \frac{1}{s}\right)\right)}, \frac{1}{4}\right)}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} \]

   10. associate-*r/ N/A

       \[\leadsto \frac{\mathsf{fma}\left(r, \mathsf{fma}\left(r, \color{blue}{\frac{\frac{5}{72} \cdot 1}{{s}^{2}}}, \mathsf{neg}\left(\frac{1}{6} \cdot \frac{1}{s}\right)\right), \frac{1}{4}\right)}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} \]

   11. metadata-eval N/A

       \[\leadsto \frac{\mathsf{fma}\left(r, \mathsf{fma}\left(r, \frac{\color{blue}{\frac{5}{72}}}{{s}^{2}}, \mathsf{neg}\left(\frac{1}{6} \cdot \frac{1}{s}\right)\right), \frac{1}{4}\right)}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} \]

   12. /-lowering-/.f32 N/A

       \[\leadsto \frac{\mathsf{fma}\left(r, \mathsf{fma}\left(r, \color{blue}{\frac{\frac{5}{72}}{{s}^{2}}}, \mathsf{neg}\left(\frac{1}{6} \cdot \frac{1}{s}\right)\right), \frac{1}{4}\right)}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} \]

   13. unpow2 N/A

       \[\leadsto \frac{\mathsf{fma}\left(r, \mathsf{fma}\left(r, \frac{\frac{5}{72}}{\color{blue}{s \cdot s}}, \mathsf{neg}\left(\frac{1}{6} \cdot \frac{1}{s}\right)\right), \frac{1}{4}\right)}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} \]

   14. *-lowering-*.f32 N/A

       \[\leadsto \frac{\mathsf{fma}\left(r, \mathsf{fma}\left(r, \frac{\frac{5}{72}}{\color{blue}{s \cdot s}}, \mathsf{neg}\left(\frac{1}{6} \cdot \frac{1}{s}\right)\right), \frac{1}{4}\right)}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} \]

   15. associate-*r/ N/A

       \[\leadsto \frac{\mathsf{fma}\left(r, \mathsf{fma}\left(r, \frac{\frac{5}{72}}{s \cdot s}, \mathsf{neg}\left(\color{blue}{\frac{\frac{1}{6} \cdot 1}{s}}\right)\right), \frac{1}{4}\right)}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} \]

   16. metadata-eval N/A

       \[\leadsto \frac{\mathsf{fma}\left(r, \mathsf{fma}\left(r, \frac{\frac{5}{72}}{s \cdot s}, \mathsf{neg}\left(\frac{\color{blue}{\frac{1}{6}}}{s}\right)\right), \frac{1}{4}\right)}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} \]

   17. distribute-neg-frac N/A

       \[\leadsto \frac{\mathsf{fma}\left(r, \mathsf{fma}\left(r, \frac{\frac{5}{72}}{s \cdot s}, \color{blue}{\frac{\mathsf{neg}\left(\frac{1}{6}\right)}{s}}\right), \frac{1}{4}\right)}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} \]

   18. metadata-eval N/A

       \[\leadsto \frac{\mathsf{fma}\left(r, \mathsf{fma}\left(r, \frac{\frac{5}{72}}{s \cdot s}, \frac{\color{blue}{\frac{-1}{6}}}{s}\right), \frac{1}{4}\right)}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} \]

   19. /-lowering-/.f32 7.4

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

10. Simplified 7.4%

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

Alternative 16: 9.2% accurate, 9.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in r around 0

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

   1. /-lowering-/.f32 N/A

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

   2. *-lowering-*.f32 N/A

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

   3. *-lowering-*.f32 N/A

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

   4. PI-lowering-PI.f32 7.2

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

5. Simplified 7.2%

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

   1. associate-/r* N/A

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

   2. div-inv N/A

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

   3. *-lowering-*.f32 N/A

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

   4. /-lowering-/.f32 N/A

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

   5. /-lowering-/.f32 N/A

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

   6. *-lowering-*.f32 N/A

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

   7. PI-lowering-PI.f32 7.2

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

7. Applied egg-rr 7.2%

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

Alternative 17: 9.2% accurate, 13.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in r around 0

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

   1. /-lowering-/.f32 N/A

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

   2. *-lowering-*.f32 N/A

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

   3. *-lowering-*.f32 N/A

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

   4. PI-lowering-PI.f32 7.2

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

5. Simplified 7.2%

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. *-lowering-*.f32 N/A

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

   4. *-lowering-*.f32 N/A

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

   5. PI-lowering-PI.f32 7.2

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

7. Applied egg-rr 7.2%

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

8. Final simplification 7.2%

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

Alternative 18: 9.2% accurate, 13.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in r around 0

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

   1. /-lowering-/.f32 N/A

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

   2. *-lowering-*.f32 N/A

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

   3. *-lowering-*.f32 N/A

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

   4. PI-lowering-PI.f32 7.2

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

5. Simplified 7.2%

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

Reproduce

herbie shell --seed 2024198 
(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))))