Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.5%

Time: 14.0s

Alternatives: 13

Speedup: N/A×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \left(\frac{0.16666666666666666}{s \cdot \pi} \cdot 0.75\right) \cdot \left(\frac{{e}^{\left(\frac{r \cdot -0.3333333333333333}{s}\right)}}{r} + \frac{e^{\frac{r}{-s}}}{r}\right) \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (*
  (* (/ 0.16666666666666666 (* s PI)) 0.75)
  (+ (/ (pow E (/ (* r -0.3333333333333333) s)) r) (/ (exp (/ r (- s))) r))))

C

float code(float s, float r) {
	return ((0.16666666666666666f / (s * ((float) M_PI))) * 0.75f) * ((powf(((float) M_E), ((r * -0.3333333333333333f) / s)) / r) + (expf((r / -s)) / r));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.16666666666666666) / Float32(s * Float32(pi))) * Float32(0.75)) * Float32(Float32((Float32(exp(1)) ^ Float32(Float32(r * Float32(-0.3333333333333333)) / s)) / r) + Float32(exp(Float32(r / Float32(-s))) / r)))
end

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.16666666666666666) / (s * single(pi))) * single(0.75)) * (((single(2.71828182845904523536) ^ ((r * single(-0.3333333333333333)) / s)) / r) + (exp((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%

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

   1. *-commutative N/A

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

   2. associate-/r* N/A

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

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

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

   4. /-lowering-/.f32 99.4

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

6. Applied egg-rr 99.4%

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

   1. clear-num N/A

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

   2. div-inv N/A

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

   3. clear-num N/A

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

   4. div-inv N/A

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

   5. metadata-eval N/A

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

   6. associate-*l/ N/A

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

   7. exp-prod N/A

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

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

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

   9. exp-1-e N/A

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

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

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

   11. associate-*l/ N/A

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

   12. metadata-eval N/A

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

   13. div-inv N/A

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

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

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

   15. div-inv N/A

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

   16. metadata-eval N/A

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

   17. *-lowering-*.f32 99.6

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

8. Applied egg-rr 99.6%

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

   1. metadata-eval N/A

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

   2. associate-*l/ N/A

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

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

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

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

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

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

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

   6. PI-lowering-PI.f32 99.6

      \[\leadsto \left(\frac{0.16666666666666666}{s \cdot \color{blue}{\pi}} \cdot 0.75\right) \cdot \left(\frac{{e}^{\left(\frac{r \cdot -0.3333333333333333}{s}\right)}}{r} + \frac{e^{\frac{r}{-s}}}{r}\right) \]

10. Applied egg-rr 99.6%

    \[\leadsto \color{blue}{\left(\frac{0.16666666666666666}{s \cdot \pi} \cdot 0.75\right)} \cdot \left(\frac{{e}^{\left(\frac{r \cdot -0.3333333333333333}{s}\right)}}{r} + \frac{e^{\frac{r}{-s}}}{r}\right) \]
11. Add Preprocessing

Alternative 2: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = (((single(2.71828182845904523536) ^ ((r * single(-0.3333333333333333)) / s)) / r) + (exp((r / -s)) / r)) * (single(0.125) / (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%

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

   1. *-commutative N/A

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

   2. associate-/r* N/A

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

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

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

   4. /-lowering-/.f32 99.4

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

6. Applied egg-rr 99.4%

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

   1. clear-num N/A

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

   2. div-inv N/A

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

   3. clear-num N/A

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

   4. div-inv N/A

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

   5. metadata-eval N/A

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

   6. associate-*l/ N/A

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

   7. exp-prod N/A

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

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

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

   9. exp-1-e N/A

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

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

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

   11. associate-*l/ N/A

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

   12. metadata-eval N/A

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

   13. div-inv N/A

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

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

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

   15. div-inv N/A

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

   16. metadata-eval N/A

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

   17. *-lowering-*.f32 99.6

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

8. Applied egg-rr 99.6%

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

9. Final simplification 99.6%

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

Alternative 3: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

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

   1. div-inv N/A

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

   2. *-commutative N/A

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

   3. associate-/r* N/A

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

   4. metadata-eval N/A

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

   5. associate-/l* N/A

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

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

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

   7. *-lowering-*.f32 99.4

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

6. Applied egg-rr 99.4%

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

7. Final simplification 99.4%

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

Alternative 4: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

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

   1. *-commutative N/A

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

   2. associate-/r* N/A

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

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

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

   4. /-lowering-/.f32 99.4

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

6. Applied egg-rr 99.4%

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

   1. clear-num N/A

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

   2. div-inv N/A

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

   3. clear-num N/A

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

   4. div-inv N/A

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

   5. metadata-eval N/A

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

   6. associate-*l/ N/A

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

   7. exp-prod N/A

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

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

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

   9. exp-1-e N/A

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

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

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

   11. associate-*l/ N/A

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

   12. metadata-eval N/A

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

   13. div-inv N/A

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

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

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

   15. div-inv N/A

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

   16. metadata-eval N/A

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

   17. *-lowering-*.f32 99.6

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

8. Applied egg-rr 99.6%

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

9. Taylor expanded in r around inf

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

    1. associate-*r/ N/A

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

    2. metadata-eval N/A

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

    3. associate-*r* N/A

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

    4. distribute-lft-out N/A

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

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

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

11. Simplified 99.4%

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

12. Final simplification 99.4%

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

Alternative 5: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

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

   1. *-commutative N/A

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

   2. associate-/r* N/A

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

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

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

   4. /-lowering-/.f32 99.4

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

6. Applied egg-rr 99.4%

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

7. Taylor expanded in r around inf

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

   1. associate-*r/ N/A

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

   2. metadata-eval N/A

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

   3. associate-*r* N/A

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

   4. distribute-lft-out N/A

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

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

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

9. Simplified 99.4%

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

10. Final simplification 99.4%

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

Alternative 6: 10.9% accurate, 1.5× speedup

Math

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

FPCore

(FPCore (s r)
 :precision binary32
 (*
  (/ 0.125 (* s PI))
  (+
   (/ (exp (/ 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((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(r / Float32(-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%

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

5. Taylor expanded in r around 0

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

   1. +-commutative N/A

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

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

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

   3. sub-neg N/A

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

   4. *-commutative N/A

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

   5. associate-*l/ N/A

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

   6. associate-*r/ N/A

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

   7. metadata-eval N/A

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

   8. associate-*r/ N/A

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

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

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

   10. associate-*r/ N/A

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

   11. metadata-eval N/A

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

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

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

   13. unpow2 N/A

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

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

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

   15. associate-*r/ N/A

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

   16. metadata-eval N/A

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

   17. distribute-neg-frac N/A

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

   18. metadata-eval N/A

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

   19. /-lowering-/.f32 7.7

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

7. Simplified 7.7%

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

8. Final simplification 7.7%

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

Alternative 7: 9.7% accurate, 1.9× speedup

Math

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

FPCore

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

C

float code(float s, float r) {
	return (0.125f / (s * ((float) M_PI))) * ((expf((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(r / Float32(-s))) / r) + Float32(Float32(1.0) / r)))
end

MATLAB

function tmp = code(s, r)
	tmp = (single(0.125) / (s * single(pi))) * ((exp((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%

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

5. Taylor expanded in r around 0

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

7. Simplified 7.5%

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

8. Final simplification 7.5%

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

Alternative 8: 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 9: 10.3% accurate, 4.0× speedup

Math

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

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/
   (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(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(Float32(fma(Float32(0.06944444444444445), Float32(r / Float32(s * Float32(pi))), Float32(Float32(-0.16666666666666666) / Float32(pi))) / Float32(s * s)) + Float32(Float32(0.25) / Float32(Float32(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 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. Final simplification 7.4%

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

Alternative 10: 9.2% accurate, 5.6× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

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

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

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

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

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

   4. PI-lowering-PI.f32 7.2

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

7. Applied egg-rr 7.2%

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. *-commutative 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. *-commutative N/A

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

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

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

   7. PI-lowering-PI.f32 7.2

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

9. Applied egg-rr 7.2%

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

    1. associate-*r* N/A

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

    2. add-sqr-sqrt N/A

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

    3. *-commutative N/A

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

    4. associate-*r* N/A

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

    5. *-commutative N/A

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

    6. associate-/r* N/A

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

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

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

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

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

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

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

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

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

    11. associate-*l* N/A

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

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

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

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

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

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

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

    15. PI-lowering-PI.f32 7.2

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

11. Applied egg-rr 7.2%

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

Alternative 11: 9.2% accurate, 9.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
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. clear-num N/A

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

   2. inv-pow N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. unpow-prod-down N/A

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

   6. inv-pow N/A

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

   7. inv-pow N/A

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

   8. clear-num N/A

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

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

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

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

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

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

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

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

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

   13. /-lowering-/.f32 7.2

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

7. Applied egg-rr 7.2%

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

Alternative 12: 9.2% accurate, 13.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

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

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

Alternative 13: 9.2% accurate, 13.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
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. Final simplification 7.2%

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