Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.6%

Time: 15.1s

Alternatives: 19

Speedup: N/A×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.6% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in s around 0

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

   1. *-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. lower-*.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. lower-PI.f32 99.7

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

5. Simplified 99.7%

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

   1. add-cube-cbrt N/A

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

   2. pow3 N/A

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

   3. lower-pow.f32 N/A

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

   4. lift-PI.f32 N/A

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

   5. lower-cbrt.f32 99.7

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

7. Applied egg-rr 99.7%

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

   1. lift-PI.f32 N/A

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

   2. rem-cube-cbrt 99.7

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

   3. rem-log-exp N/A

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

   4. lift-PI.f32 N/A

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

   5. *-un-lft-identity N/A

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

   6. lift-PI.f32 N/A

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

   7. exp-prod N/A

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

   8. log-pow N/A

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

   9. lower-*.f32 N/A

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

   10. lower-log.f32 N/A

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

   11. exp-1-e N/A

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

   12. lower-E.f32 99.7

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

9. Applied egg-rr 99.7%

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

10. Taylor expanded in r around inf

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

    1. associate-*r/ N/A

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

    2. lower-/.f32 N/A

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

    3. lower-*.f32 N/A

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

    4. lower-exp.f32 N/A

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

    5. mul-1-neg N/A

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

    6. distribute-neg-frac2 N/A

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

    7. lower-/.f32 N/A

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

    8. lower-neg.f32 N/A

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

    9. associate-*r* N/A

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

    10. *-commutative N/A

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

    11. lower-*.f32 N/A

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

    12. lower-PI.f32 N/A

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

    13. *-commutative N/A

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

    14. lower-*.f32 99.7

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

12. Simplified 99.7%

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

13. Final simplification 99.7%

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

Alternative 2: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in s around 0

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

   1. *-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. lower-*.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. lower-PI.f32 99.7

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

5. Simplified 99.7%

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

6. Final simplification 99.7%

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

Alternative 3: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{e^{\frac{r}{-s}} \cdot 0.25}{r \cdot \left(s \cdot \left(\pi \cdot 2\right)\right)} + \frac{\left(0.75 \cdot e^{\frac{r}{s \cdot -3}}\right) \cdot 0.16666666666666666}{r \cdot \left(s \cdot \pi\right)} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/ (* (exp (/ r (- s))) 0.25) (* r (* s (* PI 2.0))))
  (/ (* (* 0.75 (exp (/ r (* s -3.0)))) 0.16666666666666666) (* r (* s PI)))))

C

float code(float s, float r) {
	return ((expf((r / -s)) * 0.25f) / (r * (s * (((float) M_PI) * 2.0f)))) + (((0.75f * expf((r / (s * -3.0f)))) * 0.16666666666666666f) / (r * (s * ((float) M_PI))));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(exp(Float32(r / Float32(-s))) * Float32(0.25)) / Float32(r * Float32(s * Float32(Float32(pi) * Float32(2.0))))) + Float32(Float32(Float32(Float32(0.75) * exp(Float32(r / Float32(s * Float32(-3.0))))) * Float32(0.16666666666666666)) / Float32(r * Float32(s * Float32(pi)))))
end

MATLAB

function tmp = code(s, r)
	tmp = ((exp((r / -s)) * single(0.25)) / (r * (s * (single(pi) * single(2.0))))) + (((single(0.75) * exp((r / (s * single(-3.0))))) * single(0.16666666666666666)) / (r * (s * single(pi))));
end

Derivation

1. Initial program 99.7%

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

   1. lift-*.f32 N/A

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

   2. lift-neg.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. lift-exp.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-PI.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. lift-*.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. associate-*l* N/A

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

   12. associate-*l* N/A

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

   13. associate-/r* N/A

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

   14. lower-/.f32 N/A

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

4. Applied egg-rr 99.7%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \color{blue}{\frac{\left(0.75 \cdot e^{\frac{r}{s \cdot -3}}\right) \cdot 0.16666666666666666}{\left(s \cdot \pi\right) \cdot r}} \]

5. Final simplification 99.7%

   \[\leadsto \frac{e^{\frac{r}{-s}} \cdot 0.25}{r \cdot \left(s \cdot \left(\pi \cdot 2\right)\right)} + \frac{\left(0.75 \cdot e^{\frac{r}{s \cdot -3}}\right) \cdot 0.16666666666666666}{r \cdot \left(s \cdot \pi\right)} \]
6. Add Preprocessing

Alternative 4: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in r around inf

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

   1. distribute-lft-out N/A

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

   2. metadata-eval N/A

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

   3. associate-*r* N/A

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

   4. distribute-lft-out N/A

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

   5. lower-/.f32 N/A

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

5. Simplified 99.7%

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

6. Final simplification 99.7%

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

Alternative 5: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in s around 0

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

   1. *-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. lower-*.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. lower-PI.f32 99.7

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

5. Simplified 99.7%

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

7. Applied egg-rr 99.7%

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

8. Final simplification 99.7%

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

Alternative 6: 97.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in s around 0

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

   1. *-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. lower-*.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. lower-PI.f32 99.7

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

5. Simplified 99.7%

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

7. Applied egg-rr 97.7%

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

8. Final simplification 97.7%

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

Alternative 7: 10.7% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in s around 0

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

   1. *-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. lower-*.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. lower-PI.f32 99.7

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

5. Simplified 99.7%

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

6. Taylor expanded in s around inf

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

8. Simplified 9.7%

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

9. Final simplification 9.7%

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

Alternative 8: 10.7% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in s around inf

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

5. Simplified 9.7%

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

6. Final simplification 9.7%

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

Alternative 9: 10.6% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.7%

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

4. Applied egg-rr 97.8%

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

5. Taylor expanded in r around 0

   \[\leadsto \mathsf{fma}\left(e^{\frac{r}{\mathsf{neg}\left(s\right)}}, \frac{\frac{1}{8}}{\mathsf{PI}\left(\right) \cdot \left(r \cdot s\right)}, \color{blue}{\frac{r \cdot \left(\frac{1}{144} \cdot \frac{r}{{s}^{3} \cdot \mathsf{PI}\left(\right)} - \frac{1}{24} \cdot \frac{1}{{s}^{2} \cdot \mathsf{PI}\left(\right)}\right) + \frac{1}{8} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}}{r}}\right) \]
6. Step-by-step derivation

   1. lower-/.f32 N/A

      \[\leadsto \mathsf{fma}\left(e^{\frac{r}{\mathsf{neg}\left(s\right)}}, \frac{\frac{1}{8}}{\mathsf{PI}\left(\right) \cdot \left(r \cdot s\right)}, \color{blue}{\frac{r \cdot \left(\frac{1}{144} \cdot \frac{r}{{s}^{3} \cdot \mathsf{PI}\left(\right)} - \frac{1}{24} \cdot \frac{1}{{s}^{2} \cdot \mathsf{PI}\left(\right)}\right) + \frac{1}{8} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}}{r}}\right) \]

7. Simplified 8.4%

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

8. Taylor expanded in s around 0

   \[\leadsto \color{blue}{\frac{\frac{1}{144} \cdot \frac{r}{\mathsf{PI}\left(\right)} + s \cdot \left(s \cdot \left(\frac{1}{8} \cdot \frac{e^{-1 \cdot \frac{r}{s}}}{r \cdot \mathsf{PI}\left(\right)} + \frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}\right) - \frac{1}{24} \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{{s}^{3}}} \]
9. Step-by-step derivation

   1. lower-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{144} \cdot \frac{r}{\mathsf{PI}\left(\right)} + s \cdot \left(s \cdot \left(\frac{1}{8} \cdot \frac{e^{-1 \cdot \frac{r}{s}}}{r \cdot \mathsf{PI}\left(\right)} + \frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}\right) - \frac{1}{24} \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{{s}^{3}}} \]

10. Simplified 9.7%

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

Alternative 10: 10.1% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.7%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
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 9.4%

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

6. Taylor expanded in s around -inf

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

8. Simplified 9.4%

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

9. Final simplification 9.4%

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

Alternative 11: 10.1% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in s around 0

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

   1. *-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. lower-*.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. lower-PI.f32 99.7

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

5. Simplified 99.7%

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

6. Taylor expanded in s around inf

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

8. Simplified 9.4%

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

Alternative 12: 10.1% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.7%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
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 9.3%

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

Alternative 13: 9.2% accurate, 6.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = ((single(-0.16666666666666666) / (s * single(pi))) + (single(0.25) / (r * single(pi)))) / s;
end

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in s around -inf

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

5. Simplified 9.2%

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

6. Taylor expanded in s around inf

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

   1. sub-neg N/A

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

   2. associate-*r/ N/A

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

   3. metadata-eval N/A

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

   4. lower-+.f32 N/A

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

   5. associate-*r/ N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. lower-PI.f32 N/A

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

   10. distribute-neg-frac N/A

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

   11. metadata-eval N/A

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

   12. lower-/.f32 N/A

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

   13. lower-*.f32 N/A

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

   14. lower-PI.f32 8.8

       \[\leadsto \frac{\frac{0.25}{r \cdot \pi} + \frac{-0.16666666666666666}{s \cdot \color{blue}{\pi}}}{s} \]

8. Simplified 8.8%

   \[\leadsto \frac{\color{blue}{\frac{0.25}{r \cdot \pi} + \frac{-0.16666666666666666}{s \cdot \pi}}}{s} \]

9. Final simplification 8.8%

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

Alternative 14: 9.1% accurate, 6.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = (single(-0.16666666666666666) / (single(pi) * (s * s))) - (single(-0.25) / (single(pi) * (r * s)));
end

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in s around 0

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

   1. *-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. lower-*.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. lower-PI.f32 99.7

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

5. Simplified 99.7%

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

6. Taylor expanded in s around -inf

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

   1. mul-1-neg N/A

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

   2. distribute-neg-frac2 N/A

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

   3. mul-1-neg N/A

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

   4. div-sub N/A

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

   5. mul-1-neg N/A

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

   6. distribute-neg-frac2 N/A

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

   7. associate-*r/ N/A

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

   8. metadata-eval N/A

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

   9. associate-/l/ N/A

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

   10. metadata-eval N/A

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

   11. associate-*l* N/A

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

   12. unpow2 N/A

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

   13. associate-*r/ N/A

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

   14. mul-1-neg N/A

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

   15. distribute-neg-frac2 N/A

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

8. Simplified 8.7%

   \[\leadsto \color{blue}{\frac{-0.16666666666666666}{\pi \cdot \left(s \cdot s\right)} - \frac{-0.25}{r \cdot \left(s \cdot \pi\right)}} \]
9. Step-by-step derivation

   1. lift-PI.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-/.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. lift-/.f32 N/A

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

   9. lift--.f32 8.7

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

   10. lift-*.f32 N/A

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

   11. lift-*.f32 N/A

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

   12. associate-*r* N/A

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

   13. lift-*.f32 N/A

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

   14. *-commutative N/A

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

   15. lift-*.f32 8.7

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

10. Applied egg-rr 8.7%

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

Alternative 15: 9.1% accurate, 6.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = (single(0.25) / (r * (s * single(pi)))) + (single(-0.16666666666666666) / (s * (s * single(pi))));
end

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in s around inf

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

   1. div-sub N/A

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

   2. sub-neg N/A

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

   3. associate-/l* N/A

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

   4. associate-/l/ N/A

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

   5. associate-*l* N/A

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

   6. unpow2 N/A

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

   7. lower-+.f32 N/A

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

5. Simplified 8.7%

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

Alternative 16: 9.1% accurate, 8.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in r around 0

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 8.6

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

5. Simplified 8.6%

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

   1. lift-PI.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-/r* N/A

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

   4. lift-*.f32 N/A

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

   5. associate-/r* N/A

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

   6. lower-/.f32 N/A

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

   7. lower-/.f32 N/A

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

   8. lower-/.f32 8.6

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

7. Applied egg-rr 8.6%

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

Alternative 17: 9.1% accurate, 10.6× speedup

Math

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

FPCore

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

C

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

Julia

function code(s, r)
	return Float32(Float32(Float32(0.25) / 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 99.7%

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

3. Taylor expanded in r around 0

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 8.6

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

5. Simplified 8.6%

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

   1. lift-PI.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f32 N/A

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

   5. lower-/.f32 8.6

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

7. Applied egg-rr 8.6%

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

Alternative 18: 9.1% 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 99.7%

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

3. Taylor expanded in r around 0

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 8.6

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

5. Simplified 8.6%

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

   1. lift-PI.f32 N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. lift-*.f32 N/A

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

   5. lower-*.f32 8.6

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

7. Applied egg-rr 8.6%

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

8. Final simplification 8.6%

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

Alternative 19: 9.1% accurate, 13.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in r around 0

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 8.6

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

5. Simplified 8.6%

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

Reproduce

herbie shell --seed 2024208 
(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))))