Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.6%

Time: 14.9s

Alternatives: 15

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. times-frac N/A

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

   5. associate-*l/ N/A

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

   6. lift-*.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. associate-*l* N/A

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

   10. associate-/r* N/A

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

   11. lower-/.f32 N/A

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

4. Applied rewrites 99.5%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. times-frac N/A

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

   5. lift-*.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. associate-*l* N/A

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

   8. *-commutative N/A

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

   9. lift-*.f32 N/A

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

   10. associate-/r* N/A

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

   11. metadata-eval N/A

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

   12. lift-neg.f32 N/A

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

   13. lift-/.f32 N/A

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

   14. distribute-frac-neg N/A

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

   15. distribute-frac-neg2 N/A

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

   16. lift-neg.f32 N/A

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

   17. lift-/.f32 N/A

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

6. Applied rewrites 99.5%

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

7. Final simplification 99.5%

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

Alternative 2: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

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)))) / (r * (s * (((float) M_PI) * 6.0f))));
}

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(r / Float32(s * Float32(-Float32(3.0)))))) / Float32(r * Float32(s * Float32(Float32(pi) * Float32(6.0))))))
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))))) / (r * (s * (single(pi) * single(6.0)))));
end

Derivation

1. Initial program 99.5%

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

3. Final simplification 99.5%

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

Alternative 3: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

4. Applied rewrites 99.1%

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

5. Taylor expanded in r around inf

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

   1. lower-/.f32 N/A

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

   2. lower-+.f32 N/A

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

   3. lower-exp.f32 N/A

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

   4. mul-1-neg N/A

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

   5. distribute-neg-frac2 N/A

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

   6. lower-/.f32 N/A

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

   7. lower-neg.f32 N/A

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

   8. lower-exp.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f32 N/A

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

   11. lower-/.f32 99.0

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

7. Applied rewrites 99.0%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-*l/ N/A

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

   4. lift-*.f32 N/A

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

   5. *-commutative N/A

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

   6. associate-/r* N/A

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

   7. lower-/.f32 N/A

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

9. Applied rewrites 99.5%

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

Alternative 4: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

4. Applied rewrites 99.1%

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

5. Taylor expanded in r around inf

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

   1. associate-*r/ N/A

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

   2. metadata-eval N/A

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

   3. associate-*r* N/A

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

   4. distribute-lft-out N/A

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

   5. lower-/.f32 N/A

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

7. Applied rewrites 99.4%

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

8. Final simplification 99.4%

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

Alternative 5: 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 99.5%

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

3. Taylor expanded in s around inf

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

5. Applied rewrites 11.5%

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

6. Final simplification 11.5%

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

Alternative 6: 10.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.5%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. times-frac N/A

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

   5. associate-*l/ N/A

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

   6. lift-*.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. associate-*l* N/A

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

   10. associate-/r* N/A

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

   11. lower-/.f32 N/A

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

4. Applied rewrites 99.5%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. times-frac N/A

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

   5. lift-*.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. associate-*l* N/A

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

   8. *-commutative N/A

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

   9. lift-*.f32 N/A

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

   10. associate-/r* N/A

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

   11. metadata-eval N/A

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

   12. lift-neg.f32 N/A

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

   13. lift-/.f32 N/A

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

   14. distribute-frac-neg N/A

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

   15. distribute-frac-neg2 N/A

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

   16. lift-neg.f32 N/A

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

   17. lift-/.f32 N/A

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

6. Applied rewrites 99.5%

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

7. Taylor expanded in s around inf

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

9. Applied rewrites 11.5%

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

Alternative 7: 10.7% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.5%

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

4. Applied rewrites 99.1%

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

5. Taylor expanded in r around inf

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

   1. lower-/.f32 N/A

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

   2. lower-+.f32 N/A

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

   3. lower-exp.f32 N/A

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

   4. mul-1-neg N/A

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

   5. distribute-neg-frac2 N/A

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

   6. lower-/.f32 N/A

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

   7. lower-neg.f32 N/A

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

   8. lower-exp.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f32 N/A

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

   11. lower-/.f32 99.0

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

7. Applied rewrites 99.0%

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

9. Applied rewrites 99.1%

   \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{r}{-s}} + e^{\frac{r}{s \cdot -3}}}{r} \]

10. Taylor expanded in s around -inf

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

12. Applied rewrites 11.5%

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

Alternative 8: 10.7% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.5%

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

4. Applied rewrites 99.1%

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

5. Taylor expanded in r around inf

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

   1. lower-/.f32 N/A

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

   2. lower-+.f32 N/A

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

   3. lower-exp.f32 N/A

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

   4. mul-1-neg N/A

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

   5. distribute-neg-frac2 N/A

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

   6. lower-/.f32 N/A

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

   7. lower-neg.f32 N/A

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

   8. lower-exp.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f32 N/A

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

   11. lower-/.f32 99.0

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

7. Applied rewrites 99.0%

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

9. Applied rewrites 99.1%

   \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{r}{-s}} + e^{\frac{r}{s \cdot -3}}}{r} \]

10. Taylor expanded in r around 0

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

12. Applied rewrites 11.5%

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

Alternative 9: 10.1% accurate, 4.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.5%

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

4. Applied rewrites 99.1%

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

5. Taylor expanded in s around inf

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

7. Applied rewrites 11.1%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-*l/ N/A

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

   4. lower-/.f32 N/A

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

   5. lower-*.f32 11.1

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

9. Applied rewrites 11.1%

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

Alternative 10: 10.1% accurate, 4.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.5%

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

4. Applied rewrites 99.1%

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

5. Taylor expanded in s around inf

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

7. Applied rewrites 11.1%

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

8. Taylor expanded in s around inf

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

10. Applied rewrites 11.1%

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

Alternative 11: 9.1% accurate, 6.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = (single(0.125) / (s * single(pi))) * ((single(2.0) / r) + (single(-1.3333333333333333) / s));
end

Derivation

1. Initial program 99.5%

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

4. Applied rewrites 99.1%

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

5. Taylor expanded in s around inf

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

   1. sub-neg N/A

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

   2. lower-+.f32 N/A

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

   3. associate-*r/ N/A

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

   4. metadata-eval N/A

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

   5. lower-/.f32 N/A

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

   6. associate-*r/ N/A

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

   7. metadata-eval N/A

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

   8. distribute-neg-frac N/A

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

   9. metadata-eval N/A

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

   10. lower-/.f32 10.2

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

7. Applied rewrites 10.2%

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

Alternative 12: 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 99.5%

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

3. Taylor expanded in s around inf

   \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)} - \frac{1}{6} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}}{s}} \]
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. Applied rewrites 10.1%

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

Alternative 13: 9.1% accurate, 10.6× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
2. 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 10.0

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

5. Applied rewrites 10.0%

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

7. Applied rewrites 10.1%

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

Alternative 14: 9.1% accurate, 13.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in r around 0

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 10.0

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

5. Applied rewrites 10.0%

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

7. Applied rewrites 10.1%

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

8. Final simplification 10.1%

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

Alternative 15: 9.1% accurate, 13.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in r around 0

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 10.0

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

5. Applied rewrites 10.0%

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

Reproduce

herbie shell --seed 2024234 
(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))))