Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.5% → 99.5%

Time: 15.7s

Alternatives: 19

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: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.25) * exp(Float32(-Float32(r / s)))) / Float32(r * Float32(s * Float32(Float32(2.0) * Float32(pi))))) + 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 = ((single(0.25) * exp(-(r / s))) / (r * (s * (single(2.0) * single(pi))))) + ((single(0.75) * exp((r / (s * -single(3.0))))) / (r * (s * (single(pi) * 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. Final simplification 99.7%

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

Alternative 2: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{e^{-\frac{r}{s}} \cdot 0.125}{r \cdot \left(s \cdot \pi\right)} + \frac{0.75 \cdot e^{\frac{1}{s} \cdot \left(r \cdot -0.3333333333333333\right)}}{r \cdot \left(s \cdot \left(\pi \cdot 6\right)\right)} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/ (* (exp (- (/ r s))) 0.125) (* r (* s PI)))
  (/
   (* 0.75 (exp (* (/ 1.0 s) (* r -0.3333333333333333))))
   (* r (* s (* PI 6.0))))))

C

float code(float s, float r) {
	return ((expf(-(r / s)) * 0.125f) / (r * (s * ((float) M_PI)))) + ((0.75f * expf(((1.0f / s) * (r * -0.3333333333333333f)))) / (r * (s * (((float) M_PI) * 6.0f))));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(exp(Float32(-Float32(r / s))) * Float32(0.125)) / Float32(r * Float32(s * Float32(pi)))) + Float32(Float32(Float32(0.75) * exp(Float32(Float32(Float32(1.0) / s) * Float32(r * Float32(-0.3333333333333333))))) / Float32(r * Float32(s * Float32(Float32(pi) * Float32(6.0))))))
end

MATLAB

function tmp = code(s, r)
	tmp = ((exp(-(r / s)) * single(0.125)) / (r * (s * single(pi)))) + ((single(0.75) * exp(((single(1.0) / s) * (r * single(-0.3333333333333333))))) / (r * (s * (single(pi) * 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. 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. clear-num N/A

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

   4. inv-pow N/A

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

   5. lift-*.f32 N/A

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

   6. *-commutative N/A

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

   7. associate-/l* N/A

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

   8. unpow-prod-down N/A

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

   9. inv-pow N/A

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

   10. inv-pow N/A

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

   11. clear-num N/A

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

   12. lower-*.f32 N/A

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

   13. lower-/.f32 N/A

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

   14. frac-2neg N/A

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

   15. lift-neg.f32 N/A

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

   16. remove-double-neg N/A

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

   17. div-inv N/A

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

   18. metadata-eval N/A

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

   19. metadata-eval N/A

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

   20. metadata-eval N/A

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

   21. lower-*.f32 N/A

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

   22. metadata-eval 99.7

       \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{1}{s} \cdot \left(r \cdot \color{blue}{-0.3333333333333333}\right)}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

4. Applied egg-rr 99.7%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\color{blue}{\frac{1}{s} \cdot \left(r \cdot -0.3333333333333333\right)}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

6. Applied egg-rr 99.7%

   \[\leadsto \color{blue}{\frac{e^{\frac{r}{-s}} \cdot 0.125}{r \cdot \left(s \cdot \pi\right)}} + \frac{0.75 \cdot e^{\frac{1}{s} \cdot \left(r \cdot -0.3333333333333333\right)}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

7. Final simplification 99.7%

   \[\leadsto \frac{e^{-\frac{r}{s}} \cdot 0.125}{r \cdot \left(s \cdot \pi\right)} + \frac{0.75 \cdot e^{\frac{1}{s} \cdot \left(r \cdot -0.3333333333333333\right)}}{r \cdot \left(s \cdot \left(\pi \cdot 6\right)\right)} \]
8. Add Preprocessing

Alternative 3: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{e^{\frac{r \cdot -0.3333333333333333}{s}}}{r \cdot \left(s \cdot \left(\pi \cdot 6\right)\right)}, 0.75, \frac{e^{-\frac{r}{s}}}{r \cdot \left(\left(s \cdot \pi\right) \cdot 8\right)}\right) \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (fma
  (/ (exp (/ (* r -0.3333333333333333) s)) (* r (* s (* PI 6.0))))
  0.75
  (/ (exp (- (/ r s))) (* r (* (* s PI) 8.0)))))

C

float code(float s, float r) {
	return fmaf((expf(((r * -0.3333333333333333f) / s)) / (r * (s * (((float) M_PI) * 6.0f)))), 0.75f, (expf(-(r / s)) / (r * ((s * ((float) M_PI)) * 8.0f))));
}

Julia

function code(s, r)
	return fma(Float32(exp(Float32(Float32(r * Float32(-0.3333333333333333)) / s)) / Float32(r * Float32(s * Float32(Float32(pi) * Float32(6.0))))), Float32(0.75), Float32(exp(Float32(-Float32(r / s))) / Float32(r * Float32(Float32(s * Float32(pi)) * Float32(8.0)))))
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. 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. clear-num N/A

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

   4. inv-pow N/A

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

   5. lift-*.f32 N/A

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

   6. *-commutative N/A

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

   7. associate-/l* N/A

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

   8. unpow-prod-down N/A

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

   9. inv-pow N/A

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

   10. inv-pow N/A

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

   11. clear-num N/A

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

   12. lower-*.f32 N/A

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

   13. lower-/.f32 N/A

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

   14. frac-2neg N/A

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

   15. lift-neg.f32 N/A

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

   16. remove-double-neg N/A

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

   17. div-inv N/A

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

   18. metadata-eval N/A

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

   19. metadata-eval N/A

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

   20. metadata-eval N/A

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

   21. lower-*.f32 N/A

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

   22. metadata-eval 99.7

       \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{1}{s} \cdot \left(r \cdot \color{blue}{-0.3333333333333333}\right)}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

4. Applied egg-rr 99.7%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\color{blue}{\frac{1}{s} \cdot \left(r \cdot -0.3333333333333333\right)}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

6. Applied egg-rr 99.7%

   \[\leadsto \color{blue}{\frac{e^{\frac{r}{-s}} \cdot 0.125}{r \cdot \left(s \cdot \pi\right)}} + \frac{0.75 \cdot e^{\frac{1}{s} \cdot \left(r \cdot -0.3333333333333333\right)}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
7. Step-by-step derivation

   1. lift-PI.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f32 N/A

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

   7. lower-*.f32 99.7

      \[\leadsto \frac{e^{\frac{r}{-s}} \cdot 0.125}{r \cdot \left(s \cdot \pi\right)} + \frac{0.75 \cdot e^{\frac{1}{s} \cdot \left(r \cdot -0.3333333333333333\right)}}{\color{blue}{\left(\left(6 \cdot \pi\right) \cdot r\right)} \cdot s} \]

   8. lift-*.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f32 99.7

       \[\leadsto \frac{e^{\frac{r}{-s}} \cdot 0.125}{r \cdot \left(s \cdot \pi\right)} + \frac{0.75 \cdot e^{\frac{1}{s} \cdot \left(r \cdot -0.3333333333333333\right)}}{\left(\color{blue}{\left(\pi \cdot 6\right)} \cdot r\right) \cdot s} \]

8. Applied egg-rr 99.7%

   \[\leadsto \frac{e^{\frac{r}{-s}} \cdot 0.125}{r \cdot \left(s \cdot \pi\right)} + \frac{0.75 \cdot e^{\frac{1}{s} \cdot \left(r \cdot -0.3333333333333333\right)}}{\color{blue}{\left(\left(\pi \cdot 6\right) \cdot r\right) \cdot s}} \]

10. Applied egg-rr 99.7%

    \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{e^{\frac{r \cdot -0.3333333333333333}{s}}}{r \cdot \left(\left(\pi \cdot 6\right) \cdot s\right)}, 0.75, \frac{e^{\frac{r}{-s}}}{r \cdot \left(\left(s \cdot \pi\right) \cdot 8\right)}\right)} \]

11. Final simplification 99.7%

    \[\leadsto \mathsf{fma}\left(\frac{e^{\frac{r \cdot -0.3333333333333333}{s}}}{r \cdot \left(s \cdot \left(\pi \cdot 6\right)\right)}, 0.75, \frac{e^{-\frac{r}{s}}}{r \cdot \left(\left(s \cdot \pi\right) \cdot 8\right)}\right) \]
12. Add Preprocessing

Alternative 4: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

float code(float s, float r) {
	return (0.125f / (s * ((float) M_PI))) * ((expf((r / (s * -3.0f))) / r) + (expf(-(r / 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(-3.0)))) / r) + Float32(exp(Float32(-Float32(r / s))) / r)))
end

MATLAB

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

4. Applied egg-rr 99.7%

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

5. Final simplification 99.7%

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

Alternative 5: 10.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{e^{-\frac{r}{s}}}{r}, \frac{0.125}{s \cdot \pi}, \frac{\frac{\mathsf{fma}\left(r, \frac{0.006944444444444444}{\pi}, -0.0007716049382716049 \cdot \left(r \cdot \frac{r}{s \cdot \pi}\right)\right)}{s \cdot s} + \left(\frac{0.125}{r \cdot \pi} + \frac{-0.041666666666666664}{s \cdot \pi}\right)}{s}\right) \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (fma
  (/ (exp (- (/ r s))) r)
  (/ 0.125 (* s PI))
  (/
   (+
    (/
     (fma
      r
      (/ 0.006944444444444444 PI)
      (* -0.0007716049382716049 (* r (/ r (* s PI)))))
     (* s s))
    (+ (/ 0.125 (* r PI)) (/ -0.041666666666666664 (* s PI))))
   s)))

C

float code(float s, float r) {
	return fmaf((expf(-(r / s)) / r), (0.125f / (s * ((float) M_PI))), (((fmaf(r, (0.006944444444444444f / ((float) M_PI)), (-0.0007716049382716049f * (r * (r / (s * ((float) M_PI)))))) / (s * s)) + ((0.125f / (r * ((float) M_PI))) + (-0.041666666666666664f / (s * ((float) M_PI))))) / s));
}

Julia

function code(s, r)
	return fma(Float32(exp(Float32(-Float32(r / s))) / r), Float32(Float32(0.125) / Float32(s * Float32(pi))), Float32(Float32(Float32(fma(r, Float32(Float32(0.006944444444444444) / Float32(pi)), Float32(Float32(-0.0007716049382716049) * Float32(r * Float32(r / Float32(s * Float32(pi)))))) / Float32(s * s)) + Float32(Float32(Float32(0.125) / Float32(r * 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

   \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{\frac{-1}{144} \cdot \frac{r}{\mathsf{PI}\left(\right)} + \frac{1}{1296} \cdot \frac{{r}^{2}}{s \cdot \mathsf{PI}\left(\right)}}{s} - \frac{1}{24} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{s} - \frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s}} \]

5. Simplified 8.3%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \color{blue}{\frac{\frac{\mathsf{fma}\left(-0.0007716049382716049, \frac{r \cdot r}{s \cdot \pi}, \frac{r \cdot 0.006944444444444444}{\pi}\right)}{s \cdot s} + \left(\frac{0.125}{r \cdot \pi} + \frac{-0.041666666666666664}{s \cdot \pi}\right)}{s}} \]

7. Applied egg-rr 8.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{e^{\frac{r}{-s}}}{r}, \frac{0.125}{s \cdot \pi}, \frac{\frac{\mathsf{fma}\left(r, \frac{0.006944444444444444}{\pi}, -0.0007716049382716049 \cdot \left(r \cdot \frac{r}{s \cdot \pi}\right)\right)}{s \cdot s} + \left(\frac{0.125}{r \cdot \pi} + \frac{-0.041666666666666664}{s \cdot \pi}\right)}{s}\right)} \]

8. Final simplification 8.3%

   \[\leadsto \mathsf{fma}\left(\frac{e^{-\frac{r}{s}}}{r}, \frac{0.125}{s \cdot \pi}, \frac{\frac{\mathsf{fma}\left(r, \frac{0.006944444444444444}{\pi}, -0.0007716049382716049 \cdot \left(r \cdot \frac{r}{s \cdot \pi}\right)\right)}{s \cdot s} + \left(\frac{0.125}{r \cdot \pi} + \frac{-0.041666666666666664}{s \cdot \pi}\right)}{s}\right) \]
9. Add Preprocessing

Alternative 6: 10.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{0.125}{r \cdot \left(s \cdot \pi\right)}, e^{-\frac{r}{s}}, \frac{\frac{\mathsf{fma}\left(r, \frac{0.006944444444444444}{\pi}, -0.0007716049382716049 \cdot \left(r \cdot \frac{r}{s \cdot \pi}\right)\right)}{s \cdot s} + \left(\frac{0.125}{r \cdot \pi} + \frac{-0.041666666666666664}{s \cdot \pi}\right)}{s}\right) \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (fma
  (/ 0.125 (* r (* s PI)))
  (exp (- (/ r s)))
  (/
   (+
    (/
     (fma
      r
      (/ 0.006944444444444444 PI)
      (* -0.0007716049382716049 (* r (/ r (* s PI)))))
     (* s s))
    (+ (/ 0.125 (* r PI)) (/ -0.041666666666666664 (* s PI))))
   s)))

C

float code(float s, float r) {
	return fmaf((0.125f / (r * (s * ((float) M_PI)))), expf(-(r / s)), (((fmaf(r, (0.006944444444444444f / ((float) M_PI)), (-0.0007716049382716049f * (r * (r / (s * ((float) M_PI)))))) / (s * s)) + ((0.125f / (r * ((float) M_PI))) + (-0.041666666666666664f / (s * ((float) M_PI))))) / s));
}

Julia

function code(s, r)
	return fma(Float32(Float32(0.125) / Float32(r * Float32(s * Float32(pi)))), exp(Float32(-Float32(r / s))), Float32(Float32(Float32(fma(r, Float32(Float32(0.006944444444444444) / Float32(pi)), Float32(Float32(-0.0007716049382716049) * Float32(r * Float32(r / Float32(s * Float32(pi)))))) / Float32(s * s)) + Float32(Float32(Float32(0.125) / Float32(r * 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

   \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{\frac{-1}{144} \cdot \frac{r}{\mathsf{PI}\left(\right)} + \frac{1}{1296} \cdot \frac{{r}^{2}}{s \cdot \mathsf{PI}\left(\right)}}{s} - \frac{1}{24} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{s} - \frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s}} \]

5. Simplified 8.3%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \color{blue}{\frac{\frac{\mathsf{fma}\left(-0.0007716049382716049, \frac{r \cdot r}{s \cdot \pi}, \frac{r \cdot 0.006944444444444444}{\pi}\right)}{s \cdot s} + \left(\frac{0.125}{r \cdot \pi} + \frac{-0.041666666666666664}{s \cdot \pi}\right)}{s}} \]

7. Applied egg-rr 8.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0.125}{r \cdot \left(s \cdot \pi\right)}, e^{\frac{r}{-s}}, \frac{\frac{\mathsf{fma}\left(r, \frac{0.006944444444444444}{\pi}, -0.0007716049382716049 \cdot \left(r \cdot \frac{r}{s \cdot \pi}\right)\right)}{s \cdot s} + \left(\frac{0.125}{r \cdot \pi} + \frac{-0.041666666666666664}{s \cdot \pi}\right)}{s}\right)} \]

8. Final simplification 8.3%

   \[\leadsto \mathsf{fma}\left(\frac{0.125}{r \cdot \left(s \cdot \pi\right)}, e^{-\frac{r}{s}}, \frac{\frac{\mathsf{fma}\left(r, \frac{0.006944444444444444}{\pi}, -0.0007716049382716049 \cdot \left(r \cdot \frac{r}{s \cdot \pi}\right)\right)}{s \cdot s} + \left(\frac{0.125}{r \cdot \pi} + \frac{-0.041666666666666664}{s \cdot \pi}\right)}{s}\right) \]
9. Add Preprocessing

Alternative 7: 11.0% accurate, 1.4× speedup

Math

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

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/ (* (exp (- (/ r s))) 0.125) (* r (* s PI)))
  (/
   (+
    (/ 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)) * 0.125f) / (r * (s * ((float) M_PI)))) + (((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(-Float32(r / s))) * Float32(0.125)) / Float32(r * Float32(s * Float32(pi)))) + 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. 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. clear-num N/A

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

   4. inv-pow N/A

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

   5. lift-*.f32 N/A

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

   6. *-commutative N/A

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

   7. associate-/l* N/A

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

   8. unpow-prod-down N/A

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

   9. inv-pow N/A

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

   10. inv-pow N/A

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

   11. clear-num N/A

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

   12. lower-*.f32 N/A

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

   13. lower-/.f32 N/A

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

   14. frac-2neg N/A

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

   15. lift-neg.f32 N/A

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

   16. remove-double-neg N/A

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

   17. div-inv N/A

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

   18. metadata-eval N/A

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

   19. metadata-eval N/A

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

   20. metadata-eval N/A

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

   21. lower-*.f32 N/A

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

   22. metadata-eval 99.7

       \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{1}{s} \cdot \left(r \cdot \color{blue}{-0.3333333333333333}\right)}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

4. Applied egg-rr 99.7%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\color{blue}{\frac{1}{s} \cdot \left(r \cdot -0.3333333333333333\right)}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

6. Applied egg-rr 99.7%

   \[\leadsto \color{blue}{\frac{e^{\frac{r}{-s}} \cdot 0.125}{r \cdot \left(s \cdot \pi\right)}} + \frac{0.75 \cdot e^{\frac{1}{s} \cdot \left(r \cdot -0.3333333333333333\right)}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

7. Taylor expanded in s around -inf

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

9. Simplified 8.2%

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

10. Final simplification 8.2%

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

Alternative 8: 10.9% 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(-Float32(r / 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

3. Taylor expanded in r around 0

   \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \color{blue}{\frac{r \cdot \left(\frac{1}{144} \cdot \frac{r}{{s}^{3} \cdot \mathsf{PI}\left(\right)} - \frac{1}{24} \cdot \frac{1}{{s}^{2} \cdot \mathsf{PI}\left(\right)}\right) + \frac{1}{8} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}}{r}} \]
4. Step-by-step derivation

   1. lower-/.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \color{blue}{\frac{r \cdot \left(\frac{1}{144} \cdot \frac{r}{{s}^{3} \cdot \mathsf{PI}\left(\right)} - \frac{1}{24} \cdot \frac{1}{{s}^{2} \cdot \mathsf{PI}\left(\right)}\right) + \frac{1}{8} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}}{r}} \]

5. Simplified 8.2%

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

6. Taylor expanded in s around 0

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

   1. lower-/.f32 N/A

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

8. Simplified 8.2%

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

9. Final simplification 8.2%

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

Alternative 9: 10.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{\mathsf{fma}\left(r, \mathsf{fma}\left(r, \frac{0.05555555555555555}{s \cdot s}, \frac{-0.3333333333333333}{s}\right), 1\right)}{r \cdot \left(s \cdot \left(\pi \cdot 6\right)\right)}, 0.75, e^{-\frac{r}{s}} \cdot \frac{0.125}{\pi \cdot \left(r \cdot s\right)}\right) \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (fma
  (/
   (fma
    r
    (fma r (/ 0.05555555555555555 (* s s)) (/ -0.3333333333333333 s))
    1.0)
   (* r (* s (* PI 6.0))))
  0.75
  (* (exp (- (/ r s))) (/ 0.125 (* PI (* r s))))))

C

float code(float s, float r) {
	return fmaf((fmaf(r, fmaf(r, (0.05555555555555555f / (s * s)), (-0.3333333333333333f / s)), 1.0f) / (r * (s * (((float) M_PI) * 6.0f)))), 0.75f, (expf(-(r / s)) * (0.125f / (((float) M_PI) * (r * s)))));
}

Julia

function code(s, r)
	return fma(Float32(fma(r, fma(r, Float32(Float32(0.05555555555555555) / Float32(s * s)), Float32(Float32(-0.3333333333333333) / s)), Float32(1.0)) / Float32(r * Float32(s * Float32(Float32(pi) * Float32(6.0))))), Float32(0.75), Float32(exp(Float32(-Float32(r / s))) * Float32(Float32(0.125) / Float32(Float32(pi) * Float32(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

4. Applied egg-rr 98.2%

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

5. Taylor expanded in r around 0

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

   1. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{r \cdot \left(\frac{1}{18} \cdot \frac{r}{{s}^{2}} - \frac{1}{3} \cdot \frac{1}{s}\right) + 1}}{r \cdot \left(s \cdot \left(\mathsf{PI}\left(\right) \cdot 6\right)\right)}, \frac{3}{4}, e^{\frac{r}{\mathsf{neg}\left(s\right)}} \cdot \frac{\frac{1}{8}}{\mathsf{PI}\left(\right) \cdot \left(r \cdot s\right)}\right) \]

   2. lower-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\mathsf{fma}\left(r, \frac{1}{18} \cdot \frac{r}{{s}^{2}} - \frac{1}{3} \cdot \frac{1}{s}, 1\right)}}{r \cdot \left(s \cdot \left(\mathsf{PI}\left(\right) \cdot 6\right)\right)}, \frac{3}{4}, e^{\frac{r}{\mathsf{neg}\left(s\right)}} \cdot \frac{\frac{1}{8}}{\mathsf{PI}\left(\right) \cdot \left(r \cdot s\right)}\right) \]

   3. sub-neg N/A

      \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(r, \color{blue}{\frac{1}{18} \cdot \frac{r}{{s}^{2}} + \left(\mathsf{neg}\left(\frac{1}{3} \cdot \frac{1}{s}\right)\right)}, 1\right)}{r \cdot \left(s \cdot \left(\mathsf{PI}\left(\right) \cdot 6\right)\right)}, \frac{3}{4}, e^{\frac{r}{\mathsf{neg}\left(s\right)}} \cdot \frac{\frac{1}{8}}{\mathsf{PI}\left(\right) \cdot \left(r \cdot s\right)}\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(r, \color{blue}{\frac{r}{{s}^{2}} \cdot \frac{1}{18}} + \left(\mathsf{neg}\left(\frac{1}{3} \cdot \frac{1}{s}\right)\right), 1\right)}{r \cdot \left(s \cdot \left(\mathsf{PI}\left(\right) \cdot 6\right)\right)}, \frac{3}{4}, e^{\frac{r}{\mathsf{neg}\left(s\right)}} \cdot \frac{\frac{1}{8}}{\mathsf{PI}\left(\right) \cdot \left(r \cdot s\right)}\right) \]

   5. associate-*l/ N/A

      \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(r, \color{blue}{\frac{r \cdot \frac{1}{18}}{{s}^{2}}} + \left(\mathsf{neg}\left(\frac{1}{3} \cdot \frac{1}{s}\right)\right), 1\right)}{r \cdot \left(s \cdot \left(\mathsf{PI}\left(\right) \cdot 6\right)\right)}, \frac{3}{4}, e^{\frac{r}{\mathsf{neg}\left(s\right)}} \cdot \frac{\frac{1}{8}}{\mathsf{PI}\left(\right) \cdot \left(r \cdot s\right)}\right) \]

   6. associate-*r/ N/A

      \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(r, \color{blue}{r \cdot \frac{\frac{1}{18}}{{s}^{2}}} + \left(\mathsf{neg}\left(\frac{1}{3} \cdot \frac{1}{s}\right)\right), 1\right)}{r \cdot \left(s \cdot \left(\mathsf{PI}\left(\right) \cdot 6\right)\right)}, \frac{3}{4}, e^{\frac{r}{\mathsf{neg}\left(s\right)}} \cdot \frac{\frac{1}{8}}{\mathsf{PI}\left(\right) \cdot \left(r \cdot s\right)}\right) \]

   7. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(r, r \cdot \frac{\color{blue}{\frac{1}{18} \cdot 1}}{{s}^{2}} + \left(\mathsf{neg}\left(\frac{1}{3} \cdot \frac{1}{s}\right)\right), 1\right)}{r \cdot \left(s \cdot \left(\mathsf{PI}\left(\right) \cdot 6\right)\right)}, \frac{3}{4}, e^{\frac{r}{\mathsf{neg}\left(s\right)}} \cdot \frac{\frac{1}{8}}{\mathsf{PI}\left(\right) \cdot \left(r \cdot s\right)}\right) \]

   8. associate-*r/ N/A

      \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(r, r \cdot \color{blue}{\left(\frac{1}{18} \cdot \frac{1}{{s}^{2}}\right)} + \left(\mathsf{neg}\left(\frac{1}{3} \cdot \frac{1}{s}\right)\right), 1\right)}{r \cdot \left(s \cdot \left(\mathsf{PI}\left(\right) \cdot 6\right)\right)}, \frac{3}{4}, e^{\frac{r}{\mathsf{neg}\left(s\right)}} \cdot \frac{\frac{1}{8}}{\mathsf{PI}\left(\right) \cdot \left(r \cdot s\right)}\right) \]

   9. lower-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(r, \color{blue}{\mathsf{fma}\left(r, \frac{1}{18} \cdot \frac{1}{{s}^{2}}, \mathsf{neg}\left(\frac{1}{3} \cdot \frac{1}{s}\right)\right)}, 1\right)}{r \cdot \left(s \cdot \left(\mathsf{PI}\left(\right) \cdot 6\right)\right)}, \frac{3}{4}, e^{\frac{r}{\mathsf{neg}\left(s\right)}} \cdot \frac{\frac{1}{8}}{\mathsf{PI}\left(\right) \cdot \left(r \cdot s\right)}\right) \]

   10. associate-*r/ N/A

       \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(r, \mathsf{fma}\left(r, \color{blue}{\frac{\frac{1}{18} \cdot 1}{{s}^{2}}}, \mathsf{neg}\left(\frac{1}{3} \cdot \frac{1}{s}\right)\right), 1\right)}{r \cdot \left(s \cdot \left(\mathsf{PI}\left(\right) \cdot 6\right)\right)}, \frac{3}{4}, e^{\frac{r}{\mathsf{neg}\left(s\right)}} \cdot \frac{\frac{1}{8}}{\mathsf{PI}\left(\right) \cdot \left(r \cdot s\right)}\right) \]

   11. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(r, \mathsf{fma}\left(r, \frac{\color{blue}{\frac{1}{18}}}{{s}^{2}}, \mathsf{neg}\left(\frac{1}{3} \cdot \frac{1}{s}\right)\right), 1\right)}{r \cdot \left(s \cdot \left(\mathsf{PI}\left(\right) \cdot 6\right)\right)}, \frac{3}{4}, e^{\frac{r}{\mathsf{neg}\left(s\right)}} \cdot \frac{\frac{1}{8}}{\mathsf{PI}\left(\right) \cdot \left(r \cdot s\right)}\right) \]

   12. lower-/.f32 N/A

       \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(r, \mathsf{fma}\left(r, \color{blue}{\frac{\frac{1}{18}}{{s}^{2}}}, \mathsf{neg}\left(\frac{1}{3} \cdot \frac{1}{s}\right)\right), 1\right)}{r \cdot \left(s \cdot \left(\mathsf{PI}\left(\right) \cdot 6\right)\right)}, \frac{3}{4}, e^{\frac{r}{\mathsf{neg}\left(s\right)}} \cdot \frac{\frac{1}{8}}{\mathsf{PI}\left(\right) \cdot \left(r \cdot s\right)}\right) \]

   13. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(r, \mathsf{fma}\left(r, \frac{\frac{1}{18}}{\color{blue}{s \cdot s}}, \mathsf{neg}\left(\frac{1}{3} \cdot \frac{1}{s}\right)\right), 1\right)}{r \cdot \left(s \cdot \left(\mathsf{PI}\left(\right) \cdot 6\right)\right)}, \frac{3}{4}, e^{\frac{r}{\mathsf{neg}\left(s\right)}} \cdot \frac{\frac{1}{8}}{\mathsf{PI}\left(\right) \cdot \left(r \cdot s\right)}\right) \]

   14. lower-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(r, \mathsf{fma}\left(r, \frac{\frac{1}{18}}{\color{blue}{s \cdot s}}, \mathsf{neg}\left(\frac{1}{3} \cdot \frac{1}{s}\right)\right), 1\right)}{r \cdot \left(s \cdot \left(\mathsf{PI}\left(\right) \cdot 6\right)\right)}, \frac{3}{4}, e^{\frac{r}{\mathsf{neg}\left(s\right)}} \cdot \frac{\frac{1}{8}}{\mathsf{PI}\left(\right) \cdot \left(r \cdot s\right)}\right) \]

   15. associate-*r/ N/A

       \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(r, \mathsf{fma}\left(r, \frac{\frac{1}{18}}{s \cdot s}, \mathsf{neg}\left(\color{blue}{\frac{\frac{1}{3} \cdot 1}{s}}\right)\right), 1\right)}{r \cdot \left(s \cdot \left(\mathsf{PI}\left(\right) \cdot 6\right)\right)}, \frac{3}{4}, e^{\frac{r}{\mathsf{neg}\left(s\right)}} \cdot \frac{\frac{1}{8}}{\mathsf{PI}\left(\right) \cdot \left(r \cdot s\right)}\right) \]

   16. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(r, \mathsf{fma}\left(r, \frac{\frac{1}{18}}{s \cdot s}, \mathsf{neg}\left(\frac{\color{blue}{\frac{1}{3}}}{s}\right)\right), 1\right)}{r \cdot \left(s \cdot \left(\mathsf{PI}\left(\right) \cdot 6\right)\right)}, \frac{3}{4}, e^{\frac{r}{\mathsf{neg}\left(s\right)}} \cdot \frac{\frac{1}{8}}{\mathsf{PI}\left(\right) \cdot \left(r \cdot s\right)}\right) \]

   17. distribute-neg-frac N/A

       \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(r, \mathsf{fma}\left(r, \frac{\frac{1}{18}}{s \cdot s}, \color{blue}{\frac{\mathsf{neg}\left(\frac{1}{3}\right)}{s}}\right), 1\right)}{r \cdot \left(s \cdot \left(\mathsf{PI}\left(\right) \cdot 6\right)\right)}, \frac{3}{4}, e^{\frac{r}{\mathsf{neg}\left(s\right)}} \cdot \frac{\frac{1}{8}}{\mathsf{PI}\left(\right) \cdot \left(r \cdot s\right)}\right) \]

   18. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(r, \mathsf{fma}\left(r, \frac{\frac{1}{18}}{s \cdot s}, \frac{\color{blue}{\frac{-1}{3}}}{s}\right), 1\right)}{r \cdot \left(s \cdot \left(\mathsf{PI}\left(\right) \cdot 6\right)\right)}, \frac{3}{4}, e^{\frac{r}{\mathsf{neg}\left(s\right)}} \cdot \frac{\frac{1}{8}}{\mathsf{PI}\left(\right) \cdot \left(r \cdot s\right)}\right) \]

   19. lower-/.f32 8.1

       \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(r, \mathsf{fma}\left(r, \frac{0.05555555555555555}{s \cdot s}, \color{blue}{\frac{-0.3333333333333333}{s}}\right), 1\right)}{r \cdot \left(s \cdot \left(\pi \cdot 6\right)\right)}, 0.75, e^{\frac{r}{-s}} \cdot \frac{0.125}{\pi \cdot \left(r \cdot s\right)}\right) \]

7. Simplified 8.1%

   \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\mathsf{fma}\left(r, \mathsf{fma}\left(r, \frac{0.05555555555555555}{s \cdot s}, \frac{-0.3333333333333333}{s}\right), 1\right)}}{r \cdot \left(s \cdot \left(\pi \cdot 6\right)\right)}, 0.75, e^{\frac{r}{-s}} \cdot \frac{0.125}{\pi \cdot \left(r \cdot s\right)}\right) \]

8. Final simplification 8.1%

   \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(r, \mathsf{fma}\left(r, \frac{0.05555555555555555}{s \cdot s}, \frac{-0.3333333333333333}{s}\right), 1\right)}{r \cdot \left(s \cdot \left(\pi \cdot 6\right)\right)}, 0.75, e^{-\frac{r}{s}} \cdot \frac{0.125}{\pi \cdot \left(r \cdot s\right)}\right) \]
9. Add Preprocessing

Alternative 10: 9.7% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.25) * exp(-(r / s))) / (r * (s * (single(2.0) * single(pi))))) + ((single(0.125) / (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

   \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{\frac{-1}{144} \cdot \frac{r}{\mathsf{PI}\left(\right)} + \frac{1}{1296} \cdot \frac{{r}^{2}}{s \cdot \mathsf{PI}\left(\right)}}{s} - \frac{1}{24} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{s} - \frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s}} \]

5. Simplified 8.3%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \color{blue}{\frac{\frac{\mathsf{fma}\left(-0.0007716049382716049, \frac{r \cdot r}{s \cdot \pi}, \frac{r \cdot 0.006944444444444444}{\pi}\right)}{s \cdot s} + \left(\frac{0.125}{r \cdot \pi} + \frac{-0.041666666666666664}{s \cdot \pi}\right)}{s}} \]

6. Taylor expanded in r around 0

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-PI.f32 7.8

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

8. Simplified 7.8%

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

9. Final simplification 7.8%

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

Alternative 11: 9.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := r \cdot \left(s \cdot \pi\right)\\ \frac{e^{-\frac{r}{s}} \cdot 0.125}{t\_0} + \frac{0.125}{t\_0} \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (let* ((t_0 (* r (* s PI))))
   (+ (/ (* (exp (- (/ r s))) 0.125) t_0) (/ 0.125 t_0))))

C

float code(float s, float r) {
	float t_0 = r * (s * ((float) M_PI));
	return ((expf(-(r / s)) * 0.125f) / t_0) + (0.125f / t_0);
}

Julia

function code(s, r)
	t_0 = Float32(r * Float32(s * Float32(pi)))
	return Float32(Float32(Float32(exp(Float32(-Float32(r / s))) * Float32(0.125)) / t_0) + Float32(Float32(0.125) / t_0))
end

MATLAB

function tmp = code(s, r)
	t_0 = r * (s * single(pi));
	tmp = ((exp(-(r / s)) * single(0.125)) / t_0) + (single(0.125) / t_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. 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. clear-num N/A

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

   4. inv-pow N/A

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

   5. lift-*.f32 N/A

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

   6. *-commutative N/A

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

   7. associate-/l* N/A

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

   8. unpow-prod-down N/A

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

   9. inv-pow N/A

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

   10. inv-pow N/A

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

   11. clear-num N/A

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

   12. lower-*.f32 N/A

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

   13. lower-/.f32 N/A

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

   14. frac-2neg N/A

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

   15. lift-neg.f32 N/A

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

   16. remove-double-neg N/A

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

   17. div-inv N/A

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

   18. metadata-eval N/A

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

   19. metadata-eval N/A

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

   20. metadata-eval N/A

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

   21. lower-*.f32 N/A

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

   22. metadata-eval 99.7

       \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{1}{s} \cdot \left(r \cdot \color{blue}{-0.3333333333333333}\right)}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

4. Applied egg-rr 99.7%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\color{blue}{\frac{1}{s} \cdot \left(r \cdot -0.3333333333333333\right)}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

6. Applied egg-rr 99.7%

   \[\leadsto \color{blue}{\frac{e^{\frac{r}{-s}} \cdot 0.125}{r \cdot \left(s \cdot \pi\right)}} + \frac{0.75 \cdot e^{\frac{1}{s} \cdot \left(r \cdot -0.3333333333333333\right)}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

7. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 7.8

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

9. Simplified 7.8%

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

10. Final simplification 7.8%

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

Alternative 12: 9.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.125}{r \cdot \left(s \cdot \pi\right)}\\ \mathsf{fma}\left(t\_0, e^{-\frac{r}{s}}, t\_0\right) \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (let* ((t_0 (/ 0.125 (* r (* s PI))))) (fma t_0 (exp (- (/ r s))) t_0)))

C

float code(float s, float r) {
	float t_0 = 0.125f / (r * (s * ((float) M_PI)));
	return fmaf(t_0, expf(-(r / s)), t_0);
}

Julia

function code(s, r)
	t_0 = Float32(Float32(0.125) / Float32(r * Float32(s * Float32(pi))))
	return fma(t_0, exp(Float32(-Float32(r / s))), t_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 r around 0

   \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \color{blue}{\frac{r \cdot \left(\frac{1}{144} \cdot \frac{r}{{s}^{3} \cdot \mathsf{PI}\left(\right)} - \frac{1}{24} \cdot \frac{1}{{s}^{2} \cdot \mathsf{PI}\left(\right)}\right) + \frac{1}{8} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}}{r}} \]
4. Step-by-step derivation

   1. lower-/.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \color{blue}{\frac{r \cdot \left(\frac{1}{144} \cdot \frac{r}{{s}^{3} \cdot \mathsf{PI}\left(\right)} - \frac{1}{24} \cdot \frac{1}{{s}^{2} \cdot \mathsf{PI}\left(\right)}\right) + \frac{1}{8} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}}{r}} \]

5. Simplified 8.2%

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

7. Applied egg-rr 8.1%

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

8. Taylor expanded in r around 0

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 7.8

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

10. Simplified 7.8%

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

11. Final simplification 7.8%

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

Alternative 13: 10.2% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\mathsf{fma}\left(0.06944444444444445, \frac{r}{\pi}, -0.020833333333333332 \cdot \frac{r \cdot r}{s \cdot \pi}\right)}{s \cdot s} + \left(\frac{0.25}{r \cdot \pi} + \frac{-0.16666666666666666}{s \cdot \pi}\right)}{s} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (/
  (+
   (/
    (fma
     0.06944444444444445
     (/ r PI)
     (* -0.020833333333333332 (/ (* r r) (* s PI))))
    (* s s))
   (+ (/ 0.25 (* r PI)) (/ -0.16666666666666666 (* s PI))))
  s))

C

float code(float s, float r) {
	return ((fmaf(0.06944444444444445f, (r / ((float) M_PI)), (-0.020833333333333332f * ((r * r) / (s * ((float) M_PI))))) / (s * s)) + ((0.25f / (r * ((float) M_PI))) + (-0.16666666666666666f / (s * ((float) M_PI))))) / s;
}

Julia

function code(s, r)
	return Float32(Float32(Float32(fma(Float32(0.06944444444444445), Float32(r / Float32(pi)), Float32(Float32(-0.020833333333333332) * Float32(Float32(r * r) / Float32(s * Float32(pi))))) / Float32(s * s)) + Float32(Float32(Float32(0.25) / Float32(r * Float32(pi))) + 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 r around 0

   \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \color{blue}{\frac{r \cdot \left(\frac{1}{144} \cdot \frac{r}{{s}^{3} \cdot \mathsf{PI}\left(\right)} - \frac{1}{24} \cdot \frac{1}{{s}^{2} \cdot \mathsf{PI}\left(\right)}\right) + \frac{1}{8} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}}{r}} \]
4. Step-by-step derivation

   1. lower-/.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \color{blue}{\frac{r \cdot \left(\frac{1}{144} \cdot \frac{r}{{s}^{3} \cdot \mathsf{PI}\left(\right)} - \frac{1}{24} \cdot \frac{1}{{s}^{2} \cdot \mathsf{PI}\left(\right)}\right) + \frac{1}{8} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}}{r}} \]

5. Simplified 8.2%

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

7. Applied egg-rr 8.1%

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

8. Taylor expanded in s around -inf

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

10. Simplified 7.6%

    \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(0.06944444444444445, \frac{r}{\pi}, -0.020833333333333332 \cdot \frac{r \cdot r}{s \cdot \pi}\right)}{s \cdot s} + \left(\frac{0.25}{r \cdot \pi} + \frac{-0.16666666666666666}{s \cdot \pi}\right)}{s}} \]
11. Add Preprocessing

Alternative 14: 10.3% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.7%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
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 7.6%

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

Alternative 15: 9.2% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.25) / (r * s)) / sqrt(single(pi))) / sqrt(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 7.5

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

5. Simplified 7.5%

   \[\leadsto \color{blue}{\frac{0.25}{r \cdot \left(s \cdot \pi\right)}} \]
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. associate-*r* N/A

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

   3. lift-*.f32 N/A

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

   4. lower-*.f32 7.5

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

7. Applied egg-rr 7.5%

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

   1. lift-*.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. associate-/r* N/A

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

   4. lift-PI.f32 N/A

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

   5. add-sqr-sqrt N/A

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

   6. associate-/r* N/A

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

   7. lower-/.f32 N/A

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

   8. lower-/.f32 N/A

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

   9. lower-/.f32 N/A

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

   10. lift-PI.f32 N/A

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

   11. lower-sqrt.f32 N/A

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

   12. lift-PI.f32 N/A

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

   13. lower-sqrt.f32 7.5

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

9. Applied egg-rr 7.5%

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

Alternative 16: 9.2% accurate, 6.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = single(0.25) / (r * (sqrt(single(pi)) * (s * sqrt(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 7.5

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

5. Simplified 7.5%

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

   1. add-sqr-sqrt N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. lower-sqrt.f32 N/A

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

   7. lift-PI.f32 N/A

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

   8. lower-sqrt.f32 7.5

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

7. Applied egg-rr 7.5%

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

8. Final simplification 7.5%

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

Alternative 17: 9.2% 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 99.7%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
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 7.5

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

5. Simplified 7.5%

   \[\leadsto \color{blue}{\frac{0.25}{r \cdot \left(s \cdot \pi\right)}} \]
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. lift-*.f32 N/A

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

   4. associate-*r* N/A

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

   5. lift-*.f32 N/A

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

   6. *-commutative N/A

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

   7. associate-/r* N/A

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

   8. lower-/.f32 N/A

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

   9. lower-/.f32 7.5

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

7. Applied egg-rr 7.5%

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

Alternative 18: 9.2% 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 99.7%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
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 7.5

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

5. Simplified 7.5%

   \[\leadsto \color{blue}{\frac{0.25}{r \cdot \left(s \cdot \pi\right)}} \]
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. associate-*r* N/A

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

   3. lift-*.f32 N/A

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

   4. lower-*.f32 7.5

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

7. Applied egg-rr 7.5%

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

8. Final simplification 7.5%

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

Alternative 19: 9.2% accurate, 13.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
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 7.5

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

5. Simplified 7.5%

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

Reproduce

herbie shell --seed 2024207 
(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))))