Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.5%

Time: 15.6s

Alternatives: 17

Speedup: N/A×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = (single(0.125) * (exp(-(r / s)) / (r * (s * single(pi))))) + (((exp(((r * single(-0.3333333333333333)) / s)) * single(-0.75)) / 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. frac-2neg N/A

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

   2. *-commutative N/A

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

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

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

   4. associate-/r* N/A

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

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

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

4. Applied egg-rr 99.7%

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

   1. associate-/r* N/A

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

   2. div-inv N/A

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

   3. metadata-eval N/A

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

   4. associate-*l/ N/A

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

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

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

   6. *-lowering-*.f32 99.7

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

6. Applied egg-rr 99.7%

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

7. Taylor expanded in r around inf

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

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

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

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

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

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

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

   4. mul-1-neg N/A

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

   5. distribute-neg-frac2 N/A

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

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

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

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

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

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

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

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

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

   10. PI-lowering-PI.f32 99.7

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

9. Simplified 99.7%

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

10. Final simplification 99.7%

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

Alternative 2: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{0.125 \cdot \left(\frac{e^{-\frac{r}{s}}}{s \cdot \pi} + \frac{e^{\frac{1}{s \cdot \frac{-3}{r}}}}{s \cdot \pi}\right)}{r} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (/
  (*
   0.125
   (+
    (/ (exp (- (/ r s))) (* s PI))
    (/ (exp (/ 1.0 (* s (/ -3.0 r)))) (* s PI))))
  r))

C

float code(float s, float r) {
	return (0.125f * ((expf(-(r / s)) / (s * ((float) M_PI))) + (expf((1.0f / (s * (-3.0f / r)))) / (s * ((float) M_PI))))) / r;
}

Julia

function code(s, r)
	return Float32(Float32(Float32(0.125) * Float32(Float32(exp(Float32(-Float32(r / s))) / Float32(s * Float32(pi))) + Float32(exp(Float32(Float32(1.0) / Float32(s * Float32(Float32(-3.0) / r)))) / Float32(s * Float32(pi))))) / r)
end

MATLAB

function tmp = code(s, r)
	tmp = (single(0.125) * ((exp(-(r / s)) / (s * single(pi))) + (exp((single(1.0) / (s * (single(-3.0) / r)))) / (s * single(pi))))) / r;
end

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in r around inf

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

   1. distribute-lft-out N/A

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

   2. metadata-eval N/A

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

   3. associate-*r* N/A

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

   4. distribute-lft-out N/A

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

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

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

5. Simplified 99.7%

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

   1. metadata-eval N/A

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

   2. div-inv N/A

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

   3. associate-/r* N/A

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

   4. clear-num N/A

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

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

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

   6. associate-/l* N/A

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

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

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

   8. /-lowering-/.f32 99.7

      \[\leadsto \frac{0.125 \cdot \left(\frac{e^{\frac{r}{-s}}}{s \cdot \pi} + \frac{e^{\frac{1}{s \cdot \color{blue}{\frac{-3}{r}}}}}{s \cdot \pi}\right)}{r} \]

7. Applied egg-rr 99.7%

   \[\leadsto \frac{0.125 \cdot \left(\frac{e^{\frac{r}{-s}}}{s \cdot \pi} + \frac{e^{\color{blue}{\frac{1}{s \cdot \frac{-3}{r}}}}}{s \cdot \pi}\right)}{r} \]

8. Final simplification 99.7%

   \[\leadsto \frac{0.125 \cdot \left(\frac{e^{-\frac{r}{s}}}{s \cdot \pi} + \frac{e^{\frac{1}{s \cdot \frac{-3}{r}}}}{s \cdot \pi}\right)}{r} \]
9. Add Preprocessing

Alternative 3: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in r around inf

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

   1. distribute-lft-out N/A

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

   2. metadata-eval N/A

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

   3. associate-*r* N/A

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

   4. distribute-lft-out N/A

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

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

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

5. Simplified 99.7%

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

   1. metadata-eval N/A

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

   2. div-inv N/A

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

   3. associate-/r* N/A

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

   4. clear-num N/A

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

   5. associate-/r/ N/A

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

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

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

   7. *-commutative N/A

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

   8. associate-/r* N/A

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

   9. metadata-eval N/A

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

   10. /-lowering-/.f32 99.7

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

7. Applied egg-rr 99.7%

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

8. Final simplification 99.7%

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

Alternative 4: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in s around 0

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

   1. distribute-lft-out N/A

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

   2. *-commutative N/A

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

   3. associate-/l* N/A

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

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

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

5. Simplified 99.7%

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

6. Final simplification 99.7%

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

Alternative 5: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

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

   1. distribute-lft-out N/A

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

   2. metadata-eval N/A

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

   3. associate-*r* N/A

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

   4. distribute-lft-out N/A

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

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

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

5. Simplified 99.7%

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

6. Taylor expanded in s around 0

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

   1. associate-*r/ N/A

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

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

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

8. Simplified 99.7%

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

9. Final simplification 99.7%

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

Alternative 6: 99.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{0.125 \cdot \left(\frac{1}{s \cdot \pi} \cdot \left(e^{-\frac{r}{s}} + e^{\frac{r}{s \cdot -3}}\right)\right)}{r} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (/
  (* 0.125 (* (/ 1.0 (* s PI)) (+ (exp (- (/ r s))) (exp (/ r (* s -3.0))))))
  r))

C

float code(float s, float r) {
	return (0.125f * ((1.0f / (s * ((float) M_PI))) * (expf(-(r / s)) + expf((r / (s * -3.0f)))))) / r;
}

Julia

function code(s, r)
	return Float32(Float32(Float32(0.125) * Float32(Float32(Float32(1.0) / Float32(s * Float32(pi))) * Float32(exp(Float32(-Float32(r / s))) + exp(Float32(r / Float32(s * Float32(-3.0))))))) / r)
end

MATLAB

function tmp = code(s, r)
	tmp = (single(0.125) * ((single(1.0) / (s * single(pi))) * (exp(-(r / s)) + exp((r / (s * single(-3.0))))))) / r;
end

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in r around inf

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

   1. distribute-lft-out N/A

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

   2. metadata-eval N/A

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

   3. associate-*r* N/A

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

   4. distribute-lft-out N/A

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

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

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

5. Simplified 99.7%

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

   1. *-commutative N/A

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

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

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

7. Applied egg-rr 99.7%

   \[\leadsto \frac{\color{blue}{\left(\frac{1}{s \cdot \pi} \cdot \left(e^{\frac{r}{-s}} + e^{\frac{r}{s \cdot -3}}\right)\right) \cdot 0.125}}{r} \]

8. Final simplification 99.7%

   \[\leadsto \frac{0.125 \cdot \left(\frac{1}{s \cdot \pi} \cdot \left(e^{-\frac{r}{s}} + e^{\frac{r}{s \cdot -3}}\right)\right)}{r} \]
9. Add Preprocessing

Alternative 7: 99.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \left(\frac{1}{s \cdot \pi} \cdot \left(e^{-\frac{r}{s}} + e^{\frac{r}{s \cdot -3}}\right)\right) \cdot \frac{0.125}{r} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (*
  (* (/ 1.0 (* s PI)) (+ (exp (- (/ r s))) (exp (/ r (* s -3.0)))))
  (/ 0.125 r)))

C

float code(float s, float r) {
	return ((1.0f / (s * ((float) M_PI))) * (expf(-(r / s)) + expf((r / (s * -3.0f))))) * (0.125f / r);
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(1.0) / Float32(s * Float32(pi))) * Float32(exp(Float32(-Float32(r / s))) + exp(Float32(r / Float32(s * Float32(-3.0)))))) * Float32(Float32(0.125) / r))
end

MATLAB

function tmp = code(s, r)
	tmp = ((single(1.0) / (s * single(pi))) * (exp(-(r / s)) + exp((r / (s * single(-3.0)))))) * (single(0.125) / r);
end

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in r around inf

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

   1. distribute-lft-out N/A

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

   2. metadata-eval N/A

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

   3. associate-*r* N/A

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

   4. distribute-lft-out N/A

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

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

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

5. Simplified 99.7%

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

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

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

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

7. Applied egg-rr 99.7%

   \[\leadsto \color{blue}{\left(\frac{1}{s \cdot \pi} \cdot \left(e^{\frac{r}{-s}} + e^{\frac{r}{s \cdot -3}}\right)\right) \cdot \frac{0.125}{r}} \]

8. Final simplification 99.7%

   \[\leadsto \left(\frac{1}{s \cdot \pi} \cdot \left(e^{-\frac{r}{s}} + e^{\frac{r}{s \cdot -3}}\right)\right) \cdot \frac{0.125}{r} \]
9. Add Preprocessing

Alternative 8: 10.6% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in s around inf

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

5. Simplified 9.3%

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

6. Final simplification 9.3%

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

Alternative 9: 10.6% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in r around inf

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

   1. distribute-lft-out N/A

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

   2. metadata-eval N/A

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

   3. associate-*r* N/A

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

   4. distribute-lft-out N/A

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

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

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

5. Simplified 99.7%

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

6. Taylor expanded in r around 0

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

   1. +-commutative N/A

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

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

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

   3. sub-neg N/A

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

   4. *-commutative N/A

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

   5. associate-*l/ N/A

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

   6. associate-*r/ N/A

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

   7. metadata-eval N/A

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

   8. associate-*r/ N/A

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

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

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

   10. associate-*r/ N/A

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

   11. metadata-eval N/A

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

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

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

   13. unpow2 N/A

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

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

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

   15. associate-*r/ N/A

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

   16. metadata-eval N/A

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

   17. distribute-neg-frac N/A

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

   18. metadata-eval N/A

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

   19. /-lowering-/.f32 9.2

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

8. Simplified 9.2%

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

9. Final simplification 9.2%

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

Alternative 10: 10.6% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in r around inf

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

   1. distribute-lft-out N/A

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

   2. metadata-eval N/A

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

   3. associate-*r* N/A

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

   4. distribute-lft-out N/A

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

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

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

5. Simplified 99.7%

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

6. Taylor expanded in r around 0

   \[\leadsto \frac{\frac{1}{8} \cdot \left(\frac{e^{\frac{r}{\mathsf{neg}\left(s\right)}}}{s \cdot \mathsf{PI}\left(\right)} + \frac{\color{blue}{1 + r \cdot \left(r \cdot \left(\frac{-1}{162} \cdot \frac{r}{{s}^{3}} + \frac{1}{18} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{3} \cdot \frac{1}{s}\right)}}{s \cdot \mathsf{PI}\left(\right)}\right)}{r} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\frac{1}{8} \cdot \left(\frac{e^{\frac{r}{\mathsf{neg}\left(s\right)}}}{s \cdot \mathsf{PI}\left(\right)} + \frac{\color{blue}{r \cdot \left(r \cdot \left(\frac{-1}{162} \cdot \frac{r}{{s}^{3}} + \frac{1}{18} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{3} \cdot \frac{1}{s}\right) + 1}}{s \cdot \mathsf{PI}\left(\right)}\right)}{r} \]

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

      \[\leadsto \frac{\frac{1}{8} \cdot \left(\frac{e^{\frac{r}{\mathsf{neg}\left(s\right)}}}{s \cdot \mathsf{PI}\left(\right)} + \frac{\color{blue}{\mathsf{fma}\left(r, r \cdot \left(\frac{-1}{162} \cdot \frac{r}{{s}^{3}} + \frac{1}{18} \cdot \frac{1}{{s}^{2}}\right) - \frac{1}{3} \cdot \frac{1}{s}, 1\right)}}{s \cdot \mathsf{PI}\left(\right)}\right)}{r} \]

8. Simplified 7.6%

   \[\leadsto \frac{0.125 \cdot \left(\frac{e^{\frac{r}{-s}}}{s \cdot \pi} + \frac{\color{blue}{\mathsf{fma}\left(r, \mathsf{fma}\left(r, \mathsf{fma}\left(\frac{r}{s \cdot \left(s \cdot s\right)}, -0.006172839506172839, \frac{0.05555555555555555}{s \cdot s}\right), \frac{-0.3333333333333333}{s}\right), 1\right)}}{s \cdot \pi}\right)}{r} \]

9. Taylor expanded in r around 0

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

    1. unpow2 N/A

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

    2. associate-/r* N/A

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

    3. associate-/l* N/A

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

    4. associate-*r/ N/A

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

    5. metadata-eval N/A

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

    6. div-sub N/A

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

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

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

    8. sub-neg N/A

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

    9. associate-*r/ N/A

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

    10. *-commutative N/A

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

    11. associate-/l* N/A

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

    12. metadata-eval N/A

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

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

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

    14. /-lowering-/.f32 9.2

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

11. Simplified 9.2%

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

12. Final simplification 9.2%

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

Alternative 11: 10.0% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.25) / Float32(r * Float32(pi))) + fma(r, Float32(Float32(0.06944444444444445) / Float32(s * Float32(s * 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. Step-by-step derivation

   1. frac-2neg N/A

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

   2. *-commutative N/A

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

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

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

   4. associate-/r* N/A

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

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

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

4. Applied egg-rr 99.7%

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

5. Taylor expanded in s around inf

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

7. Simplified 8.8%

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

Alternative 12: 10.0% 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 8.7%

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

Alternative 13: 8.9% accurate, 5.6× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
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. /-lowering-/.f32 N/A

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

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

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

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

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

   4. PI-lowering-PI.f32 8.2

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

5. Simplified 8.2%

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

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

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

   4. *-commutative N/A

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

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

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

   6. PI-lowering-PI.f32 8.2

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

7. Applied egg-rr 8.2%

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

   1. add-sqr-sqrt N/A

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

   4. associate-/r* N/A

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

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

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

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

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

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

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

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

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

   9. associate-*l* N/A

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

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

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

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

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

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

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

   13. PI-lowering-PI.f32 8.2

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

9. Applied egg-rr 8.2%

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

    1. *-commutative N/A

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

    2. associate-*r* N/A

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

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

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

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

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

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

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

    6. PI-lowering-PI.f32 8.2

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

11. Applied egg-rr 8.2%

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

12. Final simplification 8.2%

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

Alternative 14: 8.9% accurate, 5.6× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = (single(0.25) / sqrt(single(pi))) / ((r * 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. /-lowering-/.f32 N/A

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

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

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

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

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

   4. PI-lowering-PI.f32 8.2

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

5. Simplified 8.2%

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

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

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

   4. *-commutative N/A

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

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

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

   6. PI-lowering-PI.f32 8.2

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

7. Applied egg-rr 8.2%

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

   1. add-sqr-sqrt N/A

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

   4. associate-/r* N/A

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

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

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

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

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

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

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

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

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

   9. associate-*l* N/A

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

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

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

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

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

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

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

   13. PI-lowering-PI.f32 8.2

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

9. Applied egg-rr 8.2%

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

    1. associate-*r* N/A

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

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

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

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

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

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

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

    5. PI-lowering-PI.f32 8.2

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

11. Applied egg-rr 8.2%

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

Alternative 15: 8.9% accurate, 10.6× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
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. /-lowering-/.f32 N/A

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

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

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

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

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

   4. PI-lowering-PI.f32 8.2

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

5. Simplified 8.2%

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

   1. *-commutative N/A

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

   2. associate-/r* N/A

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

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

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

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

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

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

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

   6. PI-lowering-PI.f32 8.2

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

7. Applied egg-rr 8.2%

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

Alternative 16: 8.9% 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. /-lowering-/.f32 N/A

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

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

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

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

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

   4. PI-lowering-PI.f32 8.2

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

5. Simplified 8.2%

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

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

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

   4. *-commutative N/A

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

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

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

   6. PI-lowering-PI.f32 8.2

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

7. Applied egg-rr 8.2%

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

8. Final simplification 8.2%

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

Alternative 17: 8.9% 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. /-lowering-/.f32 N/A

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

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

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

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

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

   4. PI-lowering-PI.f32 8.2

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

5. Simplified 8.2%

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

Reproduce

herbie shell --seed 2024204 
(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))))