Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.6%

Time: 15.7s

Alternatives: 18

Speedup: N/A×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

float code(float s, float r) {
	return (0.125f * ((expf(-(r / s)) / (s * ((float) M_PI))) + (powf(((float) M_E), (r / (s * -3.0f))) / (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((Float32(exp(1)) ^ Float32(r / Float32(s * Float32(-3.0)))) / Float32(s * Float32(pi))))) / r)
end

MATLAB

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

Derivation

1. Initial program 99.6%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
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 99.6%

   \[\leadsto \color{blue}{\frac{0.125 \cdot \left(\frac{e^{\frac{r}{-s}}}{s \cdot \pi} + \frac{e^{\frac{r}{s} \cdot -0.3333333333333333}}{s \cdot \pi}\right)}{r}} \]
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. div-inv N/A

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

   6. clear-num N/A

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

   7. exp-prod N/A

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

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

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

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

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

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

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

   11. *-lowering-*.f32 99.7

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

7. Applied egg-rr 99.7%

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

   1. exp-1-e N/A

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

   2. E-lowering-E.f32 99.7

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

9. Applied egg-rr 99.7%

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

10. Final simplification 99.7%

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

Alternative 2: 99.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
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 99.6%

   \[\leadsto \color{blue}{\frac{0.125 \cdot \left(\frac{e^{\frac{r}{-s}}}{s \cdot \pi} + \frac{e^{\frac{r}{s} \cdot -0.3333333333333333}}{s \cdot \pi}\right)}{r}} \]
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 99.6%

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

   1. *-commutative N/A

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

   2. un-div-inv N/A

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

   3. *-commutative N/A

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

   4. associate-/r* N/A

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

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

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

9. Applied egg-rr 99.7%

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

10. Final simplification 99.7%

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

Alternative 3: 99.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
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 99.6%

   \[\leadsto \color{blue}{\frac{0.125 \cdot \left(\frac{e^{\frac{r}{-s}}}{s \cdot \pi} + \frac{e^{\frac{r}{s} \cdot -0.3333333333333333}}{s \cdot \pi}\right)}{r}} \]
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 99.6%

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

8. Taylor expanded in s around 0

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

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

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

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

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

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

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

   4. mul-1-neg N/A

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

   5. distribute-neg-frac2 N/A

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

   6. mul-1-neg N/A

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

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

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

   8. mul-1-neg N/A

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

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

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

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

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

   11. associate-*r/ N/A

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

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

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

   13. *-commutative N/A

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

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

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

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

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

   16. PI-lowering-PI.f32 99.6

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

10. Simplified 99.6%

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

11. Final simplification 99.6%

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

Alternative 4: 99.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
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 99.6%

   \[\leadsto \color{blue}{\frac{0.125 \cdot \left(\frac{e^{\frac{r}{-s}}}{s \cdot \pi} + \frac{e^{\frac{r}{s} \cdot -0.3333333333333333}}{s \cdot \pi}\right)}{r}} \]
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. div-inv N/A

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

   6. clear-num N/A

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

   7. exp-prod N/A

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

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

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

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

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

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

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

   11. *-lowering-*.f32 99.7

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

7. Applied egg-rr 99.7%

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

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

9. Applied egg-rr 99.6%

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

10. Final simplification 99.6%

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

Alternative 5: 99.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
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 99.6%

   \[\leadsto \color{blue}{\frac{0.125 \cdot \left(\frac{e^{\frac{r}{-s}}}{s \cdot \pi} + \frac{e^{\frac{r}{s} \cdot -0.3333333333333333}}{s \cdot \pi}\right)}{r}} \]
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 99.6%

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

8. Taylor expanded in s around 0

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

   1. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

   6. mul-1-neg N/A

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

   7. distribute-neg-frac2 N/A

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

   8. mul-1-neg N/A

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

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

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

   10. mul-1-neg N/A

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

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

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

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

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

   13. associate-*r/ N/A

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

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

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

   15. *-commutative N/A

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

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

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

   17. *-commutative N/A

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

10. Simplified 99.6%

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

11. Final simplification 99.6%

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

Alternative 6: 97.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
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 99.6%

   \[\leadsto \color{blue}{\frac{0.125 \cdot \left(\frac{e^{\frac{r}{-s}}}{s \cdot \pi} + \frac{e^{\frac{r}{s} \cdot -0.3333333333333333}}{s \cdot \pi}\right)}{r}} \]
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. div-inv N/A

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

   6. clear-num N/A

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

   7. exp-prod N/A

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

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

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

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

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

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

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

   11. *-lowering-*.f32 99.7

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

7. Applied egg-rr 99.7%

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

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

9. Applied egg-rr 98.8%

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

10. Final simplification 98.8%

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

Alternative 7: 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 99.6%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
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 99.6%

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

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 10.8

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

8. Simplified 10.8%

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

9. Final simplification 10.8%

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

Alternative 8: 10.6% accurate, 1.5× speedup

Math

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

FPCore

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

C

float code(float s, float r) {
	return (0.125f * ((1.0f / (s * ((float) M_PI))) * (expf(-(r / s)) + fmaf(r, fmaf(0.05555555555555555f, (r / (s * s)), (-0.3333333333333333f / s)), 1.0f)))) / 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))) + fma(r, fma(Float32(0.05555555555555555), Float32(r / Float32(s * s)), Float32(Float32(-0.3333333333333333) / s)), Float32(1.0))))) / r)
end

Derivation

1. Initial program 99.6%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
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 99.6%

   \[\leadsto \color{blue}{\frac{0.125 \cdot \left(\frac{e^{\frac{r}{-s}}}{s \cdot \pi} + \frac{e^{\frac{r}{s} \cdot -0.3333333333333333}}{s \cdot \pi}\right)}{r}} \]
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 99.6%

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

8. Taylor expanded in r around 0

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

   1. +-commutative N/A

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

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

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

   3. sub-neg N/A

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

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

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

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

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

   6. unpow2 N/A

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

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

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

   8. associate-*r/ N/A

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

   9. metadata-eval N/A

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

   10. distribute-neg-frac N/A

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

   11. metadata-eval N/A

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

   12. /-lowering-/.f32 10.7

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

10. Simplified 10.7%

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

11. Final simplification 10.7%

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

Alternative 9: 9.9% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(s, \frac{\mathsf{fma}\left(0.25, \frac{s}{\pi}, r \cdot \mathsf{fma}\left(r, \mathsf{fma}\left(\frac{r}{s \cdot \left(s \cdot \pi\right)}, -0.020833333333333332, \frac{0.0625}{s \cdot \pi}\right), \frac{-0.16666666666666666}{\pi}\right)\right)}{r}, \frac{r \cdot 0.006944444444444444}{\pi}\right)}{s \cdot \left(s \cdot s\right)} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (/
  (fma
   s
   (/
    (fma
     0.25
     (/ s PI)
     (*
      r
      (fma
       r
       (fma (/ r (* s (* s PI))) -0.020833333333333332 (/ 0.0625 (* s PI)))
       (/ -0.16666666666666666 PI))))
    r)
   (/ (* r 0.006944444444444444) PI))
  (* s (* s s))))

C

float code(float s, float r) {
	return fmaf(s, (fmaf(0.25f, (s / ((float) M_PI)), (r * fmaf(r, fmaf((r / (s * (s * ((float) M_PI)))), -0.020833333333333332f, (0.0625f / (s * ((float) M_PI)))), (-0.16666666666666666f / ((float) M_PI))))) / r), ((r * 0.006944444444444444f) / ((float) M_PI))) / (s * (s * s));
}

Julia

function code(s, r)
	return Float32(fma(s, Float32(fma(Float32(0.25), Float32(s / Float32(pi)), Float32(r * fma(r, fma(Float32(r / Float32(s * Float32(s * Float32(pi)))), Float32(-0.020833333333333332), Float32(Float32(0.0625) / Float32(s * Float32(pi)))), Float32(Float32(-0.16666666666666666) / Float32(pi))))) / r), Float32(Float32(r * Float32(0.006944444444444444)) / Float32(pi))) / Float32(s * Float32(s * s)))
end

Derivation

1. Initial program 99.6%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
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 10.7%

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

6. 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. /-lowering-/.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 10.6%

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

9. Taylor expanded in r around 0

   \[\leadsto \frac{\mathsf{fma}\left(s, \color{blue}{\frac{\frac{1}{4} \cdot \frac{s}{\mathsf{PI}\left(\right)} + r \cdot \left(r \cdot \left(\frac{-1}{48} \cdot \frac{r}{{s}^{2} \cdot \mathsf{PI}\left(\right)} + \frac{1}{16} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}\right) - \frac{1}{6} \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{r}}, \frac{r \cdot \frac{1}{144}}{\mathsf{PI}\left(\right)}\right)}{s \cdot \left(s \cdot s\right)} \]
10. Step-by-step derivation

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

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

11. Simplified 10.2%

    \[\leadsto \frac{\mathsf{fma}\left(s, \color{blue}{\frac{\mathsf{fma}\left(0.25, \frac{s}{\pi}, r \cdot \mathsf{fma}\left(r, \mathsf{fma}\left(\frac{r}{s \cdot \left(s \cdot \pi\right)}, -0.020833333333333332, \frac{0.0625}{s \cdot \pi}\right), \frac{-0.16666666666666666}{\pi}\right)\right)}{r}}, \frac{r \cdot 0.006944444444444444}{\pi}\right)}{s \cdot \left(s \cdot s\right)} \]
12. Add Preprocessing

Alternative 10: 10.1% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in r around 0

   \[\leadsto \color{blue}{\frac{r \cdot \left(\frac{5}{72} \cdot \frac{r}{{s}^{3} \cdot \mathsf{PI}\left(\right)} - \frac{1}{6} \cdot \frac{1}{{s}^{2} \cdot \mathsf{PI}\left(\right)}\right) + \frac{1}{4} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}}{r}} \]

5. Simplified 10.2%

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

Alternative 11: 10.1% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.6%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
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 10.2%

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

Alternative 12: 9.1% accurate, 5.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in r around 0

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

   1. *-commutative N/A

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

   2. associate-*l/ N/A

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

   3. associate-*r/ N/A

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

   4. metadata-eval N/A

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

   5. distribute-neg-frac N/A

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

   6. metadata-eval N/A

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

   7. associate-*r/ N/A

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

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

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

5. Simplified 9.4%

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

Alternative 13: 9.1% accurate, 6.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in s around inf

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

   1. div-sub N/A

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

   2. sub-neg N/A

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

   3. associate-/l* N/A

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

   4. associate-/l/ N/A

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

   5. associate-*l* N/A

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

   6. unpow2 N/A

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

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

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

5. Simplified 9.3%

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

Alternative 14: 9.1% accurate, 7.6× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
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 9.1

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

5. Simplified 9.1%

   \[\leadsto \color{blue}{\frac{0.25}{r \cdot \left(s \cdot \pi\right)}} \]
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 9.1

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

7. Applied egg-rr 9.1%

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

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

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

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

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

   5. PI-lowering-PI.f32 9.1

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

9. Applied egg-rr 9.1%

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

    1. associate-/r* N/A

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

    2. clear-num N/A

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

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

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

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

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

    5. *-commutative N/A

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

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

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

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

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

    8. PI-lowering-PI.f32 9.1

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

11. Applied egg-rr 9.1%

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

Alternative 15: 9.1% accurate, 10.6× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
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 9.1

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

5. Simplified 9.1%

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

   1. associate-/r* N/A

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

   2. *-commutative N/A

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

   3. associate-/r* N/A

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

   4. associate-/r* N/A

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

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

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

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

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

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

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

   8. PI-lowering-PI.f32 9.1

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

7. Applied egg-rr 9.1%

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

Alternative 16: 9.1% accurate, 13.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
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 9.1

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

5. Simplified 9.1%

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

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

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

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

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

   5. PI-lowering-PI.f32 9.1

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

7. Applied egg-rr 9.1%

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

8. Final simplification 9.1%

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

Alternative 17: 9.1% accurate, 13.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
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 9.1

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

5. Simplified 9.1%

   \[\leadsto \color{blue}{\frac{0.25}{r \cdot \left(s \cdot \pi\right)}} \]
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 9.1

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

7. Applied egg-rr 9.1%

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

8. Final simplification 9.1%

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

Alternative 18: 9.1% accurate, 13.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
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 9.1

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

5. Simplified 9.1%

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

Reproduce

herbie shell --seed 2024199 
(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))))