Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.5%

Time: 15.0s

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: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

4. Applied rewrites 99.7%

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

6. Applied rewrites 99.7%

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

7. Final simplification 99.7%

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

Alternative 2: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

4. Applied rewrites 99.7%

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

6. Applied rewrites 99.7%

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

7. Taylor expanded in s around 0

   \[\leadsto \color{blue}{\frac{1}{8} \cdot \frac{e^{-1 \cdot \frac{r}{s}} + e^{\frac{-1}{3} \cdot \frac{r}{s}}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}} \]
8. 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. lower-/.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. lower-*.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. lower-+.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. lower-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. lower-/.f32 N/A

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

   9. lower-neg.f32 N/A

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

   10. lower-exp.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 N/A

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

   13. lower-/.f32 N/A

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

   14. lower-*.f32 N/A

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

   15. lower-*.f32 N/A

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

   16. lower-PI.f32 99.6

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

9. Applied rewrites 99.6%

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

10. Final simplification 99.6%

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

Alternative 3: 99.5% accurate, 1.1× speedup

Math

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

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(exp(Float32(-Float32(r / s))) + exp(Float32(r * Float32(Float32(-0.3333333333333333) / s))))) / Float32(s * Float32(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 99.7%

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

4. Applied rewrites 99.7%

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

6. Applied rewrites 99.7%

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

8. Applied rewrites 99.6%

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

9. Final simplification 99.6%

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

Alternative 4: 74.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \frac{0.125 \cdot \left(\frac{e^{-\frac{r}{s}}}{s \cdot \pi} + \frac{\frac{1}{\mathsf{fma}\left(r, \mathsf{fma}\left(r, \mathsf{fma}\left(0.006172839506172839, \frac{r}{s \cdot \left(s \cdot s\right)}, \frac{0.05555555555555555}{s \cdot s}\right), \frac{0.3333333333333333}{s}\right), 1\right)}}{s \cdot \pi}\right)}{r} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (/
  (*
   0.125
   (+
    (/ (exp (- (/ r s))) (* s PI))
    (/
     (/
      1.0
      (fma
       r
       (fma
        r
        (fma
         0.006172839506172839
         (/ r (* s (* s s)))
         (/ 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))) + ((1.0f / fmaf(r, fmaf(r, fmaf(0.006172839506172839f, (r / (s * (s * s))), (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(Float32(Float32(1.0) / fma(r, fma(r, fma(Float32(0.006172839506172839), Float32(r / Float32(s * Float32(s * s))), Float32(Float32(0.05555555555555555) / Float32(s * s))), Float32(Float32(0.3333333333333333) / s)), Float32(1.0))) / Float32(s * Float32(pi))))) / r)
end

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in r around inf

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

   1. distribute-lft-out N/A

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

   2. metadata-eval N/A

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

   3. associate-*r* N/A

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

   4. distribute-lft-out N/A

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

   5. lower-/.f32 N/A

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

5. Applied rewrites 99.7%

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

   1. lift-/.f32 N/A

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

   2. metadata-eval N/A

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

   3. div-inv N/A

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

   4. lift-/.f32 N/A

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

   5. associate-/r* N/A

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

   6. lift-*.f32 N/A

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

   7. frac-2neg N/A

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

   8. distribute-frac-neg N/A

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

   9. exp-neg N/A

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

   10. lower-/.f32 N/A

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

   11. lower-exp.f32 N/A

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

   12. lower-/.f32 N/A

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

   13. lift-*.f32 N/A

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

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

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

   15. metadata-eval N/A

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

   16. lower-*.f32 99.7

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

7. Applied rewrites 99.7%

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

8. Taylor expanded in r around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f32 N/A

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

10. Applied rewrites 72.4%

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

11. Final simplification 72.4%

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

Alternative 5: 59.4% accurate, 1.4× speedup

Math

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

FPCore

(FPCore (s r)
 :precision binary32
 (/
  (*
   0.125
   (+
    (/ (exp (- (/ r s))) (* s PI))
    (/
     (/
      1.0
      (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))) + ((1.0f / 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(Float32(Float32(1.0) / fma(r, fma(r, Float32(Float32(0.05555555555555555) / Float32(s * s)), Float32(Float32(0.3333333333333333) / s)), Float32(1.0))) / Float32(s * Float32(pi))))) / r)
end

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in r around inf

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

   1. distribute-lft-out N/A

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

   2. metadata-eval N/A

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

   3. associate-*r* N/A

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

   4. distribute-lft-out N/A

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

   5. lower-/.f32 N/A

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

5. Applied rewrites 99.7%

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

   1. lift-/.f32 N/A

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

   2. metadata-eval N/A

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

   3. div-inv N/A

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

   4. lift-/.f32 N/A

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

   5. associate-/r* N/A

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

   6. lift-*.f32 N/A

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

   7. frac-2neg N/A

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

   8. distribute-frac-neg N/A

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

   9. exp-neg N/A

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

   10. lower-/.f32 N/A

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

   11. lower-exp.f32 N/A

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

   12. lower-/.f32 N/A

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

   13. lift-*.f32 N/A

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

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

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

   15. metadata-eval N/A

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

   16. lower-*.f32 99.7

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

7. Applied rewrites 99.7%

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

8. Taylor expanded in r around 0

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

   1. +-commutative N/A

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

   2. associate-*r/ N/A

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

   3. associate-*l/ N/A

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

   4. metadata-eval N/A

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

   5. associate-*r/ N/A

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

   6. lower-fma.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f32 N/A

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

   9. associate-*r/ N/A

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

   10. metadata-eval N/A

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

   11. lower-/.f32 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f32 N/A

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

   14. associate-*r/ N/A

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

   15. metadata-eval N/A

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

   16. lower-/.f32 55.0

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

10. Applied rewrites 55.0%

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

11. Final simplification 55.0%

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

Alternative 6: 15.5% accurate, 1.5× speedup

Math

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

FPCore

(FPCore (s r)
 :precision binary32
 (/
  (*
   0.125
   (+
    (/ (exp (- (/ r s))) (* s PI))
    (/ (/ 1.0 (fma r (/ 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))) + ((1.0f / fmaf(r, (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(Float32(Float32(1.0) / fma(r, Float32(Float32(0.3333333333333333) / s), Float32(1.0))) / Float32(s * Float32(pi))))) / r)
end

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in r around inf

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

   1. distribute-lft-out N/A

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

   2. metadata-eval N/A

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

   3. associate-*r* N/A

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

   4. distribute-lft-out N/A

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

   5. lower-/.f32 N/A

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

5. Applied rewrites 99.7%

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

   1. lift-/.f32 N/A

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

   2. metadata-eval N/A

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

   3. div-inv N/A

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

   4. lift-/.f32 N/A

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

   5. associate-/r* N/A

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

   6. lift-*.f32 N/A

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

   7. frac-2neg N/A

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

   8. distribute-frac-neg N/A

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

   9. exp-neg N/A

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

   10. lower-/.f32 N/A

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

   11. lower-exp.f32 N/A

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

   12. lower-/.f32 N/A

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

   13. lift-*.f32 N/A

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

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

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

   15. metadata-eval N/A

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

   16. lower-*.f32 99.7

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

7. Applied rewrites 99.7%

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

8. Taylor expanded in r around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*l/ N/A

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

   4. associate-*r/ N/A

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

   5. metadata-eval N/A

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

   6. associate-*r/ N/A

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

   7. lower-fma.f32 N/A

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

   8. associate-*r/ N/A

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

   9. metadata-eval N/A

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

   10. lower-/.f32 15.9

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

10. Applied rewrites 15.9%

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

11. Final simplification 15.9%

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

Alternative 7: 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 99.7%

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

3. Taylor expanded in r around 0

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

5. Applied rewrites 10.0%

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

Alternative 8: 10.1% accurate, 4.0× speedup

Math

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

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/
   (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(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(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(Float32(s * Float32(pi)) * r)))
end

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in 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. Applied rewrites 10.0%

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

6. Final simplification 10.0%

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

Alternative 9: 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 99.7%

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

3. Taylor expanded in r around 0

   \[\leadsto \color{blue}{\frac{\frac{-1}{6} \cdot \frac{r}{{s}^{2} \cdot \mathsf{PI}\left(\right)} + \frac{1}{4} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}}{r}} \]
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. lower-/.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. Applied rewrites 9.1%

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

Alternative 10: 9.1% accurate, 5.6× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in s around -inf

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

5. Applied rewrites 9.8%

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

6. Taylor expanded in s around inf

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

   1. sub-neg N/A

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

   2. associate-*r/ N/A

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

   3. metadata-eval N/A

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

   4. lower-+.f32 N/A

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

   5. associate-*r/ N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

   10. lower-PI.f32 N/A

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

   11. distribute-neg-frac N/A

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

   12. metadata-eval N/A

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

   13. lower-/.f32 N/A

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

   14. lower-*.f32 N/A

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

   15. lower-PI.f32 9.0

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

8. Applied rewrites 9.0%

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

   1. lift-PI.f32 N/A

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

   2. *-commutative N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f32 N/A

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

   5. lower-/.f32 9.1

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

10. Applied rewrites 9.1%

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

Alternative 11: 9.1% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.25}{\pi \cdot r} + \frac{1}{\left(s \cdot \pi\right) \cdot -6}}{s} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (/ (+ (/ 0.25 (* PI r)) (/ 1.0 (* (* s PI) -6.0))) s))

C

float code(float s, float r) {
	return ((0.25f / (((float) M_PI) * r)) + (1.0f / ((s * ((float) M_PI)) * -6.0f))) / s;
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.25) / Float32(Float32(pi) * r)) + Float32(Float32(1.0) / Float32(Float32(s * Float32(pi)) * Float32(-6.0)))) / s)
end

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.25) / (single(pi) * r)) + (single(1.0) / ((s * single(pi)) * single(-6.0)))) / s;
end

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in s around -inf

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

5. Applied rewrites 9.8%

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

6. Taylor expanded in s around inf

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

   1. sub-neg N/A

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

   2. associate-*r/ N/A

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

   3. metadata-eval N/A

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

   4. lower-+.f32 N/A

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

   5. associate-*r/ N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

   10. lower-PI.f32 N/A

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

   11. distribute-neg-frac N/A

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

   12. metadata-eval N/A

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

   13. lower-/.f32 N/A

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

   14. lower-*.f32 N/A

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

   15. lower-PI.f32 9.0

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

8. Applied rewrites 9.0%

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

   1. lift-PI.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. clear-num N/A

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

   4. lower-/.f32 N/A

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

   5. div-inv N/A

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

   6. lower-*.f32 N/A

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

   7. metadata-eval 9.0

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

10. Applied rewrites 9.0%

    \[\leadsto \frac{\frac{0.25}{\pi \cdot r} + \color{blue}{\frac{1}{\left(s \cdot \pi\right) \cdot -6}}}{s} \]
11. Add Preprocessing

Alternative 12: 9.1% accurate, 6.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in s around -inf

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

5. Applied rewrites 9.8%

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

6. Taylor expanded in s around inf

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

   1. sub-neg N/A

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

   2. associate-*r/ N/A

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

   3. metadata-eval N/A

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

   4. lower-+.f32 N/A

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

   5. associate-*r/ N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

   10. lower-PI.f32 N/A

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

   11. distribute-neg-frac N/A

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

   12. metadata-eval N/A

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

   13. lower-/.f32 N/A

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

   14. lower-*.f32 N/A

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

   15. lower-PI.f32 9.0

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

8. Applied rewrites 9.0%

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

9. Final simplification 9.0%

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

Alternative 13: 9.1% accurate, 6.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in s around inf

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

   1. div-sub N/A

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

   2. sub-neg N/A

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

   3. associate-/l* N/A

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

   4. associate-/l/ N/A

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

   5. associate-*l* N/A

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

   6. unpow2 N/A

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

   7. lower-+.f32 N/A

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

5. Applied rewrites 9.0%

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

6. Final simplification 9.0%

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

Alternative 14: 9.0% accurate, 10.6× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in r around 0

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 8.9

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

5. Applied rewrites 8.9%

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

   1. lift-PI.f32 N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 8.9

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

7. Applied rewrites 8.9%

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

   1. lift-*.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-/r* N/A

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

   5. lower-/.f32 N/A

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

   6. lower-/.f32 8.9

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 8.9

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

9. Applied rewrites 8.9%

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

Alternative 15: 9.0% accurate, 10.6× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in r around 0

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 8.9

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

5. Applied rewrites 8.9%

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

   1. lift-PI.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f32 N/A

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

   5. lower-/.f32 8.9

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

7. Applied rewrites 8.9%

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

Alternative 16: 9.0% accurate, 13.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in r around 0

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 8.9

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

5. Applied rewrites 8.9%

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

   1. lift-PI.f32 N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. lift-*.f32 N/A

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

   5. lower-*.f32 8.9

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

7. Applied rewrites 8.9%

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

8. Final simplification 8.9%

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

Alternative 17: 9.0% accurate, 13.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in r around 0

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 8.9

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

5. Applied rewrites 8.9%

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

6. Final simplification 8.9%

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

Alternative 18: 7.1% accurate, 13.5× speedup

Math

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

FPCore

(FPCore (s r) :precision binary32 (/ 0.125 (* s (* PI r))))

C

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

Julia

function code(s, r)
	return Float32(Float32(0.125) / Float32(s * Float32(Float32(pi) * r)))
end

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in s around -inf

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

5. Applied rewrites 10.2%

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

6. Taylor expanded in r around inf

   \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\color{blue}{{r}^{2} \cdot \left(\frac{1}{144} \cdot \frac{1}{r \cdot \left({s}^{2} \cdot \mathsf{PI}\left(\right)\right)} - \frac{1}{1296} \cdot \frac{1}{{s}^{3} \cdot \mathsf{PI}\left(\right)}\right)}}{s} \]
7. Step-by-step derivation

   1. lower-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\color{blue}{{r}^{2} \cdot \left(\frac{1}{144} \cdot \frac{1}{r \cdot \left({s}^{2} \cdot \mathsf{PI}\left(\right)\right)} - \frac{1}{1296} \cdot \frac{1}{{s}^{3} \cdot \mathsf{PI}\left(\right)}\right)}}{s} \]

   2. unpow2 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\color{blue}{\left(r \cdot r\right)} \cdot \left(\frac{1}{144} \cdot \frac{1}{r \cdot \left({s}^{2} \cdot \mathsf{PI}\left(\right)\right)} - \frac{1}{1296} \cdot \frac{1}{{s}^{3} \cdot \mathsf{PI}\left(\right)}\right)}{s} \]

   3. lower-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\color{blue}{\left(r \cdot r\right)} \cdot \left(\frac{1}{144} \cdot \frac{1}{r \cdot \left({s}^{2} \cdot \mathsf{PI}\left(\right)\right)} - \frac{1}{1296} \cdot \frac{1}{{s}^{3} \cdot \mathsf{PI}\left(\right)}\right)}{s} \]

   4. sub-neg N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\left(r \cdot r\right) \cdot \color{blue}{\left(\frac{1}{144} \cdot \frac{1}{r \cdot \left({s}^{2} \cdot \mathsf{PI}\left(\right)\right)} + \left(\mathsf{neg}\left(\frac{1}{1296} \cdot \frac{1}{{s}^{3} \cdot \mathsf{PI}\left(\right)}\right)\right)\right)}}{s} \]

   5. associate-*r/ N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\left(r \cdot r\right) \cdot \left(\color{blue}{\frac{\frac{1}{144} \cdot 1}{r \cdot \left({s}^{2} \cdot \mathsf{PI}\left(\right)\right)}} + \left(\mathsf{neg}\left(\frac{1}{1296} \cdot \frac{1}{{s}^{3} \cdot \mathsf{PI}\left(\right)}\right)\right)\right)}{s} \]

   6. metadata-eval N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\left(r \cdot r\right) \cdot \left(\frac{\color{blue}{\frac{1}{144}}}{r \cdot \left({s}^{2} \cdot \mathsf{PI}\left(\right)\right)} + \left(\mathsf{neg}\left(\frac{1}{1296} \cdot \frac{1}{{s}^{3} \cdot \mathsf{PI}\left(\right)}\right)\right)\right)}{s} \]

   7. lower-+.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\left(r \cdot r\right) \cdot \color{blue}{\left(\frac{\frac{1}{144}}{r \cdot \left({s}^{2} \cdot \mathsf{PI}\left(\right)\right)} + \left(\mathsf{neg}\left(\frac{1}{1296} \cdot \frac{1}{{s}^{3} \cdot \mathsf{PI}\left(\right)}\right)\right)\right)}}{s} \]

   8. lower-/.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\left(r \cdot r\right) \cdot \left(\color{blue}{\frac{\frac{1}{144}}{r \cdot \left({s}^{2} \cdot \mathsf{PI}\left(\right)\right)}} + \left(\mathsf{neg}\left(\frac{1}{1296} \cdot \frac{1}{{s}^{3} \cdot \mathsf{PI}\left(\right)}\right)\right)\right)}{s} \]

   9. lower-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\left(r \cdot r\right) \cdot \left(\frac{\frac{1}{144}}{\color{blue}{r \cdot \left({s}^{2} \cdot \mathsf{PI}\left(\right)\right)}} + \left(\mathsf{neg}\left(\frac{1}{1296} \cdot \frac{1}{{s}^{3} \cdot \mathsf{PI}\left(\right)}\right)\right)\right)}{s} \]

   10. unpow2 N/A

       \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\left(r \cdot r\right) \cdot \left(\frac{\frac{1}{144}}{r \cdot \left(\color{blue}{\left(s \cdot s\right)} \cdot \mathsf{PI}\left(\right)\right)} + \left(\mathsf{neg}\left(\frac{1}{1296} \cdot \frac{1}{{s}^{3} \cdot \mathsf{PI}\left(\right)}\right)\right)\right)}{s} \]

   11. associate-*l* N/A

       \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\left(r \cdot r\right) \cdot \left(\frac{\frac{1}{144}}{r \cdot \color{blue}{\left(s \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)\right)}} + \left(\mathsf{neg}\left(\frac{1}{1296} \cdot \frac{1}{{s}^{3} \cdot \mathsf{PI}\left(\right)}\right)\right)\right)}{s} \]

   12. lower-*.f32 N/A

       \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\left(r \cdot r\right) \cdot \left(\frac{\frac{1}{144}}{r \cdot \color{blue}{\left(s \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)\right)}} + \left(\mathsf{neg}\left(\frac{1}{1296} \cdot \frac{1}{{s}^{3} \cdot \mathsf{PI}\left(\right)}\right)\right)\right)}{s} \]

   13. lower-*.f32 N/A

       \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\left(r \cdot r\right) \cdot \left(\frac{\frac{1}{144}}{r \cdot \left(s \cdot \color{blue}{\left(s \cdot \mathsf{PI}\left(\right)\right)}\right)} + \left(\mathsf{neg}\left(\frac{1}{1296} \cdot \frac{1}{{s}^{3} \cdot \mathsf{PI}\left(\right)}\right)\right)\right)}{s} \]

   14. lower-PI.f32 N/A

       \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\left(r \cdot r\right) \cdot \left(\frac{\frac{1}{144}}{r \cdot \left(s \cdot \left(s \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right)} + \left(\mathsf{neg}\left(\frac{1}{1296} \cdot \frac{1}{{s}^{3} \cdot \mathsf{PI}\left(\right)}\right)\right)\right)}{s} \]

8. Applied rewrites 4.2%

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

9. Taylor expanded in r around 0

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

    1. lower-/.f32 N/A

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

    2. *-commutative N/A

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

    3. associate-*l* N/A

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

    4. *-commutative N/A

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

    5. lower-*.f32 N/A

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

    6. *-commutative N/A

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

    7. lower-*.f32 N/A

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

    8. lower-PI.f32 7.0

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

11. Applied rewrites 7.0%

    \[\leadsto \color{blue}{\frac{0.125}{s \cdot \left(\pi \cdot r\right)}} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024214 
(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))))