Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.6%

Time: 5.9s

Alternatives: 17

Speedup: N/A×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in s around 0

   \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\frac{3}{4} \cdot e^{\color{blue}{\frac{-1}{3} \cdot \frac{r}{s}}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
4. Step-by-step derivation

   1. lower-*.f32 N/A

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

   2. lower-/.f32 99.5

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

5. Applied rewrites 99.5%

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

6. Taylor expanded in s around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. mul-1-neg N/A

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

   4. distribute-frac-neg N/A

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

   5. lower-/.f32 N/A

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

   6. lift-/.f32 N/A

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

   7. lift-neg.f32 N/A

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

   8. lift-exp.f32 N/A

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

   9. *-commutative N/A

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

   10. *-commutative N/A

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

   11. lift-*.f32 N/A

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

   12. lift-PI.f32 N/A

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

   13. lift-*.f32 99.5

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

8. Applied rewrites 99.5%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-*r/ N/A

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

   4. metadata-eval N/A

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

   5. lower-/.f32 N/A

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

   6. metadata-eval N/A

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

   7. lower-*.f32 99.5

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

10. Applied rewrites 99.5%

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

    1. lift-+.f32 N/A

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

    2. +-commutative N/A

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

    3. lift-/.f32 N/A

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

    4. lift-*.f32 N/A

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

    5. associate-/l* N/A

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

12. Applied rewrites 99.5%

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

Alternative 2: 10.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot \mathsf{fma}\left(\frac{r}{s \cdot s} \cdot 0.05555555555555555 - \frac{0.3333333333333333}{s}, r, 1\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/ (* 0.25 (exp (/ (- r) s))) (* (* (* 2.0 PI) s) r))
  (/
   (*
    0.75
    (fma
     (- (* (/ r (* s s)) 0.05555555555555555) (/ 0.3333333333333333 s))
     r
     1.0))
   (* (* (* 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 * fmaf((((r / (s * s)) * 0.05555555555555555f) - (0.3333333333333333f / s)), r, 1.0f)) / (((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) * fma(Float32(Float32(Float32(r / Float32(s * s)) * Float32(0.05555555555555555)) - Float32(Float32(0.3333333333333333) / s)), r, Float32(1.0))) / Float32(Float32(Float32(Float32(6.0) * Float32(pi)) * s) * r)))
end

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in r around 0

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

   1. +-commutative N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\frac{3}{4} \cdot \left(r \cdot \left(\frac{1}{18} \cdot \frac{r}{{s}^{2}} - \frac{1}{3} \cdot \frac{1}{s}\right) + \color{blue}{1}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   2. *-commutative N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\frac{3}{4} \cdot \left(\left(\frac{1}{18} \cdot \frac{r}{{s}^{2}} - \frac{1}{3} \cdot \frac{1}{s}\right) \cdot r + 1\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   3. lower-fma.f32 N/A

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

   4. lower--.f32 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. lower-/.f32 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f32 N/A

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

   10. associate-*r/ N/A

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

   11. metadata-eval N/A

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

   12. lower-/.f32 11.5

       \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot \mathsf{fma}\left(\frac{r}{s \cdot s} \cdot 0.05555555555555555 - \frac{0.3333333333333333}{s}, r, 1\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

5. Applied rewrites 11.5%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot \color{blue}{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot 0.05555555555555555 - \frac{0.3333333333333333}{s}, r, 1\right)}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
6. Add Preprocessing

Alternative 3: 10.4% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.5%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. lift-PI.f32 N/A

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

   10. lower-*.f32 99.5

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

4. Applied rewrites 99.5%

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

5. Taylor expanded in s around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. mul-1-neg N/A

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

   4. distribute-frac-neg N/A

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

   5. lower-/.f32 N/A

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

   6. lift-/.f32 N/A

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

   7. lift-neg.f32 N/A

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

   8. lift-exp.f32 N/A

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

   9. *-commutative N/A

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

   10. *-commutative N/A

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

   11. lift-*.f32 N/A

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

   12. lift-PI.f32 N/A

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

   13. lift-*.f32 99.5

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

7. Applied rewrites 99.5%

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

8. Taylor expanded in s around inf

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

   1. associate-+r+ N/A

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

   2. lower-+.f32 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

      \[\leadsto \frac{e^{\frac{-r}{s}}}{\left(\pi \cdot s\right) \cdot r} \cdot \frac{1}{8} + \frac{\frac{3}{4} \cdot \left(\left(\frac{r}{s} \cdot \frac{-1}{3} + 1\right) + \frac{1}{18} \cdot \frac{{r}^{2}}{{s}^{2}}\right)}{\left(\pi \cdot 6\right) \cdot \left(s \cdot r\right)} \]

   5. lower-fma.f32 N/A

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

   6. lift-/.f32 N/A

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

   7. associate-*r/ N/A

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

   8. lower-/.f32 N/A

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

   9. lower-*.f32 N/A

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

   10. pow2 N/A

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

   11. lift-*.f32 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f32 11.3

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

10. Applied rewrites 11.3%

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

11. Taylor expanded in s around 0

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

    1. lower-/.f32 N/A

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

    2. +-commutative N/A

       \[\leadsto \frac{e^{\frac{-r}{s}}}{\left(\pi \cdot s\right) \cdot r} \cdot \frac{1}{8} + \frac{\frac{3}{4} \cdot \frac{s \cdot \left(s + \frac{-1}{3} \cdot r\right) + \frac{1}{18} \cdot {r}^{2}}{{s}^{2}}}{\left(\pi \cdot 6\right) \cdot \left(s \cdot r\right)} \]

    3. *-commutative N/A

       \[\leadsto \frac{e^{\frac{-r}{s}}}{\left(\pi \cdot s\right) \cdot r} \cdot \frac{1}{8} + \frac{\frac{3}{4} \cdot \frac{\left(s + \frac{-1}{3} \cdot r\right) \cdot s + \frac{1}{18} \cdot {r}^{2}}{{s}^{2}}}{\left(\pi \cdot 6\right) \cdot \left(s \cdot r\right)} \]

    4. lower-fma.f32 N/A

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

    5. +-commutative N/A

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

    6. lower-fma.f32 N/A

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

    7. *-commutative N/A

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

    8. lower-*.f32 N/A

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

    9. pow2 N/A

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

    10. lift-*.f32 N/A

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

    11. pow2 N/A

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

    12. lift-*.f32 11.5

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

13. Applied rewrites 11.5%

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

Alternative 4: 10.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \frac{e^{\frac{-r}{s}}}{\left(\pi \cdot s\right) \cdot r} \cdot 0.125 + \frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot 0.041666666666666664 - \frac{0.25}{s}, r, 0.75\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (* (/ (exp (/ (- r) s)) (* (* PI s) r)) 0.125)
  (/
   (fma (- (* (/ r (* s s)) 0.041666666666666664) (/ 0.25 s)) r 0.75)
   (* (* (* 6.0 PI) s) r))))

C

float code(float s, float r) {
	return ((expf((-r / s)) / ((((float) M_PI) * s) * r)) * 0.125f) + (fmaf((((r / (s * s)) * 0.041666666666666664f) - (0.25f / s)), r, 0.75f) / (((6.0f * ((float) M_PI)) * s) * r));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(exp(Float32(Float32(-r) / s)) / Float32(Float32(Float32(pi) * s) * r)) * Float32(0.125)) + Float32(fma(Float32(Float32(Float32(r / Float32(s * s)) * Float32(0.041666666666666664)) - Float32(Float32(0.25) / s)), r, Float32(0.75)) / Float32(Float32(Float32(Float32(6.0) * Float32(pi)) * s) * r)))
end

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in s around 0

   \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\frac{3}{4} \cdot e^{\color{blue}{\frac{-1}{3} \cdot \frac{r}{s}}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
4. Step-by-step derivation

   1. lower-*.f32 N/A

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

   2. lower-/.f32 99.5

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

5. Applied rewrites 99.5%

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

6. Taylor expanded in s around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. mul-1-neg N/A

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

   4. distribute-frac-neg N/A

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

   5. lower-/.f32 N/A

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

   6. lift-/.f32 N/A

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

   7. lift-neg.f32 N/A

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

   8. lift-exp.f32 N/A

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

   9. *-commutative N/A

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

   10. *-commutative N/A

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

   11. lift-*.f32 N/A

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

   12. lift-PI.f32 N/A

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

   13. lift-*.f32 99.5

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

8. Applied rewrites 99.5%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-*r/ N/A

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

   4. metadata-eval N/A

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

   5. lower-/.f32 N/A

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

   6. metadata-eval N/A

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

   7. lower-*.f32 99.5

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

10. Applied rewrites 99.5%

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

11. Taylor expanded in r around 0

    \[\leadsto \frac{e^{\frac{-r}{s}}}{\left(\pi \cdot s\right) \cdot r} \cdot \frac{1}{8} + \frac{\color{blue}{\frac{3}{4} + r \cdot \left(\frac{1}{24} \cdot \frac{r}{{s}^{2}} - \frac{1}{4} \cdot \frac{1}{s}\right)}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
12. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \frac{e^{\frac{-r}{s}}}{\left(\pi \cdot s\right) \cdot r} \cdot \frac{1}{8} + \frac{r \cdot \left(\frac{1}{24} \cdot \frac{r}{{s}^{2}} - \frac{1}{4} \cdot \frac{1}{s}\right) + \color{blue}{\frac{3}{4}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

    2. *-commutative N/A

       \[\leadsto \frac{e^{\frac{-r}{s}}}{\left(\pi \cdot s\right) \cdot r} \cdot \frac{1}{8} + \frac{\left(\frac{1}{24} \cdot \frac{r}{{s}^{2}} - \frac{1}{4} \cdot \frac{1}{s}\right) \cdot r + \frac{3}{4}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

    3. lower-fma.f32 N/A

       \[\leadsto \frac{e^{\frac{-r}{s}}}{\left(\pi \cdot s\right) \cdot r} \cdot \frac{1}{8} + \frac{\mathsf{fma}\left(\frac{1}{24} \cdot \frac{r}{{s}^{2}} - \frac{1}{4} \cdot \frac{1}{s}, \color{blue}{r}, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

    4. lower--.f32 N/A

       \[\leadsto \frac{e^{\frac{-r}{s}}}{\left(\pi \cdot s\right) \cdot r} \cdot \frac{1}{8} + \frac{\mathsf{fma}\left(\frac{1}{24} \cdot \frac{r}{{s}^{2}} - \frac{1}{4} \cdot \frac{1}{s}, r, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

    5. *-commutative N/A

       \[\leadsto \frac{e^{\frac{-r}{s}}}{\left(\pi \cdot s\right) \cdot r} \cdot \frac{1}{8} + \frac{\mathsf{fma}\left(\frac{r}{{s}^{2}} \cdot \frac{1}{24} - \frac{1}{4} \cdot \frac{1}{s}, r, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

    6. lower-*.f32 N/A

       \[\leadsto \frac{e^{\frac{-r}{s}}}{\left(\pi \cdot s\right) \cdot r} \cdot \frac{1}{8} + \frac{\mathsf{fma}\left(\frac{r}{{s}^{2}} \cdot \frac{1}{24} - \frac{1}{4} \cdot \frac{1}{s}, r, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

    7. lower-/.f32 N/A

       \[\leadsto \frac{e^{\frac{-r}{s}}}{\left(\pi \cdot s\right) \cdot r} \cdot \frac{1}{8} + \frac{\mathsf{fma}\left(\frac{r}{{s}^{2}} \cdot \frac{1}{24} - \frac{1}{4} \cdot \frac{1}{s}, r, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

    8. pow2 N/A

       \[\leadsto \frac{e^{\frac{-r}{s}}}{\left(\pi \cdot s\right) \cdot r} \cdot \frac{1}{8} + \frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{1}{24} - \frac{1}{4} \cdot \frac{1}{s}, r, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

    9. lift-*.f32 N/A

       \[\leadsto \frac{e^{\frac{-r}{s}}}{\left(\pi \cdot s\right) \cdot r} \cdot \frac{1}{8} + \frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{1}{24} - \frac{1}{4} \cdot \frac{1}{s}, r, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

    10. associate-*r/ N/A

        \[\leadsto \frac{e^{\frac{-r}{s}}}{\left(\pi \cdot s\right) \cdot r} \cdot \frac{1}{8} + \frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{1}{24} - \frac{\frac{1}{4} \cdot 1}{s}, r, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

    11. metadata-eval N/A

        \[\leadsto \frac{e^{\frac{-r}{s}}}{\left(\pi \cdot s\right) \cdot r} \cdot \frac{1}{8} + \frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{1}{24} - \frac{\frac{1}{4}}{s}, r, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

    12. lower-/.f32 11.5

        \[\leadsto \frac{e^{\frac{-r}{s}}}{\left(\pi \cdot s\right) \cdot r} \cdot 0.125 + \frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot 0.041666666666666664 - \frac{0.25}{s}, r, 0.75\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

13. Applied rewrites 11.5%

    \[\leadsto \frac{e^{\frac{-r}{s}}}{\left(\pi \cdot s\right) \cdot r} \cdot 0.125 + \frac{\color{blue}{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot 0.041666666666666664 - \frac{0.25}{s}, r, 0.75\right)}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
14. Add Preprocessing

Alternative 5: 10.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \frac{e^{\frac{-r}{s}}}{\left(\pi \cdot s\right) \cdot r} \cdot 0.125 + \frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot 0.041666666666666664 - \frac{0.25}{s}, r, 0.75\right)}{\left(\pi \cdot 6\right) \cdot \left(s \cdot r\right)} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (* (/ (exp (/ (- r) s)) (* (* PI s) r)) 0.125)
  (/
   (fma (- (* (/ r (* s s)) 0.041666666666666664) (/ 0.25 s)) r 0.75)
   (* (* PI 6.0) (* s r)))))

C

float code(float s, float r) {
	return ((expf((-r / s)) / ((((float) M_PI) * s) * r)) * 0.125f) + (fmaf((((r / (s * s)) * 0.041666666666666664f) - (0.25f / s)), r, 0.75f) / ((((float) M_PI) * 6.0f) * (s * r)));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(exp(Float32(Float32(-r) / s)) / Float32(Float32(Float32(pi) * s) * r)) * Float32(0.125)) + Float32(fma(Float32(Float32(Float32(r / Float32(s * s)) * Float32(0.041666666666666664)) - Float32(Float32(0.25) / s)), r, Float32(0.75)) / Float32(Float32(Float32(pi) * Float32(6.0)) * Float32(s * r))))
end

Derivation

1. Initial program 99.5%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. lift-PI.f32 N/A

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

   10. lower-*.f32 99.5

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

4. Applied rewrites 99.5%

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

5. Taylor expanded in s around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. mul-1-neg N/A

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

   4. distribute-frac-neg N/A

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

   5. lower-/.f32 N/A

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

   6. lift-/.f32 N/A

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

   7. lift-neg.f32 N/A

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

   8. lift-exp.f32 N/A

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

   9. *-commutative N/A

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

   10. *-commutative N/A

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

   11. lift-*.f32 N/A

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

   12. lift-PI.f32 N/A

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

   13. lift-*.f32 99.5

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

7. Applied rewrites 99.5%

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

8. Taylor expanded in r around 0

   \[\leadsto \frac{e^{\frac{-r}{s}}}{\left(\pi \cdot s\right) \cdot r} \cdot \frac{1}{8} + \frac{\color{blue}{\frac{3}{4} + r \cdot \left(\frac{1}{24} \cdot \frac{r}{{s}^{2}} - \frac{1}{4} \cdot \frac{1}{s}\right)}}{\left(\pi \cdot 6\right) \cdot \left(s \cdot r\right)} \]
9. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{e^{\frac{-r}{s}}}{\left(\pi \cdot s\right) \cdot r} \cdot \frac{1}{8} + \frac{r \cdot \left(\frac{1}{24} \cdot \frac{r}{{s}^{2}} - \frac{1}{4} \cdot \frac{1}{s}\right) + \color{blue}{\frac{3}{4}}}{\left(\pi \cdot 6\right) \cdot \left(s \cdot r\right)} \]

   2. *-commutative N/A

      \[\leadsto \frac{e^{\frac{-r}{s}}}{\left(\pi \cdot s\right) \cdot r} \cdot \frac{1}{8} + \frac{\left(\frac{1}{24} \cdot \frac{r}{{s}^{2}} - \frac{1}{4} \cdot \frac{1}{s}\right) \cdot r + \frac{3}{4}}{\left(\pi \cdot 6\right) \cdot \left(s \cdot r\right)} \]

   3. lower-fma.f32 N/A

      \[\leadsto \frac{e^{\frac{-r}{s}}}{\left(\pi \cdot s\right) \cdot r} \cdot \frac{1}{8} + \frac{\mathsf{fma}\left(\frac{1}{24} \cdot \frac{r}{{s}^{2}} - \frac{1}{4} \cdot \frac{1}{s}, \color{blue}{r}, \frac{3}{4}\right)}{\left(\pi \cdot 6\right) \cdot \left(s \cdot r\right)} \]

   4. lower--.f32 N/A

      \[\leadsto \frac{e^{\frac{-r}{s}}}{\left(\pi \cdot s\right) \cdot r} \cdot \frac{1}{8} + \frac{\mathsf{fma}\left(\frac{1}{24} \cdot \frac{r}{{s}^{2}} - \frac{1}{4} \cdot \frac{1}{s}, r, \frac{3}{4}\right)}{\left(\pi \cdot 6\right) \cdot \left(s \cdot r\right)} \]

   5. *-commutative N/A

      \[\leadsto \frac{e^{\frac{-r}{s}}}{\left(\pi \cdot s\right) \cdot r} \cdot \frac{1}{8} + \frac{\mathsf{fma}\left(\frac{r}{{s}^{2}} \cdot \frac{1}{24} - \frac{1}{4} \cdot \frac{1}{s}, r, \frac{3}{4}\right)}{\left(\pi \cdot 6\right) \cdot \left(s \cdot r\right)} \]

   6. lower-*.f32 N/A

      \[\leadsto \frac{e^{\frac{-r}{s}}}{\left(\pi \cdot s\right) \cdot r} \cdot \frac{1}{8} + \frac{\mathsf{fma}\left(\frac{r}{{s}^{2}} \cdot \frac{1}{24} - \frac{1}{4} \cdot \frac{1}{s}, r, \frac{3}{4}\right)}{\left(\pi \cdot 6\right) \cdot \left(s \cdot r\right)} \]

   7. lower-/.f32 N/A

      \[\leadsto \frac{e^{\frac{-r}{s}}}{\left(\pi \cdot s\right) \cdot r} \cdot \frac{1}{8} + \frac{\mathsf{fma}\left(\frac{r}{{s}^{2}} \cdot \frac{1}{24} - \frac{1}{4} \cdot \frac{1}{s}, r, \frac{3}{4}\right)}{\left(\pi \cdot 6\right) \cdot \left(s \cdot r\right)} \]

   8. unpow2 N/A

      \[\leadsto \frac{e^{\frac{-r}{s}}}{\left(\pi \cdot s\right) \cdot r} \cdot \frac{1}{8} + \frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{1}{24} - \frac{1}{4} \cdot \frac{1}{s}, r, \frac{3}{4}\right)}{\left(\pi \cdot 6\right) \cdot \left(s \cdot r\right)} \]

   9. lower-*.f32 N/A

      \[\leadsto \frac{e^{\frac{-r}{s}}}{\left(\pi \cdot s\right) \cdot r} \cdot \frac{1}{8} + \frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{1}{24} - \frac{1}{4} \cdot \frac{1}{s}, r, \frac{3}{4}\right)}{\left(\pi \cdot 6\right) \cdot \left(s \cdot r\right)} \]

   10. associate-*r/ N/A

       \[\leadsto \frac{e^{\frac{-r}{s}}}{\left(\pi \cdot s\right) \cdot r} \cdot \frac{1}{8} + \frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{1}{24} - \frac{\frac{1}{4} \cdot 1}{s}, r, \frac{3}{4}\right)}{\left(\pi \cdot 6\right) \cdot \left(s \cdot r\right)} \]

   11. metadata-eval N/A

       \[\leadsto \frac{e^{\frac{-r}{s}}}{\left(\pi \cdot s\right) \cdot r} \cdot \frac{1}{8} + \frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{1}{24} - \frac{\frac{1}{4}}{s}, r, \frac{3}{4}\right)}{\left(\pi \cdot 6\right) \cdot \left(s \cdot r\right)} \]

   12. lower-/.f32 11.5

       \[\leadsto \frac{e^{\frac{-r}{s}}}{\left(\pi \cdot s\right) \cdot r} \cdot 0.125 + \frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot 0.041666666666666664 - \frac{0.25}{s}, r, 0.75\right)}{\left(\pi \cdot 6\right) \cdot \left(s \cdot r\right)} \]

10. Applied rewrites 11.5%

    \[\leadsto \frac{e^{\frac{-r}{s}}}{\left(\pi \cdot s\right) \cdot r} \cdot 0.125 + \frac{\color{blue}{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot 0.041666666666666664 - \frac{0.25}{s}, r, 0.75\right)}}{\left(\pi \cdot 6\right) \cdot \left(s \cdot r\right)} \]
11. Add Preprocessing

Alternative 6: 9.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot 0.125 - \frac{0.25}{s}, r, 0.25\right)}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left(0.3333333333333333, r, -0.05555555555555555 \cdot \frac{r \cdot r}{s}\right)}{s}, -1, 1\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/
   (fma (- (* (/ r (* s s)) 0.125) (/ 0.25 s)) r 0.25)
   (* (* (* 2.0 PI) s) r))
  (/
   (*
    0.75
    (fma
     (/ (fma 0.3333333333333333 r (* -0.05555555555555555 (/ (* r r) s))) s)
     -1.0
     1.0))
   (* (* (* 6.0 PI) s) r))))

C

float code(float s, float r) {
	return (fmaf((((r / (s * s)) * 0.125f) - (0.25f / s)), r, 0.25f) / (((2.0f * ((float) M_PI)) * s) * r)) + ((0.75f * fmaf((fmaf(0.3333333333333333f, r, (-0.05555555555555555f * ((r * r) / s))) / s), -1.0f, 1.0f)) / (((6.0f * ((float) M_PI)) * s) * r));
}

Julia

function code(s, r)
	return Float32(Float32(fma(Float32(Float32(Float32(r / Float32(s * s)) * Float32(0.125)) - Float32(Float32(0.25) / s)), r, Float32(0.25)) / Float32(Float32(Float32(Float32(2.0) * Float32(pi)) * s) * r)) + Float32(Float32(Float32(0.75) * fma(Float32(fma(Float32(0.3333333333333333), r, Float32(Float32(-0.05555555555555555) * Float32(Float32(r * r) / s))) / s), Float32(-1.0), Float32(1.0))) / Float32(Float32(Float32(Float32(6.0) * Float32(pi)) * s) * r)))
end

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in r around 0

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

   1. +-commutative N/A

      \[\leadsto \frac{r \cdot \left(\frac{1}{8} \cdot \frac{r}{{s}^{2}} - \frac{1}{4} \cdot \frac{1}{s}\right) + \color{blue}{\frac{1}{4}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\frac{3}{4} \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   2. *-commutative N/A

      \[\leadsto \frac{\left(\frac{1}{8} \cdot \frac{r}{{s}^{2}} - \frac{1}{4} \cdot \frac{1}{s}\right) \cdot r + \frac{1}{4}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\frac{3}{4} \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   3. lower-fma.f32 N/A

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

   4. lower--.f32 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. lower-/.f32 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f32 N/A

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

   10. associate-*r/ N/A

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

   11. metadata-eval N/A

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

   12. lower-/.f32 10.8

       \[\leadsto \frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot 0.125 - \frac{0.25}{s}, r, 0.25\right)}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

5. Applied rewrites 10.8%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot 0.125 - \frac{0.25}{s}, r, 0.25\right)}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

6. Taylor expanded in s around -inf

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

   4. lower-/.f32 N/A

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

   5. +-commutative N/A

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

   6. lower-fma.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. lower-/.f32 N/A

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

   9. unpow2 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{1}{8} - \frac{\frac{1}{4}}{s}, r, \frac{1}{4}\right)}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\frac{3}{4} \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{1}{3}, r, \frac{-1}{18} \cdot \frac{r \cdot r}{s}\right)}{s}, -1, 1\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   10. lower-*.f32 10.8

       \[\leadsto \frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot 0.125 - \frac{0.25}{s}, r, 0.25\right)}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left(0.3333333333333333, r, -0.05555555555555555 \cdot \frac{r \cdot r}{s}\right)}{s}, -1, 1\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

8. Applied rewrites 10.8%

   \[\leadsto \frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot 0.125 - \frac{0.25}{s}, r, 0.25\right)}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot \color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.3333333333333333, r, -0.05555555555555555 \cdot \frac{r \cdot r}{s}\right)}{s}, -1, 1\right)}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
9. Add Preprocessing

Alternative 7: 9.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot 0.125 - \frac{0.25}{s}, r, 0.25\right)}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{r \cdot r}{s}, -0.041666666666666664, r \cdot 0.25\right)}{s}, -1, 0.75\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/
   (fma (- (* (/ r (* s s)) 0.125) (/ 0.25 s)) r 0.25)
   (* (* (* 2.0 PI) s) r))
  (/
   (fma (/ (fma (/ (* r r) s) -0.041666666666666664 (* r 0.25)) s) -1.0 0.75)
   (* (* (* 6.0 PI) s) r))))

C

float code(float s, float r) {
	return (fmaf((((r / (s * s)) * 0.125f) - (0.25f / s)), r, 0.25f) / (((2.0f * ((float) M_PI)) * s) * r)) + (fmaf((fmaf(((r * r) / s), -0.041666666666666664f, (r * 0.25f)) / s), -1.0f, 0.75f) / (((6.0f * ((float) M_PI)) * s) * r));
}

Julia

function code(s, r)
	return Float32(Float32(fma(Float32(Float32(Float32(r / Float32(s * s)) * Float32(0.125)) - Float32(Float32(0.25) / s)), r, Float32(0.25)) / Float32(Float32(Float32(Float32(2.0) * Float32(pi)) * s) * r)) + Float32(fma(Float32(fma(Float32(Float32(r * r) / s), Float32(-0.041666666666666664), Float32(r * Float32(0.25))) / s), Float32(-1.0), Float32(0.75)) / Float32(Float32(Float32(Float32(6.0) * Float32(pi)) * s) * r)))
end

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in r around 0

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

   1. +-commutative N/A

      \[\leadsto \frac{r \cdot \left(\frac{1}{8} \cdot \frac{r}{{s}^{2}} - \frac{1}{4} \cdot \frac{1}{s}\right) + \color{blue}{\frac{1}{4}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\frac{3}{4} \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   2. *-commutative N/A

      \[\leadsto \frac{\left(\frac{1}{8} \cdot \frac{r}{{s}^{2}} - \frac{1}{4} \cdot \frac{1}{s}\right) \cdot r + \frac{1}{4}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\frac{3}{4} \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   3. lower-fma.f32 N/A

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

   4. lower--.f32 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. lower-/.f32 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f32 N/A

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

   10. associate-*r/ N/A

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

   11. metadata-eval N/A

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

   12. lower-/.f32 10.8

       \[\leadsto \frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot 0.125 - \frac{0.25}{s}, r, 0.25\right)}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

5. Applied rewrites 10.8%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot 0.125 - \frac{0.25}{s}, r, 0.25\right)}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

6. Taylor expanded in s around -inf

   \[\leadsto \frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{1}{8} - \frac{\frac{1}{4}}{s}, r, \frac{1}{4}\right)}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\color{blue}{\frac{3}{4} + -1 \cdot \frac{\frac{-1}{24} \cdot \frac{{r}^{2}}{s} + \frac{1}{4} \cdot r}{s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{1}{8} - \frac{\frac{1}{4}}{s}, r, \frac{1}{4}\right)}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{-1 \cdot \frac{\frac{-1}{24} \cdot \frac{{r}^{2}}{s} + \frac{1}{4} \cdot r}{s} + \color{blue}{\frac{3}{4}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   2. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{1}{8} - \frac{\frac{1}{4}}{s}, r, \frac{1}{4}\right)}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\frac{\frac{-1}{24} \cdot \frac{{r}^{2}}{s} + \frac{1}{4} \cdot r}{s} \cdot -1 + \frac{3}{4}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   3. lower-fma.f32 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{1}{8} - \frac{\frac{1}{4}}{s}, r, \frac{1}{4}\right)}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\frac{-1}{24} \cdot \frac{{r}^{2}}{s} + \frac{1}{4} \cdot r}{s}, \color{blue}{-1}, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   4. lower-/.f32 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{1}{8} - \frac{\frac{1}{4}}{s}, r, \frac{1}{4}\right)}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\frac{-1}{24} \cdot \frac{{r}^{2}}{s} + \frac{1}{4} \cdot r}{s}, -1, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   5. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{1}{8} - \frac{\frac{1}{4}}{s}, r, \frac{1}{4}\right)}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\frac{{r}^{2}}{s} \cdot \frac{-1}{24} + \frac{1}{4} \cdot r}{s}, -1, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   6. lower-fma.f32 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{1}{8} - \frac{\frac{1}{4}}{s}, r, \frac{1}{4}\right)}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{{r}^{2}}{s}, \frac{-1}{24}, \frac{1}{4} \cdot r\right)}{s}, -1, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   7. lower-/.f32 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{1}{8} - \frac{\frac{1}{4}}{s}, r, \frac{1}{4}\right)}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{{r}^{2}}{s}, \frac{-1}{24}, \frac{1}{4} \cdot r\right)}{s}, -1, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   8. unpow2 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{1}{8} - \frac{\frac{1}{4}}{s}, r, \frac{1}{4}\right)}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{r \cdot r}{s}, \frac{-1}{24}, \frac{1}{4} \cdot r\right)}{s}, -1, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   9. lower-*.f32 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{1}{8} - \frac{\frac{1}{4}}{s}, r, \frac{1}{4}\right)}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{r \cdot r}{s}, \frac{-1}{24}, \frac{1}{4} \cdot r\right)}{s}, -1, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   10. *-commutative N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{1}{8} - \frac{\frac{1}{4}}{s}, r, \frac{1}{4}\right)}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{r \cdot r}{s}, \frac{-1}{24}, r \cdot \frac{1}{4}\right)}{s}, -1, \frac{3}{4}\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   11. lower-*.f32 10.8

       \[\leadsto \frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot 0.125 - \frac{0.25}{s}, r, 0.25\right)}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{r \cdot r}{s}, -0.041666666666666664, r \cdot 0.25\right)}{s}, -1, 0.75\right)}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

8. Applied rewrites 10.8%

   \[\leadsto \frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot 0.125 - \frac{0.25}{s}, r, 0.25\right)}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{r \cdot r}{s}, -0.041666666666666664, r \cdot 0.25\right)}{s}, -1, 0.75\right)}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
9. Add Preprocessing

Alternative 8: 9.9% accurate, 3.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in s around -inf

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

5. Applied rewrites 10.6%

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

6. Taylor expanded in r around 0

   \[\leadsto -\frac{\frac{r \cdot \left(\frac{-5}{72} \cdot \frac{r}{{s}^{2} \cdot \mathsf{PI}\left(\right)} + \frac{1}{6} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}\right) - \frac{1}{4} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{r}}{s} \]
7. Step-by-step derivation

   1. lower-/.f32 N/A

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

8. Applied rewrites 10.7%

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

9. Final simplification 10.7%

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

Alternative 9: 9.9% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in s around -inf

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

5. Applied rewrites 10.6%

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

6. Taylor expanded in r around 0

   \[\leadsto -\frac{\frac{r \cdot \left(\frac{-5}{72} \cdot \frac{r}{{s}^{2} \cdot \mathsf{PI}\left(\right)} + \frac{1}{6} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}\right) - \frac{1}{4} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{r}}{s} \]
7. Step-by-step derivation

   1. lower-/.f32 N/A

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

8. Applied rewrites 10.7%

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

9. Taylor expanded in s around 0

   \[\leadsto -\frac{\frac{\frac{-5}{72} \cdot \frac{r}{\mathsf{PI}\left(\right)} + s \cdot \left(\frac{-1}{4} \cdot \frac{s}{r \cdot \mathsf{PI}\left(\right)} + \frac{1}{6} \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{{s}^{2}}}{s} \]
10. Step-by-step derivation

    1. lower-/.f32 N/A

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

11. Applied rewrites 10.7%

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

12. Final simplification 10.7%

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

Alternative 10: 9.9% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in s around -inf

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

5. Applied rewrites 10.6%

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

6. Taylor expanded in r around 0

   \[\leadsto -\frac{\left(\frac{-5}{72} \cdot \frac{r}{{s}^{2} \cdot \mathsf{PI}\left(\right)} + \frac{1}{6} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}\right) - \frac{\frac{1}{4}}{\pi \cdot r}}{s} \]
7. Step-by-step derivation

   1. *-commutative N/A

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

   2. associate-*r/ N/A

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

   3. metadata-eval N/A

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

   4. lower-fma.f32 N/A

      \[\leadsto -\frac{\mathsf{fma}\left(\frac{r}{{s}^{2} \cdot \mathsf{PI}\left(\right)}, \frac{-5}{72}, \frac{\frac{1}{6}}{s \cdot \mathsf{PI}\left(\right)}\right) - \frac{\frac{1}{4}}{\pi \cdot r}}{s} \]

   5. lower-/.f32 N/A

      \[\leadsto -\frac{\mathsf{fma}\left(\frac{r}{{s}^{2} \cdot \mathsf{PI}\left(\right)}, \frac{-5}{72}, \frac{\frac{1}{6}}{s \cdot \mathsf{PI}\left(\right)}\right) - \frac{\frac{1}{4}}{\pi \cdot r}}{s} \]

   6. lower-*.f32 N/A

      \[\leadsto -\frac{\mathsf{fma}\left(\frac{r}{{s}^{2} \cdot \mathsf{PI}\left(\right)}, \frac{-5}{72}, \frac{\frac{1}{6}}{s \cdot \mathsf{PI}\left(\right)}\right) - \frac{\frac{1}{4}}{\pi \cdot r}}{s} \]

   7. unpow2 N/A

      \[\leadsto -\frac{\mathsf{fma}\left(\frac{r}{\left(s \cdot s\right) \cdot \mathsf{PI}\left(\right)}, \frac{-5}{72}, \frac{\frac{1}{6}}{s \cdot \mathsf{PI}\left(\right)}\right) - \frac{\frac{1}{4}}{\pi \cdot r}}{s} \]

   8. lower-*.f32 N/A

      \[\leadsto -\frac{\mathsf{fma}\left(\frac{r}{\left(s \cdot s\right) \cdot \mathsf{PI}\left(\right)}, \frac{-5}{72}, \frac{\frac{1}{6}}{s \cdot \mathsf{PI}\left(\right)}\right) - \frac{\frac{1}{4}}{\pi \cdot r}}{s} \]

   9. lift-PI.f32 N/A

      \[\leadsto -\frac{\mathsf{fma}\left(\frac{r}{\left(s \cdot s\right) \cdot \pi}, \frac{-5}{72}, \frac{\frac{1}{6}}{s \cdot \mathsf{PI}\left(\right)}\right) - \frac{\frac{1}{4}}{\pi \cdot r}}{s} \]

   10. lower-/.f32 N/A

       \[\leadsto -\frac{\mathsf{fma}\left(\frac{r}{\left(s \cdot s\right) \cdot \pi}, \frac{-5}{72}, \frac{\frac{1}{6}}{s \cdot \mathsf{PI}\left(\right)}\right) - \frac{\frac{1}{4}}{\pi \cdot r}}{s} \]

   11. *-commutative N/A

       \[\leadsto -\frac{\mathsf{fma}\left(\frac{r}{\left(s \cdot s\right) \cdot \pi}, \frac{-5}{72}, \frac{\frac{1}{6}}{\mathsf{PI}\left(\right) \cdot s}\right) - \frac{\frac{1}{4}}{\pi \cdot r}}{s} \]

   12. lift-*.f32 N/A

       \[\leadsto -\frac{\mathsf{fma}\left(\frac{r}{\left(s \cdot s\right) \cdot \pi}, \frac{-5}{72}, \frac{\frac{1}{6}}{\mathsf{PI}\left(\right) \cdot s}\right) - \frac{\frac{1}{4}}{\pi \cdot r}}{s} \]

   13. lift-PI.f32 10.7

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

8. Applied rewrites 10.7%

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

9. Final simplification 10.7%

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

Alternative 11: 9.9% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in s around -inf

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

5. Applied rewrites 10.6%

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

6. Taylor expanded in s around inf

   \[\leadsto \frac{\left(\frac{5}{72} \cdot \frac{r}{{s}^{2} \cdot \mathsf{PI}\left(\right)} + \frac{1}{4} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}\right) - \frac{\frac{1}{6}}{s \cdot \mathsf{PI}\left(\right)}}{\color{blue}{s}} \]
7. Step-by-step derivation

   1. lower-/.f32 N/A

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

8. Applied rewrites 10.7%

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

Alternative 12: 9.0% accurate, 5.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.5%

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

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. lower-/.f32 N/A

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

5. Applied rewrites 9.3%

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

Alternative 13: 9.0% accurate, 6.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

5. Applied rewrites 9.3%

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

Alternative 14: 8.9% accurate, 8.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in 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 10.6%

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

6. Taylor expanded in r around 0

   \[\leadsto -\frac{\frac{r \cdot \left(\frac{-5}{72} \cdot \frac{r}{{s}^{2} \cdot \mathsf{PI}\left(\right)} + \frac{1}{6} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}\right) - \frac{1}{4} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{r}}{s} \]
7. Step-by-step derivation

   1. lower-/.f32 N/A

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

8. Applied rewrites 10.7%

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

9. Taylor expanded in s around inf

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

    1. lower-/.f32 N/A

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

    2. lift-PI.f32 9.1

       \[\leadsto -\frac{\frac{\frac{-0.25}{\pi}}{r}}{s} \]

11. Applied rewrites 9.1%

    \[\leadsto -\frac{\frac{\frac{-0.25}{\pi}}{r}}{s} \]

12. Final simplification 9.1%

    \[\leadsto \frac{\frac{\frac{-0.25}{\pi}}{r}}{-s} \]
13. Add Preprocessing

Alternative 15: 8.9% accurate, 8.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lift-PI.f32 9.1

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

5. Applied rewrites 9.1%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-PI.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. *-commutative N/A

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

   6. *-commutative N/A

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

   7. associate-/r* N/A

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

   8. metadata-eval N/A

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

   9. associate-*r/ N/A

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

   10. lower-/.f32 N/A

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

   11. associate-*r/ N/A

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

   12. metadata-eval N/A

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

   13. lower-/.f32 N/A

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

   14. *-commutative N/A

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

   15. lift-*.f32 N/A

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

   16. lift-PI.f32 9.1

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

7. Applied rewrites 9.1%

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

   1. lift-/.f32 N/A

      \[\leadsto \frac{\frac{\frac{1}{4}}{r}}{\color{blue}{\pi} \cdot s} \]

   2. lift-/.f32 N/A

      \[\leadsto \frac{\frac{\frac{1}{4}}{r}}{\color{blue}{\pi \cdot s}} \]

   3. lift-PI.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. associate-/r* N/A

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

   6. lower-/.f32 N/A

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

   7. lower-/.f32 N/A

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

   8. lift-/.f32 N/A

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

   9. lift-PI.f32 9.1

      \[\leadsto \frac{\frac{\frac{0.25}{r}}{\pi}}{s} \]

9. Applied rewrites 9.1%

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

Alternative 16: 8.9% accurate, 10.6× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lift-PI.f32 9.1

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

5. Applied rewrites 9.1%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-PI.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. *-commutative N/A

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

   6. *-commutative N/A

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

   7. associate-/r* N/A

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

   8. metadata-eval N/A

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

   9. associate-*r/ N/A

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

   10. lower-/.f32 N/A

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

   11. associate-*r/ N/A

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

   12. metadata-eval N/A

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

   13. lower-/.f32 N/A

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

   14. *-commutative N/A

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

   15. lift-*.f32 N/A

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

   16. lift-PI.f32 9.1

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

7. Applied rewrites 9.1%

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

Alternative 17: 8.9% accurate, 13.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lift-PI.f32 9.1

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

5. Applied rewrites 9.1%

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

   1. lift-*.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. lift-PI.f32 9.1

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

7. Applied rewrites 9.1%

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

Reproduce

herbie shell --seed 2025057 
(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))))