Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.6%

Time: 4.9s

Alternatives: 16

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 99.6% 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

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

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (* (/ (exp (/ (- r) s)) (* (* PI s) r)) 0.125)
  (/ (* (exp (/ (- r) (* 3.0 s))) (/ 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) + ((expf((-r / (3.0f * s))) * (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(Float32(exp(Float32(Float32(-r) / Float32(Float32(3.0) * s))) * Float32(Float32(0.75) / Float32(Float32(Float32(pi) * Float32(6.0)) * s))) / r))
end

MATLAB

function tmp = code(s, r)
	tmp = ((exp((-r / s)) / ((single(pi) * s) * r)) * single(0.125)) + ((exp((-r / (single(3.0) * s))) * (single(0.75) / ((single(pi) * single(6.0)) * s))) / r);
end

Derivation

1. Initial program 99.6%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-exp.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. lift-neg.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. associate-/r* N/A

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

   9. lower-/.f32 N/A

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

3. Applied rewrites 99.6%

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

4. Taylor expanded in s around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. lower-exp.f32 N/A

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

   5. mul-1-neg N/A

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

   6. distribute-frac-neg N/A

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

   7. lift-neg.f32 N/A

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

   8. lift-/.f32 N/A

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

   9. *-commutative N/A

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

   10. *-commutative N/A

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

   11. lift-*.f32 N/A

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

   12. lift-PI.f32 N/A

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

   13. lift-*.f32 99.6

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

6. Applied rewrites 99.6%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-exp.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. lift-neg.f32 N/A

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

   7. associate-/l* N/A

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

   8. lower-*.f32 N/A

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

   9. lift-neg.f32 N/A

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

   10. lift-/.f32 N/A

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

   11. lift-*.f32 N/A

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

   12. lift-exp.f32 N/A

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

   13. lower-/.f32 99.6

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

8. Applied rewrites 99.6%

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

Alternative 2: 99.6% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (* (/ (exp (/ (- r) s)) (* (* PI s) r)) 0.125)
  (/ (* 0.75 (exp (/ (- r) (* 3.0 s)))) (* (* (* 6.0 PI) s) r))))

C

float code(float s, float r) {
	return ((expf((-r / s)) / ((((float) M_PI) * s) * r)) * 0.125f) + ((0.75f * expf((-r / (3.0f * s)))) / (((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(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 = ((exp((-r / s)) / ((single(pi) * s) * r)) * single(0.125)) + ((single(0.75) * exp((-r / (single(3.0) * s)))) / (((single(6.0) * single(pi)) * s) * r));
end

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in s around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. lower-exp.f32 N/A

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

   5. mul-1-neg N/A

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

   6. distribute-frac-neg N/A

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

   7. lift-neg.f32 N/A

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

   8. lift-/.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 N/A

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

   13. lift-PI.f32 99.6

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

4. Applied rewrites 99.6%

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

Alternative 3: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-exp.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. lift-neg.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. associate-/r* N/A

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

   9. lower-/.f32 N/A

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

3. Applied rewrites 99.6%

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

4. Taylor expanded in s around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. lower-exp.f32 N/A

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

   5. mul-1-neg N/A

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

   6. distribute-frac-neg N/A

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

   7. lift-neg.f32 N/A

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

   8. lift-/.f32 N/A

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

   9. *-commutative N/A

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

   10. *-commutative N/A

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

   11. lift-*.f32 N/A

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

   12. lift-PI.f32 N/A

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

   13. lift-*.f32 99.6

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

6. Applied rewrites 99.6%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-exp.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. lift-neg.f32 N/A

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

   7. associate-/l* N/A

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

   8. lower-*.f32 N/A

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

   9. lift-neg.f32 N/A

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

   10. lift-/.f32 N/A

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

   11. lift-*.f32 N/A

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

   12. lift-exp.f32 N/A

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

   13. lower-/.f32 99.6

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

8. Applied rewrites 99.6%

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

9. Taylor expanded in s around 0

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

    1. lower-/.f32 N/A

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

    2. *-commutative N/A

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

    3. lift-*.f32 N/A

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

    4. lift-PI.f32 99.5

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

11. Applied rewrites 99.5%

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

Alternative 4: 99.5% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in s around 0

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

   1. lower-/.f32 N/A

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

4. Applied rewrites 99.5%

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

   1. lift-+.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-exp.f32 N/A

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

   4. lift-/.f32 N/A

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

   5. lift-neg.f32 N/A

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

   6. lift-PI.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. lift-/.f32 N/A

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

   9. lift-exp.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. lift-/.f32 N/A

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

   12. lift-PI.f32 N/A

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

   13. lift-*.f32 N/A

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

   14. +-commutative N/A

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

6. Applied rewrites 99.5%

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

Alternative 5: 99.5% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = (single(0.125) * ((exp((single(-0.3333333333333333) * (r / s))) + exp((-r / s))) * (single(1.0) / (single(pi) * r)))) / s;
end

Derivation

1. Initial program 99.6%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-exp.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. lift-neg.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. associate-/r* N/A

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

   9. lower-/.f32 N/A

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

3. Applied rewrites 99.6%

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

4. Taylor expanded in s around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. lower-exp.f32 N/A

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

   5. mul-1-neg N/A

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

   6. distribute-frac-neg N/A

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

   7. lift-neg.f32 N/A

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

   8. lift-/.f32 N/A

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

   9. *-commutative N/A

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

   10. *-commutative N/A

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

   11. lift-*.f32 N/A

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

   12. lift-PI.f32 N/A

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

   13. lift-*.f32 99.6

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

6. Applied rewrites 99.6%

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

7. Taylor expanded in s around 0

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

   1. lower-/.f32 N/A

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

9. Applied rewrites 99.5%

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

    1. lift-/.f32 N/A

       \[\leadsto \frac{\frac{1}{8} \cdot \frac{e^{\frac{-r}{s}} + e^{\frac{-1}{3} \cdot \frac{r}{s}}}{r \cdot \pi}}{s} \]

    2. lift-+.f32 N/A

       \[\leadsto \frac{\frac{1}{8} \cdot \frac{e^{\frac{-r}{s}} + e^{\frac{-1}{3} \cdot \frac{r}{s}}}{r \cdot \pi}}{s} \]

    3. lift-exp.f32 N/A

       \[\leadsto \frac{\frac{1}{8} \cdot \frac{e^{\frac{-r}{s}} + e^{\frac{-1}{3} \cdot \frac{r}{s}}}{r \cdot \pi}}{s} \]

    4. lift-/.f32 N/A

       \[\leadsto \frac{\frac{1}{8} \cdot \frac{e^{\frac{-r}{s}} + e^{\frac{-1}{3} \cdot \frac{r}{s}}}{r \cdot \pi}}{s} \]

    5. lift-neg.f32 N/A

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

    6. lift-exp.f32 N/A

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

    7. lift-*.f32 N/A

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

    8. lift-/.f32 N/A

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

    9. lift-PI.f32 N/A

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

    10. lift-*.f32 N/A

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

    11. mult-flip N/A

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

    12. lower-*.f32 N/A

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

11. Applied rewrites 99.5%

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

Alternative 6: 99.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in s around 0

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

   1. lower-/.f32 N/A

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

4. Applied rewrites 99.5%

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

6. Applied rewrites 99.5%

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

Alternative 7: 99.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in s around 0

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

   1. lower-/.f32 N/A

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

4. Applied rewrites 99.5%

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

   1. lift-+.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-exp.f32 N/A

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

   4. lift-/.f32 N/A

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

   5. lift-neg.f32 N/A

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

   6. lift-PI.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. associate-/r* N/A

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

   9. distribute-frac-neg N/A

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

   10. mul-1-neg N/A

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

   11. lift-/.f32 N/A

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

   12. lift-exp.f32 N/A

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

   13. lift-*.f32 N/A

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

   14. lift-/.f32 N/A

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

   15. lift-PI.f32 N/A

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

   16. lift-*.f32 N/A

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

6. Applied rewrites 99.5%

   \[\leadsto \frac{0.125 \cdot \frac{1}{\frac{r}{\frac{e^{\frac{-r}{s}} + e^{\frac{r}{s} \cdot -0.3333333333333333}}{\pi}}}}{s} \]

7. Taylor expanded in s around 0

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

   1. lower-*.f32 N/A

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

   2. lower-/.f32 N/A

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

   3. mul-1-neg N/A

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

   4. distribute-frac-neg N/A

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

   5. lower-+.f32 N/A

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

   6. lift-neg.f32 N/A

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

   7. lift-/.f32 N/A

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

   8. lift-exp.f32 N/A

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

   9. lower-exp.f32 N/A

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

   10. lower-*.f32 N/A

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

   11. lift-/.f32 N/A

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

   12. lower-*.f32 N/A

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

9. Applied rewrites 99.5%

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

Alternative 8: 74.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 38:\\ \;\;\;\;\frac{0.125 \cdot \frac{1}{r \cdot \mathsf{fma}\left(0.5, \pi, r \cdot \left(\left(-r \cdot \left(\frac{\pi}{s \cdot s} \cdot -0.08333333333333333\right)\right) - -0.3333333333333333 \cdot \frac{\pi}{s}\right)\right)}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.25}{\log \left({\left(e^{\pi}\right)}^{r}\right)}}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (if (<= r 38.0)
   (/
    (*
     0.125
     (/
      1.0
      (*
       r
       (fma
        0.5
        PI
        (*
         r
         (-
          (- (* r (* (/ PI (* s s)) -0.08333333333333333)))
          (* -0.3333333333333333 (/ PI s))))))))
    s)
   (/ (/ 0.25 (log (pow (exp PI) r))) s)))

C

float code(float s, float r) {
	float tmp;
	if (r <= 38.0f) {
		tmp = (0.125f * (1.0f / (r * fmaf(0.5f, ((float) M_PI), (r * (-(r * ((((float) M_PI) / (s * s)) * -0.08333333333333333f)) - (-0.3333333333333333f * (((float) M_PI) / s)))))))) / s;
	} else {
		tmp = (0.25f / logf(powf(expf(((float) M_PI)), r))) / s;
	}
	return tmp;
}

Julia

function code(s, r)
	tmp = Float32(0.0)
	if (r <= Float32(38.0))
		tmp = Float32(Float32(Float32(0.125) * Float32(Float32(1.0) / Float32(r * fma(Float32(0.5), Float32(pi), Float32(r * Float32(Float32(-Float32(r * Float32(Float32(Float32(pi) / Float32(s * s)) * Float32(-0.08333333333333333)))) - Float32(Float32(-0.3333333333333333) * Float32(Float32(pi) / s)))))))) / s);
	else
		tmp = Float32(Float32(Float32(0.25) / log((exp(Float32(pi)) ^ r))) / s);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if r < 38

   1. Initial program 99.4%

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

   2. Taylor expanded in s around 0

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

      1. lower-/.f32 N/A

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

   4. Applied rewrites 99.3%

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

      1. lift-+.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. lift-exp.f32 N/A

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

      4. lift-/.f32 N/A

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

      5. lift-neg.f32 N/A

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

      6. lift-PI.f32 N/A

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

      7. lift-*.f32 N/A

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

      8. associate-/r* N/A

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

      9. distribute-frac-neg N/A

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

      10. mul-1-neg N/A

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

      11. lift-/.f32 N/A

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

      12. lift-exp.f32 N/A

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

      13. lift-*.f32 N/A

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

      14. lift-/.f32 N/A

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

      15. lift-PI.f32 N/A

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

      16. lift-*.f32 N/A

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

   6. Applied rewrites 99.3%

      \[\leadsto \frac{0.125 \cdot \frac{1}{\frac{r}{\frac{e^{\frac{-r}{s}} + e^{\frac{r}{s} \cdot -0.3333333333333333}}{\pi}}}}{s} \]

   7. Taylor expanded in r around 0

      \[\leadsto \frac{\frac{1}{8} \cdot \frac{1}{r \cdot \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) + r \cdot \left(-1 \cdot \left(r \cdot \left(\frac{-2}{9} \cdot \frac{\mathsf{PI}\left(\right)}{{s}^{2}} + \frac{5}{36} \cdot \frac{\mathsf{PI}\left(\right)}{{s}^{2}}\right)\right) - \frac{-1}{3} \cdot \frac{\mathsf{PI}\left(\right)}{s}\right)\right)}}{s} \]
   8. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \frac{\frac{1}{8} \cdot \frac{1}{r \cdot \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) + r \cdot \left(-1 \cdot \left(r \cdot \left(\frac{-2}{9} \cdot \frac{\mathsf{PI}\left(\right)}{{s}^{2}} + \frac{5}{36} \cdot \frac{\mathsf{PI}\left(\right)}{{s}^{2}}\right)\right) - \frac{-1}{3} \cdot \frac{\mathsf{PI}\left(\right)}{s}\right)\right)}}{s} \]

      2. lower-fma.f32 N/A

         \[\leadsto \frac{\frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{fma}\left(\frac{1}{2}, \mathsf{PI}\left(\right), r \cdot \left(-1 \cdot \left(r \cdot \left(\frac{-2}{9} \cdot \frac{\mathsf{PI}\left(\right)}{{s}^{2}} + \frac{5}{36} \cdot \frac{\mathsf{PI}\left(\right)}{{s}^{2}}\right)\right) - \frac{-1}{3} \cdot \frac{\mathsf{PI}\left(\right)}{s}\right)\right)}}{s} \]

      3. lift-PI.f32 N/A

         \[\leadsto \frac{\frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{fma}\left(\frac{1}{2}, \pi, r \cdot \left(-1 \cdot \left(r \cdot \left(\frac{-2}{9} \cdot \frac{\mathsf{PI}\left(\right)}{{s}^{2}} + \frac{5}{36} \cdot \frac{\mathsf{PI}\left(\right)}{{s}^{2}}\right)\right) - \frac{-1}{3} \cdot \frac{\mathsf{PI}\left(\right)}{s}\right)\right)}}{s} \]

      4. lower-*.f32 N/A

         \[\leadsto \frac{\frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{fma}\left(\frac{1}{2}, \pi, r \cdot \left(-1 \cdot \left(r \cdot \left(\frac{-2}{9} \cdot \frac{\mathsf{PI}\left(\right)}{{s}^{2}} + \frac{5}{36} \cdot \frac{\mathsf{PI}\left(\right)}{{s}^{2}}\right)\right) - \frac{-1}{3} \cdot \frac{\mathsf{PI}\left(\right)}{s}\right)\right)}}{s} \]

      5. lower--.f32 N/A

         \[\leadsto \frac{\frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{fma}\left(\frac{1}{2}, \pi, r \cdot \left(-1 \cdot \left(r \cdot \left(\frac{-2}{9} \cdot \frac{\mathsf{PI}\left(\right)}{{s}^{2}} + \frac{5}{36} \cdot \frac{\mathsf{PI}\left(\right)}{{s}^{2}}\right)\right) - \frac{-1}{3} \cdot \frac{\mathsf{PI}\left(\right)}{s}\right)\right)}}{s} \]

   9. Applied rewrites 61.2%

      \[\leadsto \frac{0.125 \cdot \frac{1}{r \cdot \mathsf{fma}\left(0.5, \pi, r \cdot \left(\left(-r \cdot \left(\frac{\pi}{s \cdot s} \cdot -0.08333333333333333\right)\right) - -0.3333333333333333 \cdot \frac{\pi}{s}\right)\right)}}{s} \]

   if 38 < r

   1. Initial program 99.8%

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

   2. Taylor expanded in s around 0

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

      1. lower-/.f32 N/A

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

   4. Applied rewrites 99.8%

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

      1. lift-+.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. lift-exp.f32 N/A

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

      4. lift-/.f32 N/A

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

      5. lift-neg.f32 N/A

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

      6. lift-PI.f32 N/A

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

      7. lift-*.f32 N/A

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

      8. associate-/r* N/A

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

      9. distribute-frac-neg N/A

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

      10. mul-1-neg N/A

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

      11. lift-/.f32 N/A

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

      12. lift-exp.f32 N/A

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

      13. lift-*.f32 N/A

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

      14. lift-/.f32 N/A

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

      15. lift-PI.f32 N/A

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

      16. lift-*.f32 N/A

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

   6. Applied rewrites 99.8%

      \[\leadsto \frac{0.125 \cdot \frac{1}{\frac{r}{\frac{e^{\frac{-r}{s}} + e^{\frac{r}{s} \cdot -0.3333333333333333}}{\pi}}}}{s} \]

   7. Taylor expanded in s around inf

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

      1. lower-/.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lift-PI.f32 5.3

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

   9. Applied rewrites 5.3%

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

       1. lift-PI.f32 N/A

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

       2. lift-*.f32 N/A

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

       3. add-log-exp N/A

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

       4. log-pow-rev N/A

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

       5. lower-log.f32 N/A

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

       6. lower-pow.f32 N/A

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

       7. lower-exp.f32 N/A

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

       8. lift-PI.f32 97.5

          \[\leadsto \frac{\frac{0.25}{\log \left({\left(e^{\pi}\right)}^{r}\right)}}{s} \]

   11. Applied rewrites 97.5%

       \[\leadsto \frac{\frac{0.25}{\log \left({\left(e^{\pi}\right)}^{r}\right)}}{s} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 45.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 38:\\ \;\;\;\;\frac{0.125 \cdot \frac{1}{\mathsf{fma}\left(0.25, \frac{r \cdot \left(\pi \cdot \left(1.3333333333333333 \cdot r\right)\right)}{s}, 0.5 \cdot \left(r \cdot \pi\right)\right)}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.25}{\log \left({\left(e^{\pi}\right)}^{r}\right)}}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (if (<= r 38.0)
   (/
    (*
     0.125
     (/
      1.0
      (fma 0.25 (/ (* r (* PI (* 1.3333333333333333 r))) s) (* 0.5 (* r PI)))))
    s)
   (/ (/ 0.25 (log (pow (exp PI) r))) s)))

C

float code(float s, float r) {
	float tmp;
	if (r <= 38.0f) {
		tmp = (0.125f * (1.0f / fmaf(0.25f, ((r * (((float) M_PI) * (1.3333333333333333f * r))) / s), (0.5f * (r * ((float) M_PI)))))) / s;
	} else {
		tmp = (0.25f / logf(powf(expf(((float) M_PI)), r))) / s;
	}
	return tmp;
}

Julia

function code(s, r)
	tmp = Float32(0.0)
	if (r <= Float32(38.0))
		tmp = Float32(Float32(Float32(0.125) * Float32(Float32(1.0) / fma(Float32(0.25), Float32(Float32(r * Float32(Float32(pi) * Float32(Float32(1.3333333333333333) * r))) / s), Float32(Float32(0.5) * Float32(r * Float32(pi)))))) / s);
	else
		tmp = Float32(Float32(Float32(0.25) / log((exp(Float32(pi)) ^ r))) / s);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if r < 38

   1. Initial program 99.4%

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

   2. Taylor expanded in s around 0

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

      1. lower-/.f32 N/A

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

   4. Applied rewrites 99.3%

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

      1. lift-+.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. lift-exp.f32 N/A

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

      4. lift-/.f32 N/A

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

      5. lift-neg.f32 N/A

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

      6. lift-PI.f32 N/A

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

      7. lift-*.f32 N/A

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

      8. associate-/r* N/A

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

      9. distribute-frac-neg N/A

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

      10. mul-1-neg N/A

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

      11. lift-/.f32 N/A

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

      12. lift-exp.f32 N/A

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

      13. lift-*.f32 N/A

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

      14. lift-/.f32 N/A

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

      15. lift-PI.f32 N/A

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

      16. lift-*.f32 N/A

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

   6. Applied rewrites 99.3%

      \[\leadsto \frac{0.125 \cdot \frac{1}{\frac{r}{\frac{e^{\frac{-r}{s}} + e^{\frac{r}{s} \cdot -0.3333333333333333}}{\pi}}}}{s} \]

   7. Taylor expanded in s around -inf

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

      1. lower-fma.f32 N/A

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

      2. lower-/.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lift-PI.f32 N/A

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

      6. distribute-rgt1-in N/A

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

      7. metadata-eval N/A

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

      8. lower-*.f32 N/A

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

      9. lower-*.f32 N/A

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

      10. lower-*.f32 N/A

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

      11. lift-PI.f32 16.4

          \[\leadsto \frac{0.125 \cdot \frac{1}{\mathsf{fma}\left(0.25, \frac{r \cdot \left(\pi \cdot \left(1.3333333333333333 \cdot r\right)\right)}{s}, 0.5 \cdot \left(r \cdot \pi\right)\right)}}{s} \]

   9. Applied rewrites 16.4%

      \[\leadsto \frac{0.125 \cdot \frac{1}{\mathsf{fma}\left(0.25, \frac{r \cdot \left(\pi \cdot \left(1.3333333333333333 \cdot r\right)\right)}{s}, 0.5 \cdot \left(r \cdot \pi\right)\right)}}{s} \]

   if 38 < r

   1. Initial program 99.8%

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

   2. Taylor expanded in s around 0

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

      1. lower-/.f32 N/A

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

   4. Applied rewrites 99.8%

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

      1. lift-+.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. lift-exp.f32 N/A

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

      4. lift-/.f32 N/A

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

      5. lift-neg.f32 N/A

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

      6. lift-PI.f32 N/A

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

      7. lift-*.f32 N/A

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

      8. associate-/r* N/A

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

      9. distribute-frac-neg N/A

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

      10. mul-1-neg N/A

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

      11. lift-/.f32 N/A

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

      12. lift-exp.f32 N/A

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

      13. lift-*.f32 N/A

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

      14. lift-/.f32 N/A

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

      15. lift-PI.f32 N/A

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

      16. lift-*.f32 N/A

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

   6. Applied rewrites 99.8%

      \[\leadsto \frac{0.125 \cdot \frac{1}{\frac{r}{\frac{e^{\frac{-r}{s}} + e^{\frac{r}{s} \cdot -0.3333333333333333}}{\pi}}}}{s} \]

   7. Taylor expanded in s around inf

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

      1. lower-/.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lift-PI.f32 5.3

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

   9. Applied rewrites 5.3%

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

       1. lift-PI.f32 N/A

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

       2. lift-*.f32 N/A

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

       3. add-log-exp N/A

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

       4. log-pow-rev N/A

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

       5. lower-log.f32 N/A

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

       6. lower-pow.f32 N/A

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

       7. lower-exp.f32 N/A

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

       8. lift-PI.f32 97.5

          \[\leadsto \frac{\frac{0.25}{\log \left({\left(e^{\pi}\right)}^{r}\right)}}{s} \]

   11. Applied rewrites 97.5%

       \[\leadsto \frac{\frac{0.25}{\log \left({\left(e^{\pi}\right)}^{r}\right)}}{s} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 19.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{0.125 \cdot \frac{1}{\mathsf{fma}\left(0.25, \frac{r \cdot \left(\pi \cdot \left(1.3333333333333333 \cdot r\right)\right)}{s}, 0.5 \cdot \left(r \cdot \pi\right)\right)}}{s} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (/
  (*
   0.125
   (/
    1.0
    (fma 0.25 (/ (* r (* PI (* 1.3333333333333333 r))) s) (* 0.5 (* r PI)))))
  s))

C

float code(float s, float r) {
	return (0.125f * (1.0f / fmaf(0.25f, ((r * (((float) M_PI) * (1.3333333333333333f * r))) / s), (0.5f * (r * ((float) M_PI)))))) / s;
}

Julia

function code(s, r)
	return Float32(Float32(Float32(0.125) * Float32(Float32(1.0) / fma(Float32(0.25), Float32(Float32(r * Float32(Float32(pi) * Float32(Float32(1.3333333333333333) * r))) / s), Float32(Float32(0.5) * Float32(r * Float32(pi)))))) / s)
end

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in s around 0

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

   1. lower-/.f32 N/A

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

4. Applied rewrites 99.5%

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

   1. lift-+.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-exp.f32 N/A

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

   4. lift-/.f32 N/A

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

   5. lift-neg.f32 N/A

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

   6. lift-PI.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. associate-/r* N/A

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

   9. distribute-frac-neg N/A

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

   10. mul-1-neg N/A

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

   11. lift-/.f32 N/A

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

   12. lift-exp.f32 N/A

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

   13. lift-*.f32 N/A

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

   14. lift-/.f32 N/A

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

   15. lift-PI.f32 N/A

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

   16. lift-*.f32 N/A

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

6. Applied rewrites 99.5%

   \[\leadsto \frac{0.125 \cdot \frac{1}{\frac{r}{\frac{e^{\frac{-r}{s}} + e^{\frac{r}{s} \cdot -0.3333333333333333}}{\pi}}}}{s} \]

7. Taylor expanded in s around -inf

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

   1. lower-fma.f32 N/A

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

   2. lower-/.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. distribute-rgt1-in N/A

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

   7. metadata-eval N/A

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

   8. lower-*.f32 N/A

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

   9. lower-*.f32 N/A

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

   10. lower-*.f32 N/A

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

   11. lift-PI.f32 19.6

       \[\leadsto \frac{0.125 \cdot \frac{1}{\mathsf{fma}\left(0.25, \frac{r \cdot \left(\pi \cdot \left(1.3333333333333333 \cdot r\right)\right)}{s}, 0.5 \cdot \left(r \cdot \pi\right)\right)}}{s} \]

9. Applied rewrites 19.6%

   \[\leadsto \frac{0.125 \cdot \frac{1}{\mathsf{fma}\left(0.25, \frac{r \cdot \left(\pi \cdot \left(1.3333333333333333 \cdot r\right)\right)}{s}, 0.5 \cdot \left(r \cdot \pi\right)\right)}}{s} \]
10. Add Preprocessing

Alternative 11: 19.0% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \frac{0.125 \cdot \frac{1}{r \cdot \mathsf{fma}\left(0.3333333333333333, \frac{r \cdot \pi}{s}, 0.5 \cdot \pi\right)}}{s} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (/
  (* 0.125 (/ 1.0 (* r (fma 0.3333333333333333 (/ (* r PI) s) (* 0.5 PI)))))
  s))

C

float code(float s, float r) {
	return (0.125f * (1.0f / (r * fmaf(0.3333333333333333f, ((r * ((float) M_PI)) / s), (0.5f * ((float) M_PI)))))) / s;
}

Julia

function code(s, r)
	return Float32(Float32(Float32(0.125) * Float32(Float32(1.0) / Float32(r * fma(Float32(0.3333333333333333), Float32(Float32(r * Float32(pi)) / s), Float32(Float32(0.5) * Float32(pi)))))) / s)
end

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in s around 0

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

   1. lower-/.f32 N/A

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

4. Applied rewrites 99.5%

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

   1. lift-+.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-exp.f32 N/A

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

   4. lift-/.f32 N/A

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

   5. lift-neg.f32 N/A

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

   6. lift-PI.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. associate-/r* N/A

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

   9. distribute-frac-neg N/A

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

   10. mul-1-neg N/A

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

   11. lift-/.f32 N/A

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

   12. lift-exp.f32 N/A

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

   13. lift-*.f32 N/A

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

   14. lift-/.f32 N/A

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

   15. lift-PI.f32 N/A

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

   16. lift-*.f32 N/A

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

6. Applied rewrites 99.5%

   \[\leadsto \frac{0.125 \cdot \frac{1}{\frac{r}{\frac{e^{\frac{-r}{s}} + e^{\frac{r}{s} \cdot -0.3333333333333333}}{\pi}}}}{s} \]

7. Taylor expanded in r around 0

   \[\leadsto \frac{\frac{1}{8} \cdot \frac{1}{r \cdot \left(\frac{1}{3} \cdot \frac{r \cdot \mathsf{PI}\left(\right)}{s} + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}}{s} \]
8. Step-by-step derivation

   1. lower-*.f32 N/A

      \[\leadsto \frac{\frac{1}{8} \cdot \frac{1}{r \cdot \left(\frac{1}{3} \cdot \frac{r \cdot \mathsf{PI}\left(\right)}{s} + \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}}{s} \]

   2. lower-fma.f32 N/A

      \[\leadsto \frac{\frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{fma}\left(\frac{1}{3}, \frac{r \cdot \mathsf{PI}\left(\right)}{s}, \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}}{s} \]

   3. lower-/.f32 N/A

      \[\leadsto \frac{\frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{fma}\left(\frac{1}{3}, \frac{r \cdot \mathsf{PI}\left(\right)}{s}, \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}}{s} \]

   4. lower-*.f32 N/A

      \[\leadsto \frac{\frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{fma}\left(\frac{1}{3}, \frac{r \cdot \mathsf{PI}\left(\right)}{s}, \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}}{s} \]

   5. lift-PI.f32 N/A

      \[\leadsto \frac{\frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{fma}\left(\frac{1}{3}, \frac{r \cdot \pi}{s}, \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}}{s} \]

   6. lower-*.f32 N/A

      \[\leadsto \frac{\frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{fma}\left(\frac{1}{3}, \frac{r \cdot \pi}{s}, \frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}}{s} \]

   7. lift-PI.f32 19.0

      \[\leadsto \frac{0.125 \cdot \frac{1}{r \cdot \mathsf{fma}\left(0.3333333333333333, \frac{r \cdot \pi}{s}, 0.5 \cdot \pi\right)}}{s} \]

9. Applied rewrites 19.0%

   \[\leadsto \frac{0.125 \cdot \frac{1}{r \cdot \mathsf{fma}\left(0.3333333333333333, \frac{r \cdot \pi}{s}, 0.5 \cdot \pi\right)}}{s} \]
10. Add Preprocessing

Alternative 12: 9.5% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in s around -inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f32 N/A

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

   3. lower-/.f32 N/A

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

4. Applied rewrites 10.4%

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

5. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

   2. lift-PI.f32 9.5

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

7. Applied rewrites 9.5%

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

Alternative 13: 9.5% accurate, 3.3× 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 99.6%

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

2. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

   2. lower--.f32 N/A

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

   3. mult-flip-rev N/A

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

   4. lower-/.f32 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. lift-PI.f32 N/A

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

   8. mult-flip-rev N/A

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

   9. lower-/.f32 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f32 N/A

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

   12. lift-PI.f32 9.5

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

4. Applied rewrites 9.5%

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

Alternative 14: 9.3% accurate, 5.7× 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) / s)
end

MATLAB

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

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in s around 0

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

   1. lower-/.f32 N/A

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

4. Applied rewrites 99.5%

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

   1. lift-+.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-exp.f32 N/A

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

   4. lift-/.f32 N/A

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

   5. lift-neg.f32 N/A

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

   6. lift-PI.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. associate-/r* N/A

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

   9. distribute-frac-neg N/A

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

   10. mul-1-neg N/A

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

   11. lift-/.f32 N/A

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

   12. lift-exp.f32 N/A

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

   13. lift-*.f32 N/A

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

   14. lift-/.f32 N/A

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

   15. lift-PI.f32 N/A

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

   16. lift-*.f32 N/A

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

6. Applied rewrites 99.5%

   \[\leadsto \frac{0.125 \cdot \frac{1}{\frac{r}{\frac{e^{\frac{-r}{s}} + e^{\frac{r}{s} \cdot -0.3333333333333333}}{\pi}}}}{s} \]

7. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lift-PI.f32 9.3

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

9. Applied rewrites 9.3%

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

    1. lift-/.f32 N/A

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

    2. lift-PI.f32 N/A

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

    3. lift-*.f32 N/A

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

    4. *-commutative N/A

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

    5. associate-/l/ N/A

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

    6. lift-/.f32 N/A

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

    7. lift-PI.f32 N/A

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

    8. lift-/.f32 9.3

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

11. Applied rewrites 9.3%

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

Alternative 15: 9.3% accurate, 6.0× speedup

Math

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

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(0.25) / Float32(Float32(pi) * r)) / s)
end

MATLAB

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

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in s around 0

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

   1. lower-/.f32 N/A

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

4. Applied rewrites 99.5%

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

5. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. lift-PI.f32 9.3

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

7. Applied rewrites 9.3%

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

Alternative 16: 9.3% accurate, 6.4× 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 99.6%

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

2. Taylor expanded in s around inf

   \[\leadsto \color{blue}{\frac{\frac{1}{4}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}} \]
3. 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 9.3

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

4. Applied rewrites 9.3%

   \[\leadsto \color{blue}{\frac{0.25}{\left(\pi \cdot s\right) \cdot r}} \]
5. 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

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. lower-*.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

   10. lift-PI.f32 9.3

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

6. Applied rewrites 9.3%

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

Reproduce

herbie shell --seed 2025127 
(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))))