Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.5% → 99.5%

Time: 5.9s

Alternatives: 15

Speedup: N/A×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = (single(0.125) / ((s * single(pi)) * (exp((r / s)) * r))) + ((exp((single(-0.3333333333333333) * (r / s))) * single(0.75)) / (((single(6.0) * single(pi)) * s) * r));
end

Derivation

1. Initial program 99.5%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. frac-times N/A

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

   5. lift-*.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. associate-*l* N/A

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

   8. associate-/r* N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. associate-/r* N/A

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

   12. associate-*l* N/A

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

   13. lift-*.f32 N/A

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

   14. lift-*.f32 N/A

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

   15. associate-*r/ N/A

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

   16. lower-/.f32 N/A

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

3. Applied rewrites 99.5%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. associate-/r* N/A

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

   6. lift-neg.f32 N/A

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

   7. metadata-eval N/A

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

   8. frac-2neg N/A

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

   9. mult-flip N/A

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

   10. metadata-eval N/A

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

   11. associate-*l/ N/A

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

   12. lift-/.f32 N/A

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

   13. *-commutative N/A

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

   14. lift-*.f32 N/A

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

   15. lower-*.f32 99.5

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

5. Applied rewrites 99.5%

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

   1. lift-/.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-/l/ N/A

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

   4. lower-/.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lift-*.f32 N/A

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

   9. associate-*l* N/A

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

   10. lower-*.f32 N/A

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

   11. lower-*.f32 99.5

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

7. Applied rewrites 99.5%

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

Alternative 2: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. frac-times N/A

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

   5. lift-*.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. associate-*l* N/A

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

   8. associate-/r* N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. associate-/r* N/A

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

   12. associate-*l* N/A

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

   13. lift-*.f32 N/A

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

   14. lift-*.f32 N/A

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

   15. associate-*r/ N/A

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

   16. lower-/.f32 N/A

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

3. Applied rewrites 99.5%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. associate-/r* N/A

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

   6. lift-neg.f32 N/A

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

   7. metadata-eval N/A

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

   8. frac-2neg N/A

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

   9. mult-flip N/A

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

   10. metadata-eval N/A

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

   11. associate-*l/ N/A

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

   12. lift-/.f32 N/A

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

   13. *-commutative N/A

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

   14. lift-*.f32 N/A

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

   15. lower-*.f32 99.5

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

5. Applied rewrites 99.5%

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

   1. lift-/.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-/l/ N/A

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

   4. lower-/.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lift-*.f32 N/A

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

   9. associate-*l* N/A

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

   10. lower-*.f32 N/A

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

   11. lower-*.f32 99.5

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

7. Applied rewrites 99.5%

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

9. Applied rewrites 99.5%

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

Alternative 3: 99.5% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.5%

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

3. Applied rewrites 99.5%

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

Alternative 4: 99.5% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Applied rewrites 99.5%

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

5. Applied rewrites 99.4%

   \[\leadsto \color{blue}{\frac{1}{\frac{s}{\frac{e^{\frac{-r}{s}} + e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r \cdot \pi} \cdot 0.125}}} \]
6. Add Preprocessing

Alternative 5: 99.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Applied rewrites 99.5%

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

5. Applied rewrites 99.5%

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

Alternative 6: 99.4% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{e^{\frac{-r}{s}}}{\pi}, 0.125, \frac{e^{\frac{r}{-3 \cdot s}}}{\pi} \cdot 0.125\right)}{s \cdot r}} \]
4. Step-by-step derivation

   1. lift-fma.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. distribute-rgt-out N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

5. Applied rewrites 99.5%

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

Alternative 7: 46.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 25:\\ \;\;\;\;\mathsf{fma}\left(0.125, \frac{e^{\frac{r}{-3 \cdot s}}}{\pi \cdot \left(s \cdot r\right)}, \frac{\frac{0.125}{\mathsf{fma}\left(r, \pi, s \cdot \pi\right)}}{r}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{\log \left({\left(e^{r \cdot \pi}\right)}^{s}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (if (<= r 25.0)
   (fma
    0.125
    (/ (exp (/ r (* -3.0 s))) (* PI (* s r)))
    (/ (/ 0.125 (fma r PI (* s PI))) r))
   (/ 0.25 (log (pow (exp (* r PI)) s)))))

C

float code(float s, float r) {
	float tmp;
	if (r <= 25.0f) {
		tmp = fmaf(0.125f, (expf((r / (-3.0f * s))) / (((float) M_PI) * (s * r))), ((0.125f / fmaf(r, ((float) M_PI), (s * ((float) M_PI)))) / r));
	} else {
		tmp = 0.25f / logf(powf(expf((r * ((float) M_PI))), s));
	}
	return tmp;
}

Julia

function code(s, r)
	tmp = Float32(0.0)
	if (r <= Float32(25.0))
		tmp = fma(Float32(0.125), Float32(exp(Float32(r / Float32(Float32(-3.0) * s))) / Float32(Float32(pi) * Float32(s * r))), Float32(Float32(Float32(0.125) / fma(r, Float32(pi), Float32(s * Float32(pi)))) / r));
	else
		tmp = Float32(Float32(0.25) / log((exp(Float32(r * Float32(pi))) ^ s)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if r < 25

   1. Initial program 99.5%

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

   3. Applied rewrites 99.5%

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

   4. Taylor expanded in r around 0

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

      1. lower-fma.f32 N/A

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

      2. lower-PI.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-PI.f32 12.1

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

   6. Applied rewrites 12.1%

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

   if 25 < r

   1. Initial program 99.5%

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

   2. 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. lower-*.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-PI.f32 8.8

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

   4. Applied rewrites 8.8%

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

      1. lift-*.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lift-PI.f32 N/A

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

      7. add-log-exp N/A

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

      8. log-pow-rev N/A

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

      9. log-pow-rev N/A

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

      10. lower-log.f32 N/A

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

      11. lower-pow.f32 N/A

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

      12. lift-PI.f32 N/A

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

      13. pow-exp N/A

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

      14. *-commutative N/A

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

      15. lift-*.f32 N/A

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

      16. lower-exp.f32 42.2

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

   6. Applied rewrites 42.2%

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

Alternative 8: 43.4% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

2. 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. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 8.8

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

4. Applied rewrites 8.8%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lift-*.f32 N/A

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

   6. lower-*.f32 8.8

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 8.8

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

6. Applied rewrites 8.8%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f32 N/A

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

   6. lower-*.f32 8.8

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

8. Applied rewrites 8.8%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-PI.f32 N/A

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

   4. add-log-exp N/A

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

   5. log-pow-rev N/A

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

   6. lower-log.f32 N/A

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

   7. lower-pow.f32 N/A

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

   8. lift-PI.f32 N/A

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

   9. lower-exp.f32 43.4

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

10. Applied rewrites 43.4%

    \[\leadsto \frac{0.25}{\log \left({\left(e^{\pi}\right)}^{r}\right) \cdot s} \]
11. Add Preprocessing

Alternative 9: 42.2% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

2. 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. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 8.8

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

4. Applied rewrites 8.8%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. lift-PI.f32 N/A

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

   7. add-log-exp N/A

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

   8. log-pow-rev N/A

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

   9. log-pow-rev N/A

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

   10. lower-log.f32 N/A

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

   11. lower-pow.f32 N/A

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

   12. lift-PI.f32 N/A

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

   13. pow-exp N/A

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

   14. *-commutative N/A

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

   15. lift-*.f32 N/A

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

   16. lower-exp.f32 42.2

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

6. Applied rewrites 42.2%

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

Alternative 10: 9.8% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

2. 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. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 8.8

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

4. Applied rewrites 8.8%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lift-*.f32 N/A

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

   6. lift-PI.f32 N/A

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

   7. add-log-exp N/A

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

   8. log-pow-rev N/A

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

   9. lower-log.f32 N/A

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

   10. lift-PI.f32 N/A

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

   11. pow-exp N/A

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

   12. *-commutative N/A

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

   13. lift-*.f32 N/A

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

   14. *-commutative N/A

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

   15. associate-*r* N/A

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

   16. lift-*.f32 N/A

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

   17. lift-*.f32 N/A

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

   18. lower-exp.f32 9.8

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

6. Applied rewrites 9.8%

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

Alternative 11: 8.8% accurate, 6.0× speedup

Math

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

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

2. 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. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 8.8

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

4. Applied rewrites 8.8%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lift-*.f32 N/A

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

   6. lower-*.f32 8.8

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 8.8

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

6. Applied rewrites 8.8%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f32 N/A

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

   6. lower-*.f32 8.8

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

8. Applied rewrites 8.8%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lift-*.f32 N/A

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

   5. *-commutative N/A

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

   6. lift-*.f32 N/A

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

   7. associate-/r* N/A

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

   8. lower-/.f32 N/A

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

   9. lower-/.f32 8.8

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

10. Applied rewrites 8.8%

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

Alternative 12: 8.8% accurate, 6.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

2. 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. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 8.8

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

4. Applied rewrites 8.8%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. lift-*.f32 N/A

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

   7. associate-/r* N/A

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

   8. lower-/.f32 N/A

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

   9. lower-/.f32 8.8

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

   10. lift-*.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 8.8

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

6. Applied rewrites 8.8%

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

Alternative 13: 8.8% accurate, 6.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

2. 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. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 8.8

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

4. Applied rewrites 8.8%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. lift-*.f32 N/A

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

   6. lower-*.f32 8.8

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

6. Applied rewrites 8.8%

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

Alternative 14: 8.8% accurate, 6.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

2. 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. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 8.8

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

4. Applied rewrites 8.8%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lift-*.f32 N/A

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

   6. lower-*.f32 8.8

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 8.8

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

6. Applied rewrites 8.8%

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

Alternative 15: 8.8% accurate, 6.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

2. 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. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 8.8

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

4. Applied rewrites 8.8%

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

Reproduce

herbie shell --seed 2025155 
(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))))