Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.6%

Time: 6.0s

Alternatives: 25

Speedup: 0.7×

Specification

\[\left(0 \leq s \land s \leq 256\right) \land \left(\frac{1}{1000000} < r \land r < 1000000\right)\]

Math

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

FPCore

(FPCore (s r)
  :precision binary32
  (+
 (/ (* 1/4 (exp (/ (- r) s))) (* (* (* 2 PI) s) r))
 (/ (* 3/4 (exp (/ (- r) (* 3 s)))) (* (* (* 6 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

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

FPCore

(FPCore (s r)
  :precision binary32
  (+
 (/ (* 1/4 (exp (/ (- r) s))) (* (* (* 2 PI) s) r))
 (/ (* 3/4 (exp (/ (- r) (* 3 s)))) (* (* (* 6 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, 0.7× speedup

Math

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

FPCore

(FPCore (s r)
  :precision binary32
  (+
 (* 1/8 (/ (exp (/ (- r) s)) (* PI (* s r))))
 (/
  3/4
  (*
   (- (cosh (* (/ 1/3 s) r)) (sinh (/ r (* -3 s))))
   (* (* (* 6 PI) s) r)))))

C

float code(float s, float r) {
	return (0.125f * (expf((-r / s)) / (((float) M_PI) * (s * r)))) + (0.75f / ((coshf(((0.3333333333333333f / s) * r)) - sinhf((r / (-3.0f * s)))) * (((6.0f * ((float) M_PI)) * s) * r)));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(0.125) * Float32(exp(Float32(Float32(-r) / s)) / Float32(Float32(pi) * Float32(s * r)))) + Float32(Float32(0.75) / Float32(Float32(cosh(Float32(Float32(Float32(0.3333333333333333) / s) * r)) - sinh(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(0.125) * (exp((-r / s)) / (single(pi) * (s * r)))) + (single(0.75) / ((cosh(((single(0.3333333333333333) / s) * r)) - sinh((r / (single(-3.0) * s)))) * (((single(6.0) * single(pi)) * s) * r)));
end

Derivation

1. Initial program 99.6%

   \[\frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\frac{3}{4} \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
2. 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. lift-*.f32 N/A

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

   5. associate-*l* N/A

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

   6. lift-*.f32 N/A

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

   7. associate-*l* N/A

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

   8. times-frac N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. lower-*.f32 N/A

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

   12. metadata-eval N/A

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

   13. lower-/.f32 N/A

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

   14. lower-*.f32 N/A

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

   15. lower-*.f32 99.6%

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

3. Applied rewrites 99.6%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

5. Applied rewrites 99.5%

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

   1. lift-exp.f32 N/A

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

   2. sinh-+-cosh-rev N/A

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

   3. add-flip N/A

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

   4. lift-*.f32 N/A

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

   5. *-commutative N/A

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

   6. lift-/.f32 N/A

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

   7. metadata-eval N/A

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

   8. associate-/r* N/A

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

   9. mult-flip N/A

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

   10. cosh-neg N/A

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

   11. distribute-frac-neg N/A

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

   12. lift-*.f32 N/A

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

   13. *-commutative N/A

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

   14. lift-/.f32 N/A

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

   15. metadata-eval N/A

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

   16. associate-/r* N/A

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

   17. mult-flip N/A

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

   18. sinh-neg N/A

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

   19. distribute-frac-neg N/A

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

7. Applied rewrites 99.5%

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

Alternative 2: 99.5% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (s r)
  :precision binary32
  (+
 (* 1/8 (/ (exp (/ (- r) s)) (* PI (* s r))))
 (/ (* 3/4 (exp (/ (- r) (* 3 s)))) (* (* (* 6 PI) s) r))))

C

float code(float s, float r) {
	return (0.125f * (expf((-r / s)) / (((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(0.125) * Float32(exp(Float32(Float32(-r) / s)) / Float32(Float32(pi) * Float32(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(0.125) * (exp((-r / s)) / (single(pi) * (s * r)))) + ((single(0.75) * exp((-r / (single(3.0) * s)))) / (((single(6.0) * single(pi)) * s) * r));
end

Derivation

1. Initial program 99.6%

   \[\frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\frac{3}{4} \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
2. 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. lift-*.f32 N/A

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

   5. associate-*l* N/A

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

   6. lift-*.f32 N/A

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

   7. associate-*l* N/A

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

   8. times-frac N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. lower-*.f32 N/A

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

   12. metadata-eval N/A

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

   13. lower-/.f32 N/A

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

   14. lower-*.f32 N/A

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

   15. lower-*.f32 99.6%

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

3. Applied rewrites 99.6%

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

Alternative 3: 99.5% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (s r)
  :precision binary32
  (/
 (+
  (/ 1/8 (* (* PI s) (exp (/ r s))))
  (* 3/4 (/ (exp (* (/ -1/3 s) r)) (* (* 6 PI) s))))
 r))

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Applied rewrites 99.5%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-*l/ N/A

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

   4. associate-/l* N/A

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

   5. metadata-eval N/A

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

   6. lift-/.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. associate-/r* N/A

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

   9. frac-2neg N/A

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

   10. lift-neg.f32 N/A

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

   11. metadata-eval N/A

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

   12. associate-/r* N/A

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

   13. lift-*.f32 N/A

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

   14. lift-/.f32 N/A

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

   15. times-frac N/A

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

   16. lift-*.f32 N/A

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

   17. associate-*l* N/A

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

   18. lift-*.f32 N/A

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

5. Applied rewrites 99.5%

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

Alternative 4: 99.5% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (s r)
  :precision binary32
  (/
 (+
  (* (/ (exp (/ (- r) s)) PI) 1/8)
  (* (/ (exp (* -1/3 (/ r s))) PI) 1/8))
 (* s r)))

C

float code(float s, float r) {
	return (((expf((-r / s)) / ((float) M_PI)) * 0.125f) + ((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)) / Float32(pi)) * Float32(0.125)) + Float32(Float32(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)) / single(pi)) * single(0.125)) + ((exp((single(-0.3333333333333333) * (r / s))) / single(pi)) * single(0.125))) / (s * r);
end

Derivation

1. Initial program 99.6%

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

3. Applied rewrites 99.6%

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

4. Taylor expanded in s around 0

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

   1. lower-/.f32 N/A

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

   2. lower-exp.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-/.f32 N/A

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

   5. lower-PI.f32 99.5%

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

6. Applied rewrites 99.5%

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

Alternative 5: 99.5% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (s r)
  :precision binary32
  (/
 1
 (/
  (* r s)
  (* (/ (+ (exp (/ (- r) s)) (exp (* (/ -1/3 s) r))) PI) 1/8))))

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Applied rewrites 99.6%

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

   1. lift-/.f32 N/A

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

   2. div-flip N/A

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

   3. lower-unsound-/.f32 N/A

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

   4. lower-unsound-/.f32 99.5%

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

   5. lift-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 99.5%

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

   8. lift-+.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. distribute-rgt-out N/A

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

   12. *-commutative N/A

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

5. Applied rewrites 99.5%

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

Alternative 6: 99.5% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (s r)
  :precision binary32
  (*
 (/ 1/8 s)
 (/ (/ (+ (exp (/ (- r) s)) (exp (* (/ -1/3 s) r))) PI) r)))

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Applied rewrites 99.6%

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

   1. lift-/.f32 N/A

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

   2. lift-+.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. distribute-rgt-out N/A

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

   6. lift-*.f32 N/A

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

   7. times-frac N/A

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

   8. lower-*.f32 N/A

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

   9. lower-/.f32 N/A

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

   10. lower-/.f32 N/A

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

5. Applied rewrites 99.4%

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

Alternative 7: 99.4% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (s r)
  :precision binary32
  (/
 (* (/ (+ (exp (/ (- r) s)) (exp (* (/ -1/3 s) r))) PI) 1/8)
 (* s r)))

C

float code(float s, float r) {
	return (((expf((-r / s)) + expf(((-0.3333333333333333f / s) * r))) / ((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(Float32(-0.3333333333333333) / s) * r))) / Float32(pi)) * Float32(0.125)) / Float32(s * r))
end

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Applied rewrites 99.6%

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

   1. lift-+.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

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

   3. lift-*.f32 N/A

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

   4. distribute-rgt-out N/A

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

   5. *-commutative N/A

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

   6. 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%

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

Alternative 8: 49.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \mathbf{if}\;r \leq 26:\\ \;\;\;\;\frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\frac{3}{4}}{r \cdot \left(6 \cdot \left(s \cdot \pi\right) + r \cdot \left(\frac{1}{3} \cdot \frac{r \cdot \pi}{s} + 2 \cdot \pi\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{4}}{\log \left(e^{\pi \cdot r}\right) \cdot s}\\ \end{array} \]

FPCore

(FPCore (s r)
  :precision binary32
  (if (<= r 26)
  (+
   (/ (* 1/4 (exp (/ (- r) s))) (* (* (* 2 PI) s) r))
   (/
    3/4
    (*
     r
     (+ (* 6 (* s PI)) (* r (+ (* 1/3 (/ (* r PI) s)) (* 2 PI)))))))
  (/ 1/4 (* (log (exp (* PI r))) s))))

C

float code(float s, float r) {
	float tmp;
	if (r <= 26.0f) {
		tmp = ((0.25f * expf((-r / s))) / (((2.0f * ((float) M_PI)) * s) * r)) + (0.75f / (r * ((6.0f * (s * ((float) M_PI))) + (r * ((0.3333333333333333f * ((r * ((float) M_PI)) / s)) + (2.0f * ((float) M_PI)))))));
	} else {
		tmp = 0.25f / (logf(expf((((float) M_PI) * r))) * s);
	}
	return tmp;
}

Julia

function code(s, r)
	tmp = Float32(0.0)
	if (r <= Float32(26.0))
		tmp = Float32(Float32(Float32(Float32(0.25) * exp(Float32(Float32(-r) / s))) / Float32(Float32(Float32(Float32(2.0) * Float32(pi)) * s) * r)) + Float32(Float32(0.75) / Float32(r * Float32(Float32(Float32(6.0) * Float32(s * Float32(pi))) + Float32(r * Float32(Float32(Float32(0.3333333333333333) * Float32(Float32(r * Float32(pi)) / s)) + Float32(Float32(2.0) * Float32(pi))))))));
	else
		tmp = Float32(Float32(0.25) / Float32(log(exp(Float32(Float32(pi) * r))) * s));
	end
	return tmp
end

MATLAB

function tmp_2 = code(s, r)
	tmp = single(0.0);
	if (r <= single(26.0))
		tmp = ((single(0.25) * exp((-r / s))) / (((single(2.0) * single(pi)) * s) * r)) + (single(0.75) / (r * ((single(6.0) * (s * single(pi))) + (r * ((single(0.3333333333333333) * ((r * single(pi)) / s)) + (single(2.0) * single(pi)))))));
	else
		tmp = single(0.25) / (log(exp((single(pi) * r))) * s);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if r < 26

   1. Initial program 99.6%

      \[\frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\frac{3}{4} \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
   2. 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. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lift-exp.f32 N/A

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

      6. lift-/.f32 N/A

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

      7. lift-neg.f32 N/A

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

      8. distribute-frac-neg N/A

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

      9. exp-neg N/A

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

      10. frac-times N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f32 N/A

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

      13. lower-*.f32 N/A

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

   3. Applied rewrites 99.5%

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

   4. Taylor expanded in r around 0

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

      1. lower-*.f32 N/A

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

      2. lower-+.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lower-PI.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. lower-*.f32 N/A

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

      8. lower-PI.f32 13.0%

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

   6. Applied rewrites 13.0%

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

   7. Taylor expanded in r around 0

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

      1. lower-*.f32 N/A

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

      2. lower-+.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lower-PI.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. lower-+.f32 N/A

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

      8. lower-*.f32 N/A

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

      9. lower-/.f32 N/A

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

      10. lower-*.f32 N/A

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

      11. lower-PI.f32 N/A

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

      12. lower-*.f32 N/A

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

      13. lower-PI.f32 26.5%

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

   9. Applied rewrites 26.5%

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

   if 26 < r

   1. Initial program 99.6%

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

   2. Taylor expanded in s around inf

      \[\leadsto \color{blue}{\frac{\frac{1}{4}}{r \cdot \left(s \cdot \pi\right)}} \]
   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 9.2%

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

   4. Applied rewrites 9.2%

      \[\leadsto \color{blue}{\frac{\frac{1}{4}}{r \cdot \left(s \cdot \pi\right)}} \]
   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 9.2%

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

   6. Applied rewrites 9.2%

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

      1. lift-*.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. associate-*l* N/A

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

      4. lift-*.f32 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f32 9.2%

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

   8. Applied rewrites 9.2%

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

      1. lift-*.f32 N/A

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

      2. lift-PI.f32 N/A

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

      3. add-log-exp N/A

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

      4. log-pow-rev N/A

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

      5. lower-log.f32 N/A

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

      6. lift-PI.f32 N/A

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

      7. pow-exp N/A

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

      8. *-commutative N/A

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

      9. lift-*.f32 N/A

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

      10. lower-exp.f32 43.1%

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

      11. lift-*.f32 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f32 43.1%

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

   10. Applied rewrites 43.1%

       \[\leadsto \frac{\frac{1}{4}}{\log \left(e^{\pi \cdot r}\right) \cdot s} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 26.5% accurate, 1.4× speedup

Math

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

FPCore

(FPCore (s r)
  :precision binary32
  (+
 (/ (* 1/4 (exp (/ (- r) s))) (* (* (* 2 PI) s) r))
 (/
  3/4
  (* r (+ (* 6 (* s PI)) (* r (+ (* 1/3 (/ (* r PI) s)) (* 2 PI))))))))

C

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

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(0.75) / Float32(r * Float32(Float32(Float32(6.0) * Float32(s * Float32(pi))) + Float32(r * Float32(Float32(Float32(0.3333333333333333) * Float32(Float32(r * Float32(pi)) / s)) + Float32(Float32(2.0) * Float32(pi))))))))
end

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.25) * exp((-r / s))) / (((single(2.0) * single(pi)) * s) * r)) + (single(0.75) / (r * ((single(6.0) * (s * single(pi))) + (r * ((single(0.3333333333333333) * ((r * single(pi)) / s)) + (single(2.0) * single(pi)))))));
end

Derivation

1. Initial program 99.6%

   \[\frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\frac{3}{4} \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
2. 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. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. lift-exp.f32 N/A

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

   6. lift-/.f32 N/A

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

   7. lift-neg.f32 N/A

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

   8. distribute-frac-neg N/A

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

   9. exp-neg N/A

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

   10. frac-times N/A

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

   11. metadata-eval N/A

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

   12. lower-/.f32 N/A

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

   13. lower-*.f32 N/A

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

3. Applied rewrites 99.5%

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

4. Taylor expanded in r around 0

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

   1. lower-*.f32 N/A

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

   2. lower-+.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-PI.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. lower-PI.f32 13.0%

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

6. Applied rewrites 13.0%

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

7. Taylor expanded in r around 0

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

   1. lower-*.f32 N/A

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

   2. lower-+.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-PI.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower-+.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. lower-/.f32 N/A

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

   10. lower-*.f32 N/A

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

   11. lower-PI.f32 N/A

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

   12. lower-*.f32 N/A

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

   13. lower-PI.f32 26.5%

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

9. Applied rewrites 26.5%

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

Alternative 10: 26.5% accurate, 1.4× speedup

Math

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

FPCore

(FPCore (s r)
  :precision binary32
  (+
 (* 1/8 (/ (exp (/ (- r) s)) (* PI (* s r))))
 (/
  3/4
  (* r (+ (* 6 (* s PI)) (* r (+ (* 1/3 (/ (* r PI) s)) (* 2 PI))))))))

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = (single(0.125) * (exp((-r / s)) / (single(pi) * (s * r)))) + (single(0.75) / (r * ((single(6.0) * (s * single(pi))) + (r * ((single(0.3333333333333333) * ((r * single(pi)) / s)) + (single(2.0) * single(pi)))))));
end

Derivation

1. Initial program 99.6%

   \[\frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\frac{3}{4} \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
2. 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. lift-*.f32 N/A

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

   5. associate-*l* N/A

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

   6. lift-*.f32 N/A

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

   7. associate-*l* N/A

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

   8. times-frac N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. lower-*.f32 N/A

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

   12. metadata-eval N/A

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

   13. lower-/.f32 N/A

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

   14. lower-*.f32 N/A

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

   15. lower-*.f32 99.6%

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

3. Applied rewrites 99.6%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

5. Applied rewrites 99.5%

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

6. Taylor expanded in r around 0

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

   1. lower-*.f32 N/A

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

   2. lower-+.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-PI.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower-+.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. lower-/.f32 N/A

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

   10. lower-*.f32 N/A

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

   11. lower-PI.f32 N/A

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

   12. lower-*.f32 N/A

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

   13. lower-PI.f32 26.5%

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

8. Applied rewrites 26.5%

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

Alternative 11: 25.5% accurate, 1.4× speedup

Math

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

FPCore

(FPCore (s r)
  :precision binary32
  (/
 (+
  (/ 1/8 (+ (* r (+ PI (* 1/2 (/ (* r PI) s)))) (* s PI)))
  (* 3/4 (/ (exp (* (/ -1/3 s) r)) (* (* 6 PI) s))))
 r))

C

float code(float s, float r) {
	return ((0.125f / ((r * (((float) M_PI) + (0.5f * ((r * ((float) M_PI)) / s)))) + (s * ((float) M_PI)))) + (0.75f * (expf(((-0.3333333333333333f / s) * r)) / ((6.0f * ((float) M_PI)) * s)))) / r;
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.125) / Float32(Float32(r * Float32(Float32(pi) + Float32(Float32(0.5) * Float32(Float32(r * Float32(pi)) / s)))) + Float32(s * Float32(pi)))) + Float32(Float32(0.75) * Float32(exp(Float32(Float32(Float32(-0.3333333333333333) / s) * r)) / Float32(Float32(Float32(6.0) * Float32(pi)) * s)))) / r)
end

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.125) / ((r * (single(pi) + (single(0.5) * ((r * single(pi)) / s)))) + (s * single(pi)))) + (single(0.75) * (exp(((single(-0.3333333333333333) / s) * r)) / ((single(6.0) * single(pi)) * s)))) / r;
end

Derivation

1. Initial program 99.6%

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

3. Applied rewrites 99.5%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-*l/ N/A

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

   4. associate-/l* N/A

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

   5. metadata-eval N/A

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

   6. lift-/.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. associate-/r* N/A

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

   9. frac-2neg N/A

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

   10. lift-neg.f32 N/A

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

   11. metadata-eval N/A

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

   12. associate-/r* N/A

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

   13. lift-*.f32 N/A

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

   14. lift-/.f32 N/A

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

   15. times-frac N/A

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

   16. lift-*.f32 N/A

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

   17. associate-*l* N/A

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

   18. lift-*.f32 N/A

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

5. Applied rewrites 99.5%

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

6. Taylor expanded in r around 0

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

   1. lower-+.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-+.f32 N/A

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

   4. lower-PI.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-/.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. lower-PI.f32 N/A

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

   9. lower-*.f32 N/A

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

   10. lower-PI.f32 25.5%

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

8. Applied rewrites 25.5%

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

Alternative 12: 25.4% accurate, 1.5× speedup

Math

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

FPCore

(FPCore (s r)
  :precision binary32
  (/
 (+
  (/ 1/8 (+ (* r (+ PI (* 1/2 (/ (* r PI) s)))) (* s PI)))
  (* (/ 1/8 (* PI s)) (exp (/ r (* -3 s)))))
 r))

C

float code(float s, float r) {
	return ((0.125f / ((r * (((float) M_PI) + (0.5f * ((r * ((float) M_PI)) / s)))) + (s * ((float) M_PI)))) + ((0.125f / (((float) M_PI) * s)) * expf((r / (-3.0f * s))))) / r;
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.125) / Float32(Float32(r * Float32(Float32(pi) + Float32(Float32(0.5) * Float32(Float32(r * Float32(pi)) / s)))) + Float32(s * Float32(pi)))) + Float32(Float32(Float32(0.125) / Float32(Float32(pi) * s)) * exp(Float32(r / Float32(Float32(-3.0) * s))))) / r)
end

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.125) / ((r * (single(pi) + (single(0.5) * ((r * single(pi)) / s)))) + (s * single(pi)))) + ((single(0.125) / (single(pi) * s)) * exp((r / (single(-3.0) * s))))) / r;
end

Derivation

1. Initial program 99.6%

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

3. Applied rewrites 99.5%

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

4. Taylor expanded in r around 0

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

   1. lower-+.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-+.f32 N/A

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

   4. lower-PI.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-/.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. lower-PI.f32 N/A

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

   9. lower-*.f32 N/A

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

   10. lower-PI.f32 25.4%

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

6. Applied rewrites 25.4%

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

Alternative 13: 16.9% accurate, 1.5× speedup

Math

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

FPCore

(FPCore (s r)
  :precision binary32
  (+
 (/ (* 1/4 (exp (/ (- r) s))) (* (* (* 2 PI) s) r))
 (/ 3/4 (* r (* s (+ (* 2 (/ (* r PI) s)) (* 6 PI)))))))

C

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

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(0.75) / Float32(r * Float32(s * Float32(Float32(Float32(2.0) * Float32(Float32(r * Float32(pi)) / s)) + Float32(Float32(6.0) * Float32(pi)))))))
end

MATLAB

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

Derivation

1. Initial program 99.6%

   \[\frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\frac{3}{4} \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
2. 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. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. lift-exp.f32 N/A

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

   6. lift-/.f32 N/A

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

   7. lift-neg.f32 N/A

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

   8. distribute-frac-neg N/A

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

   9. exp-neg N/A

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

   10. frac-times N/A

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

   11. metadata-eval N/A

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

   12. lower-/.f32 N/A

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

   13. lower-*.f32 N/A

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

3. Applied rewrites 99.5%

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

4. Taylor expanded in r around 0

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

   1. lower-*.f32 N/A

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

   2. lower-+.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-PI.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. lower-PI.f32 13.0%

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

6. Applied rewrites 13.0%

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

7. Taylor expanded in s around inf

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

   1. lower-*.f32 N/A

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

   2. lower-+.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-/.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-PI.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. lower-PI.f32 16.9%

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

9. Applied rewrites 16.9%

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

Alternative 14: 16.1% accurate, 1.6× speedup

Math

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

FPCore

(FPCore (s r)
  :precision binary32
  (/
 (+
  (/ 1/8 (* (* PI s) (+ 1 (/ r s))))
  (* (/ 1/8 (* PI s)) (exp (/ r (* -3 s)))))
 r))

C

float code(float s, float r) {
	return ((0.125f / ((((float) M_PI) * s) * (1.0f + (r / s)))) + ((0.125f / (((float) M_PI) * s)) * expf((r / (-3.0f * s))))) / r;
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.125) / Float32(Float32(Float32(pi) * s) * Float32(Float32(1.0) + Float32(r / s)))) + Float32(Float32(Float32(0.125) / Float32(Float32(pi) * s)) * exp(Float32(r / Float32(Float32(-3.0) * s))))) / r)
end

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.125) / ((single(pi) * s) * (single(1.0) + (r / s)))) + ((single(0.125) / (single(pi) * s)) * exp((r / (single(-3.0) * s))))) / r;
end

Derivation

1. Initial program 99.6%

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

3. Applied rewrites 99.5%

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

4. Taylor expanded in s around inf

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

   1. lower-+.f32 N/A

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

   2. lower-/.f32 15.4%

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

6. Applied rewrites 15.4%

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

Alternative 15: 15.4% accurate, 1.6× speedup

Math

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

FPCore

(FPCore (s r)
  :precision binary32
  (/
 (+
  (/ 1/8 (* s (+ PI (/ (* r PI) s))))
  (* (/ 1/8 (* PI s)) (exp (/ r (* -3 s)))))
 r))

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Applied rewrites 99.5%

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

4. Taylor expanded in s around inf

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

   1. lower-*.f32 N/A

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

   2. lower-+.f32 N/A

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

   3. lower-PI.f32 N/A

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

   4. lower-/.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-PI.f32 16.1%

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

6. Applied rewrites 16.1%

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

Alternative 16: 13.0% accurate, 1.6× speedup

Math

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

FPCore

(FPCore (s r)
  :precision binary32
  (+
 (/ (* 1/4 (exp (/ (- r) s))) (* (* (* 2 PI) s) r))
 (/ 3/4 (* r (+ (* 2 (* r PI)) (* 6 (* s PI)))))))

C

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

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(0.75) / Float32(r * Float32(Float32(Float32(2.0) * Float32(r * Float32(pi))) + Float32(Float32(6.0) * Float32(s * Float32(pi)))))))
end

MATLAB

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

Derivation

1. Initial program 99.6%

   \[\frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\frac{3}{4} \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
2. 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. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. lift-exp.f32 N/A

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

   6. lift-/.f32 N/A

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

   7. lift-neg.f32 N/A

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

   8. distribute-frac-neg N/A

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

   9. exp-neg N/A

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

   10. frac-times N/A

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

   11. metadata-eval N/A

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

   12. lower-/.f32 N/A

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

   13. lower-*.f32 N/A

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

3. Applied rewrites 99.5%

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

4. Taylor expanded in r around 0

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

   1. lower-*.f32 N/A

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

   2. lower-+.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-PI.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. lower-PI.f32 13.0%

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

6. Applied rewrites 13.0%

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

Alternative 17: 13.0% accurate, 1.7× speedup

Math

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

FPCore

(FPCore (s r)
  :precision binary32
  (-
 (* (/ (exp (/ (- r) s)) (* (* s r) PI)) 1/8)
 (/ -3/4 (* (- (* r (+ PI PI)) (* (* PI s) -6)) r))))

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

   \[\frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\frac{3}{4} \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
2. 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. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. lift-exp.f32 N/A

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

   6. lift-/.f32 N/A

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

   7. lift-neg.f32 N/A

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

   8. distribute-frac-neg N/A

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

   9. exp-neg N/A

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

   10. frac-times N/A

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

   11. metadata-eval N/A

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

   12. lower-/.f32 N/A

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

   13. lower-*.f32 N/A

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

3. Applied rewrites 99.5%

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

4. Taylor expanded in r around 0

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

   1. lower-*.f32 N/A

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

   2. lower-+.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-PI.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. lower-PI.f32 13.0%

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

6. Applied rewrites 13.0%

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

   1. lift-+.f32 N/A

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

   2. add-flip N/A

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

   3. lift-*.f32 N/A

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

   4. count-2-rev N/A

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

   5. lift-*.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. *-commutative N/A

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

   8. associate-*l* N/A

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

   9. lift-*.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. associate--l+ N/A

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

   12. lower-+.f32 N/A

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

   13. lower--.f32 N/A

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

   14. lift-*.f32 N/A

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

   15. lift-*.f32 N/A

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

   16. associate-*l* N/A

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

   17. lift-*.f32 N/A

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

   18. distribute-lft-neg-out N/A

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

   19. lower-*.f32 N/A

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

8. Applied rewrites 13.0%

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

10. Applied rewrites 13.0%

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

Alternative 18: 13.0% accurate, 1.7× speedup

Math

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

FPCore

(FPCore (s r)
  :precision binary32
  (-
 (/ 1/8 (* (* (exp (/ r s)) (* s PI)) r))
 (/ -3/4 (* (* PI (+ (+ r r) (* 6 s))) r))))

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

   \[\frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\frac{3}{4} \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
2. 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. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. lift-exp.f32 N/A

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

   6. lift-/.f32 N/A

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

   7. lift-neg.f32 N/A

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

   8. distribute-frac-neg N/A

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

   9. exp-neg N/A

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

   10. frac-times N/A

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

   11. metadata-eval N/A

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

   12. lower-/.f32 N/A

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

   13. lower-*.f32 N/A

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

3. Applied rewrites 99.5%

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

4. Taylor expanded in r around 0

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

   1. lower-*.f32 N/A

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

   2. lower-+.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-PI.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. lower-PI.f32 13.0%

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

6. Applied rewrites 13.0%

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

8. Applied rewrites 13.0%

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

Alternative 19: 10.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} t_0 := \frac{r}{s \cdot \pi}\\ -1 \cdot \frac{-1 \cdot \frac{\left(\frac{1}{144} \cdot t\_0 + \frac{1}{16} \cdot t\_0\right) - \frac{1}{6} \cdot \frac{1}{\pi}}{s} - \frac{1}{4} \cdot \frac{1}{r \cdot \pi}}{s} \end{array} \]

FPCore

(FPCore (s r)
  :precision binary32
  (let* ((t_0 (/ r (* s PI))))
  (*
   -1
   (/
    (-
     (* -1 (/ (- (+ (* 1/144 t_0) (* 1/16 t_0)) (* 1/6 (/ 1 PI))) s))
     (* 1/4 (/ 1 (* r PI))))
    s))))

C

float code(float s, float r) {
	float t_0 = r / (s * ((float) M_PI));
	return -1.0f * (((-1.0f * ((((0.006944444444444444f * t_0) + (0.0625f * t_0)) - (0.16666666666666666f * (1.0f / ((float) M_PI)))) / s)) - (0.25f * (1.0f / (r * ((float) M_PI))))) / s);
}

Julia

function code(s, r)
	t_0 = Float32(r / Float32(s * Float32(pi)))
	return Float32(Float32(-1.0) * Float32(Float32(Float32(Float32(-1.0) * Float32(Float32(Float32(Float32(Float32(0.006944444444444444) * t_0) + Float32(Float32(0.0625) * t_0)) - Float32(Float32(0.16666666666666666) * Float32(Float32(1.0) / Float32(pi)))) / s)) - Float32(Float32(0.25) * Float32(Float32(1.0) / Float32(r * Float32(pi))))) / s))
end

MATLAB

function tmp = code(s, r)
	t_0 = r / (s * single(pi));
	tmp = single(-1.0) * (((single(-1.0) * ((((single(0.006944444444444444) * t_0) + (single(0.0625) * t_0)) - (single(0.16666666666666666) * (single(1.0) / single(pi)))) / s)) - (single(0.25) * (single(1.0) / (r * single(pi))))) / s);
end

Derivation

1. Initial program 99.6%

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

3. Applied rewrites 99.6%

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

4. Taylor expanded in s around -inf

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

   1. lower-*.f32 N/A

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

   2. lower-/.f32 N/A

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

6. Applied rewrites 10.3%

   \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{\left(\frac{1}{144} \cdot \frac{r}{s \cdot \pi} + \frac{1}{16} \cdot \frac{r}{s \cdot \pi}\right) - \frac{1}{6} \cdot \frac{1}{\pi}}{s} - \frac{1}{4} \cdot \frac{1}{r \cdot \pi}}{s}} \]
7. Add Preprocessing

Alternative 20: 9.3% accurate, 3.2× speedup

Math

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

FPCore

(FPCore (s r)
  :precision binary32
  (+
 (/ (* 1/4 (+ 1 (* -1 (/ r s)))) (* (* (* 2 PI) s) r))
 (/ 3/4 (* r (+ (* 2 (* r PI)) (* 6 (* s PI)))))))

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.25) * (single(1.0) + (single(-1.0) * (r / s)))) / (((single(2.0) * single(pi)) * s) * r)) + (single(0.75) / (r * ((single(2.0) * (r * single(pi))) + (single(6.0) * (s * single(pi))))));
end

Derivation

1. Initial program 99.6%

   \[\frac{\frac{1}{4} \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{\frac{3}{4} \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
2. 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. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. lift-exp.f32 N/A

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

   6. lift-/.f32 N/A

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

   7. lift-neg.f32 N/A

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

   8. distribute-frac-neg N/A

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

   9. exp-neg N/A

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

   10. frac-times N/A

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

   11. metadata-eval N/A

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

   12. lower-/.f32 N/A

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

   13. lower-*.f32 N/A

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

3. Applied rewrites 99.5%

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

4. Taylor expanded in r around 0

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

   1. lower-*.f32 N/A

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

   2. lower-+.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-PI.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. lower-PI.f32 13.0%

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

6. Applied rewrites 13.0%

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

7. Taylor expanded in s around inf

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

   1. lower-+.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-/.f32 9.3%

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

9. Applied rewrites 9.3%

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

Alternative 21: 9.3% accurate, 5.2× speedup

Math

\[\frac{\frac{1}{4} \cdot \frac{1}{r \cdot \pi} - \frac{1}{6} \cdot \frac{1}{s \cdot \pi}}{s} \]

FPCore

(FPCore (s r)
  :precision binary32
  (/ (- (* 1/4 (/ 1 (* r PI))) (* 1/6 (/ 1 (* s PI)))) s))

C

float code(float s, float r) {
	return ((0.25f * (1.0f / (r * ((float) M_PI)))) - (0.16666666666666666f * (1.0f / (s * ((float) M_PI))))) / s;
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.25) * Float32(Float32(1.0) / Float32(r * Float32(pi)))) - Float32(Float32(0.16666666666666666) * Float32(Float32(1.0) / Float32(s * Float32(pi))))) / s)
end

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.25) * (single(1.0) / (r * single(pi)))) - (single(0.16666666666666666) * (single(1.0) / (s * single(pi))))) / s;
end

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in s around inf

   \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot \frac{1}{r \cdot \pi} - \frac{1}{6} \cdot \frac{1}{s \cdot \pi}}{s}} \]
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. lower-*.f32 N/A

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

   4. lower-/.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-PI.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. lower-/.f32 N/A

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

   9. lower-*.f32 N/A

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

   10. lower-PI.f32 9.3%

       \[\leadsto \frac{\frac{1}{4} \cdot \frac{1}{r \cdot \pi} - \frac{1}{6} \cdot \frac{1}{s \cdot \pi}}{s} \]

4. Applied rewrites 9.3%

   \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot \frac{1}{r \cdot \pi} - \frac{1}{6} \cdot \frac{1}{s \cdot \pi}}{s}} \]
5. Add Preprocessing

Alternative 22: 9.2% accurate, 10.6× speedup

Math

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

FPCore

(FPCore (s r)
  :precision binary32
  (/ (/ 1/4 (* 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%

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

2. Taylor expanded in s around inf

   \[\leadsto \color{blue}{\frac{\frac{1}{4}}{r \cdot \left(s \cdot \pi\right)}} \]
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 9.2%

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

4. Applied rewrites 9.2%

   \[\leadsto \color{blue}{\frac{\frac{1}{4}}{r \cdot \left(s \cdot \pi\right)}} \]
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 9.2%

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

6. Applied rewrites 9.2%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*l* N/A

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

   4. lift-*.f32 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 9.2%

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

8. Applied rewrites 9.2%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f32 N/A

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

   5. lower-/.f32 9.2%

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

   6. lift-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 9.2%

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

10. Applied rewrites 9.2%

    \[\leadsto \frac{\frac{\frac{1}{4}}{\pi \cdot r}}{\color{blue}{s}} \]
11. Add Preprocessing

Alternative 23: 9.2% accurate, 10.6× speedup

Math

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

FPCore

(FPCore (s r)
  :precision binary32
  (/ (/ 1/4 r) (* PI s)))

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in s around inf

   \[\leadsto \color{blue}{\frac{\frac{1}{4}}{r \cdot \left(s \cdot \pi\right)}} \]
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 9.2%

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

4. Applied rewrites 9.2%

   \[\leadsto \color{blue}{\frac{\frac{1}{4}}{r \cdot \left(s \cdot \pi\right)}} \]
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. associate-/r* N/A

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

   4. lift-*.f32 N/A

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

   5. *-commutative N/A

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

   6. lift-*.f32 N/A

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

   7. lower-/.f32 N/A

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

   8. lower-/.f32 9.2%

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

6. Applied rewrites 9.2%

   \[\leadsto \frac{\frac{\frac{1}{4}}{r}}{\color{blue}{\pi \cdot s}} \]
7. Add Preprocessing

Alternative 24: 9.2% accurate, 13.5× speedup

Math

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

FPCore

(FPCore (s r)
  :precision binary32
  (/ 1/4 (* (* 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%

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

2. Taylor expanded in s around inf

   \[\leadsto \color{blue}{\frac{\frac{1}{4}}{r \cdot \left(s \cdot \pi\right)}} \]
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 9.2%

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

4. Applied rewrites 9.2%

   \[\leadsto \color{blue}{\frac{\frac{1}{4}}{r \cdot \left(s \cdot \pi\right)}} \]
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 9.2%

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

6. Applied rewrites 9.2%

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

Alternative 25: 9.2% accurate, 13.5× speedup

Math

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

FPCore

(FPCore (s r)
  :precision binary32
  (/ 1/4 (* r (* s PI))))

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in s around inf

   \[\leadsto \color{blue}{\frac{\frac{1}{4}}{r \cdot \left(s \cdot \pi\right)}} \]
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 9.2%

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

4. Applied rewrites 9.2%

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

Reproduce

herbie shell --seed 2025271 -o generate:evaluate
(FPCore (s r)
  :name "Disney BSSRDF, PDF of scattering profile"
  :precision binary32
  :pre (and (and (<= 0 s) (<= s 256)) (and (< 1/1000000 r) (< r 1000000)))
  (+ (/ (* 1/4 (exp (/ (- r) s))) (* (* (* 2 PI) s) r)) (/ (* 3/4 (exp (/ (- r) (* 3 s)))) (* (* (* 6 PI) s) r))))