Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.5% → 99.5%

Time: 4.8s

Alternatives: 12

Speedup: N/A×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.5% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/ (* 0.25 (exp (/ (- r) s))) (* (* (* 2.0 PI) s) r))
  (/
   0.75
   (*
    (+
     (sinh (* (/ 0.3333333333333333 s) r))
     (cosh (* (/ r s) 0.3333333333333333)))
    (* (* (* 6.0 PI) s) r)))))

C

float code(float s, float r) {
	return ((0.25f * expf((-r / s))) / (((2.0f * ((float) M_PI)) * s) * r)) + (0.75f / ((sinhf(((0.3333333333333333f / s) * r)) + coshf(((r / s) * 0.3333333333333333f))) * (((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(0.75) / Float32(Float32(sinh(Float32(Float32(Float32(0.3333333333333333) / s) * r)) + cosh(Float32(Float32(r / s) * Float32(0.3333333333333333)))) * 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) / ((sinh(((single(0.3333333333333333) / s) * r)) + cosh(((r / s) * single(0.3333333333333333)))) * (((single(6.0) * single(pi)) * s) * r)));
end

Derivation

1. Initial program 99.5%

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

   1. lift-/.f32 N/A

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

   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%

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

   1. lift-exp.f32 N/A

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

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

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

   3. +-commutative N/A

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

   4. cosh-neg-rev N/A

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

   5. lift-*.f32 N/A

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

   6. lift-/.f32 N/A

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

   7. metadata-eval N/A

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

   8. associate-/r* N/A

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

   9. lift-*.f32 N/A

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

   10. mult-flip N/A

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

   11. distribute-neg-frac2 N/A

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

   12. lift-*.f32 N/A

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

   13. *-commutative N/A

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

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

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

   15. metadata-eval N/A

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

   16. *-commutative N/A

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

   17. lift-*.f32 N/A

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

   18. lift-/.f32 N/A

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

   19. lower-+.f32 N/A

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

5. Applied rewrites 99.5%

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

Alternative 2: 99.5% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/ (* 0.25 (exp (/ (- r) s))) (* (* (* 2.0 PI) s) r))
  (/ (* 0.75 (exp (* -0.3333333333333333 (/ r s)))) (* (* (* 6.0 PI) s) r))))

C

float code(float s, float r) {
	return ((0.25f * expf((-r / s))) / (((2.0f * ((float) M_PI)) * s) * r)) + ((0.75f * expf((-0.3333333333333333f * (r / 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(-0.3333333333333333) * Float32(r / 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((single(-0.3333333333333333) * (r / s)))) / (((single(6.0) * single(pi)) * s) * r));
end

Derivation

1. Initial program 99.5%

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

2. Taylor expanded in s around 0

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\color{blue}{\frac{-1}{3} \cdot \frac{r}{s}}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
3. Step-by-step derivation

   1. lower-*.f32 N/A

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

   2. lower-/.f32 99.5%

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

4. Applied rewrites 99.5%

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

Alternative 3: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Applied rewrites 99.5%

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

   1. lift-/.f32 N/A

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

   2. div-flip N/A

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

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

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

   4. lower-unsound-/.f32 99.5%

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

   5. lift-fma.f32 N/A

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

   6. *-commutative N/A

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

   7. lift-/.f32 N/A

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

   8. mult-flip N/A

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

   9. distribute-lft-out N/A

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

   10. lower-*.f32 N/A

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

5. Applied rewrites 99.5%

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

Alternative 4: 99.5% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Applied rewrites 99.5%

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

   1. lift-fma.f32 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f32 N/A

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

   4. mult-flip N/A

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

   5. *-commutative N/A

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

   6. distribute-lft-out N/A

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

   7. lower-*.f32 N/A

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

   8. lower-+.f32 N/A

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

5. Applied rewrites 99.5%

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

Alternative 5: 99.5% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Derivation

1. Initial program 99.5%

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

3. Applied rewrites 99.5%

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

4. Taylor expanded in s around 0

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

   1. lower-/.f32 N/A

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

6. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.125, \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{\pi}, 0.125 \cdot \frac{1}{\pi \cdot e^{\frac{r}{s}}}\right)}{r \cdot s}} \]
7. Step-by-step derivation

8. Applied rewrites 99.5%

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

Alternative 6: 43.5% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

2. Taylor expanded in s around inf

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

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

4. Applied rewrites 9.2%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f32 N/A

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

   6. lower-*.f32 9.2%

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

6. Applied rewrites 9.2%

   \[\leadsto \color{blue}{\frac{0.25}{\left(r \cdot \pi\right) \cdot s}} \]
7. 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. lower-pow.f32 N/A

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

   7. lift-PI.f32 N/A

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

   8. lower-exp.f32 43.5%

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

8. Applied rewrites 43.5%

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

Alternative 7: 10.7% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

2. Taylor expanded in s around inf

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

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

4. Applied rewrites 9.2%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*r* N/A

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

   4. lift-PI.f32 N/A

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

   5. add-log-exp N/A

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

   6. log-pow-rev N/A

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

   7. lower-log.f32 N/A

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

   8. lift-PI.f32 N/A

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

   9. *-commutative N/A

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

   10. pow-exp N/A

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

   11. *-commutative N/A

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

   12. *-commutative N/A

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

   13. associate-*r* N/A

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

   14. lift-*.f32 N/A

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

   15. lift-*.f32 N/A

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

   16. lower-exp.f32 10.7%

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

   17. lift-*.f32 N/A

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

   18. lift-*.f32 N/A

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

   19. *-commutative N/A

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

   20. associate-*r* N/A

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

   21. lower-*.f32 N/A

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

   22. lower-*.f32 10.7%

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

6. Applied rewrites 10.7%

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

Alternative 8: 9.4% accurate, 2.6× speedup

Math

\[\frac{\mathsf{fma}\left(-0.16666666666666666, \frac{r}{s \cdot \pi}, 0.25 \cdot \frac{1}{\pi}\right)}{r \cdot s} \]

FPCore

(FPCore (s r)
 :precision binary32
 (/ (fma -0.16666666666666666 (/ r (* s PI)) (* 0.25 (/ 1.0 PI))) (* r s)))

C

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

Julia

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

Derivation

1. Initial program 99.5%

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

3. Applied rewrites 99.5%

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

4. Taylor expanded in s around 0

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

   1. lower-/.f32 N/A

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

6. Applied rewrites 99.5%

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

7. Taylor expanded in r around 0

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

   1. lower-fma.f32 N/A

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

   2. lower-/.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-/.f32 N/A

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

   7. lower-PI.f32 9.4%

      \[\leadsto \frac{\mathsf{fma}\left(-0.16666666666666666, \frac{r}{s \cdot \pi}, 0.25 \cdot \frac{1}{\pi}\right)}{r \cdot s} \]

9. Applied rewrites 9.4%

   \[\leadsto \frac{\mathsf{fma}\left(-0.16666666666666666, \frac{r}{s \cdot \pi}, 0.25 \cdot \frac{1}{\pi}\right)}{\color{blue}{r} \cdot s} \]
10. Add Preprocessing

Alternative 9: 9.2% accurate, 6.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Applied rewrites 99.5%

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

4. Taylor expanded in s around 0

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

   1. lower-/.f32 N/A

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

6. Applied rewrites 99.5%

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

7. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

   2. lower-PI.f32 9.2%

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

9. Applied rewrites 9.2%

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

Alternative 10: 9.2% accurate, 6.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

2. 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%

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

4. Applied rewrites 9.2%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lower-*.f32 9.2%

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

6. Applied rewrites 9.2%

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

Alternative 11: 9.2% accurate, 6.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

2. Taylor expanded in s around inf

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

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

4. Applied rewrites 9.2%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f32 N/A

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

   6. lower-*.f32 9.2%

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

6. Applied rewrites 9.2%

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

Alternative 12: 9.2% accurate, 6.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

2. Taylor expanded in s around inf

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

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

4. Applied rewrites 9.2%

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

Reproduce

herbie shell --seed 2025187 
(FPCore (s r)
  :name "Disney BSSRDF, PDF of scattering profile"
  :precision binary32
  :pre (and (and (<= 0.0 s) (<= s 256.0)) (and (< 1e-6 r) (< r 1000000.0)))
  (+ (/ (* 0.25 (exp (/ (- r) s))) (* (* (* 2.0 PI) s) r)) (/ (* 0.75 (exp (/ (- r) (* 3.0 s)))) (* (* (* 6.0 PI) s) r))))