Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.5% → 99.5%

Time: 6.0s

Alternatives: 13

Speedup: N/A×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.5% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/ (* 0.125 (exp (/ (- r) s))) (* (* PI s) r))
  (/ (* 0.75 (exp (/ (- r) (* 3.0 s)))) (* (* PI 6.0) (* 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)))) / ((((float) M_PI) * 6.0f) * (s * r)));
}

Julia

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

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

   2. lift-*.f32 N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. lift-PI.f32 N/A

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

   10. lower-*.f32 99.5

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

3. Applied rewrites 99.5%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-exp.f32 N/A

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

   4. lift-/.f32 N/A

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

   5. lift-neg.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. times-frac N/A

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

   8. metadata-eval N/A

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

   9. lift-*.f32 N/A

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

   10. lift-PI.f32 N/A

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

   11. lift-*.f32 N/A

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

   12. associate-*l* N/A

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

   13. *-commutative N/A

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

   14. times-frac N/A

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

   15. metadata-eval N/A

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

   16. associate-*r/ N/A

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

   17. metadata-eval N/A

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

   18. *-commutative N/A

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

5. Applied rewrites 99.5%

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

Alternative 2: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

Julia

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

Derivation

1. Initial program 99.5%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-exp.f32 N/A

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

   4. lift-/.f32 N/A

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

   5. lift-neg.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. associate-/r* N/A

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

3. Applied rewrites 99.5%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-neg.f32 N/A

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

   4. distribute-frac-neg N/A

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

   5. distribute-neg-frac2 N/A

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

   6. lower-/.f32 N/A

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

   7. distribute-lft-neg-in N/A

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

   8. lower-*.f32 N/A

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

   9. metadata-eval 99.5

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

5. Applied rewrites 99.5%

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

   1. lift-*.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. lower-*.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. lift-PI.f32 99.5

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

7. Applied rewrites 99.5%

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

Alternative 3: 99.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\pi \cdot s\right) \cdot r\\ \frac{0.125 \cdot e^{\frac{-r}{s}}}{t\_0} + \frac{0.75 \cdot e^{\frac{r}{-3 \cdot s}}}{t\_0 \cdot 6} \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (let* ((t_0 (* (* PI s) r)))
   (+
    (/ (* 0.125 (exp (/ (- r) s))) t_0)
    (/ (* 0.75 (exp (/ r (* -3.0 s)))) (* t_0 6.0)))))

C

float code(float s, float r) {
	float t_0 = (((float) M_PI) * s) * r;
	return ((0.125f * expf((-r / s))) / t_0) + ((0.75f * expf((r / (-3.0f * s)))) / (t_0 * 6.0f));
}

Julia

function code(s, r)
	t_0 = Float32(Float32(Float32(pi) * s) * r)
	return Float32(Float32(Float32(Float32(0.125) * exp(Float32(Float32(-r) / s))) / t_0) + Float32(Float32(Float32(0.75) * exp(Float32(r / Float32(Float32(-3.0) * s)))) / Float32(t_0 * Float32(6.0))))
end

MATLAB

function tmp = code(s, r)
	t_0 = (single(pi) * s) * r;
	tmp = ((single(0.125) * exp((-r / s))) / t_0) + ((single(0.75) * exp((r / (single(-3.0) * s)))) / (t_0 * single(6.0)));
end

Derivation

1. Initial program 99.5%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-exp.f32 N/A

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

   4. lift-/.f32 N/A

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

   5. lift-neg.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. associate-/r* N/A

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

3. Applied rewrites 99.5%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-neg.f32 N/A

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

   4. distribute-frac-neg N/A

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

   5. distribute-neg-frac2 N/A

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

   6. lower-/.f32 N/A

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

   7. distribute-lft-neg-in N/A

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

   8. lower-*.f32 N/A

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

   9. metadata-eval 99.5

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

5. Applied rewrites 99.5%

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

6. Taylor expanded in s around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. *-commutative N/A

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

   6. lift-*.f32 N/A

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

   7. lift-PI.f32 99.5

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

8. Applied rewrites 99.5%

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

Alternative 4: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.5%

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

3. Applied rewrites 99.4%

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

   1. lift-/.f32 N/A

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

   2. lift-exp.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. lift-neg.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-PI.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. associate-/r* N/A

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

   9. distribute-frac-neg N/A

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

   10. mul-1-neg N/A

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

   11. *-commutative N/A

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

   12. lower-/.f32 N/A

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

5. Applied rewrites 99.4%

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

Alternative 5: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.5%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*l* N/A

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

   4. lower-*.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. lift-PI.f32 N/A

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

   10. lower-*.f32 99.5

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

3. Applied rewrites 99.5%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-exp.f32 N/A

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

   4. lift-/.f32 N/A

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

   5. lift-neg.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. times-frac N/A

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

   8. metadata-eval N/A

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

   9. lift-*.f32 N/A

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

   10. lift-PI.f32 N/A

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

   11. lift-*.f32 N/A

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

   12. associate-*l* N/A

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

   13. *-commutative N/A

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

   14. times-frac N/A

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

   15. metadata-eval N/A

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

   16. associate-*r/ N/A

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

   17. metadata-eval N/A

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

   18. *-commutative N/A

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

5. Applied rewrites 99.5%

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

7. Applied rewrites 99.5%

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

Alternative 6: 99.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(s \cdot r\right) \cdot \pi\\ \mathsf{fma}\left(\frac{e^{\frac{-r}{s}}}{t\_0}, 0.125, \frac{0.125 \cdot e^{-0.3333333333333333 \cdot \frac{r}{s}}}{t\_0}\right) \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (let* ((t_0 (* (* s r) PI)))
   (fma
    (/ (exp (/ (- r) s)) t_0)
    0.125
    (/ (* 0.125 (exp (* -0.3333333333333333 (/ r s)))) t_0))))

C

float code(float s, float r) {
	float t_0 = (s * r) * ((float) M_PI);
	return fmaf((expf((-r / s)) / t_0), 0.125f, ((0.125f * expf((-0.3333333333333333f * (r / s)))) / t_0));
}

Julia

function code(s, r)
	t_0 = Float32(Float32(s * r) * Float32(pi))
	return fma(Float32(exp(Float32(Float32(-r) / s)) / t_0), Float32(0.125), Float32(Float32(Float32(0.125) * exp(Float32(Float32(-0.3333333333333333) * Float32(r / s)))) / t_0))
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 99.4%

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

   1. lift-*.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. lift-PI.f32 99.4

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

5. Applied rewrites 99.4%

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

   1. lift-*.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. lift-PI.f32 99.4

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

7. Applied rewrites 99.4%

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

Alternative 7: 99.4% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

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

Alternative 8: 99.4% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Applied rewrites 99.5%

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

Alternative 9: 43.9% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

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

4. Applied rewrites 10.1%

   \[\leadsto \color{blue}{\frac{\frac{\frac{\frac{r}{\pi} \cdot 0.06944444444444445}{s} - \frac{0.16666666666666666}{\pi}}{s} + \frac{0.25}{\pi \cdot r}}{s}} \]

5. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. lift-PI.f32 9.1

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

7. Applied rewrites 9.1%

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

   1. lift-PI.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. add-log-exp N/A

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

   5. log-pow-rev N/A

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

   6. lower-log.f32 N/A

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

   7. pow-to-exp N/A

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

   8. add-log-exp N/A

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

   9. *-commutative N/A

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

   10. lower-exp.f32 N/A

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

   11. *-commutative N/A

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

   12. lift-*.f32 N/A

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

   13. lift-PI.f32 43.9

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

9. Applied rewrites 43.9%

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

Alternative 10: 9.1% accurate, 6.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

2. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lift-PI.f32 9.1

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

4. Applied rewrites 9.1%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-PI.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. *-commutative N/A

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

   6. *-commutative N/A

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

   7. associate-/r* N/A

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

   8. lower-/.f32 N/A

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

   9. lower-/.f32 N/A

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

   10. *-commutative N/A

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

   11. lift-*.f32 N/A

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

   12. lift-PI.f32 9.1

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

6. Applied rewrites 9.1%

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

Alternative 11: 9.1% accurate, 6.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = (single(0.25) / (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 inf

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

   1. lower-/.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lift-PI.f32 9.1

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

4. Applied rewrites 9.1%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-PI.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. associate-/r* N/A

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

   6. metadata-eval N/A

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

   7. *-commutative N/A

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

   8. associate-*r/ N/A

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

   9. lower-/.f32 N/A

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

   10. associate-*r/ N/A

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

   11. metadata-eval N/A

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

   12. lower-/.f32 N/A

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

   13. *-commutative N/A

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

   14. lift-*.f32 N/A

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

   15. lift-PI.f32 9.1

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

6. Applied rewrites 9.1%

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

Alternative 12: 9.1% accurate, 6.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = single(0.25) / ((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 inf

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

   1. lower-/.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lift-PI.f32 9.1

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

4. Applied rewrites 9.1%

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

Alternative 13: 9.1% accurate, 6.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

2. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lift-PI.f32 9.1

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

4. Applied rewrites 9.1%

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

   1. lift-*.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. lift-PI.f32 9.1

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

6. Applied rewrites 9.1%

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

Reproduce

herbie shell --seed 2025134 
(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))))