Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.5%

Time: 4.9s

Alternatives: 10

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: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.5% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.6%

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

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

   3. lift-*.f32 N/A

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

   4. associate-/r* N/A

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

   5. lift-/.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. associate-/r* N/A

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

   8. div-add-rev N/A

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

   9. lower-/.f32 N/A

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

3. Applied rewrites 99.5%

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

5. Applied rewrites 99.6%

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

   1. lift-fma.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. frac-times N/A

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

   5. lift-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f32 N/A

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

   8. lift-/.f32 N/A

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

   9. +-commutative N/A

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

   10. lift-*.f32 N/A

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

7. Applied rewrites 99.5%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. lift-*.f32 N/A

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

   7. lift-*.f32 99.5

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

9. Applied rewrites 99.5%

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

Alternative 2: 99.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

   1. lift-+.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. associate-/r* N/A

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

   5. lift-/.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. associate-/r* N/A

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

   8. div-add-rev N/A

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

   9. lower-/.f32 N/A

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

3. Applied rewrites 99.5%

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

5. Applied rewrites 99.6%

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

   1. lift-fma.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. frac-times N/A

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

   5. lift-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f32 N/A

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

   8. lift-/.f32 N/A

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

   9. +-commutative N/A

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

   10. lift-*.f32 N/A

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

7. Applied rewrites 99.5%

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

   1. lift-fma.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lift-*.f32 N/A

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

   5. *-commutative N/A

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

   6. lift-*.f32 N/A

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

   7. distribute-rgt-out N/A

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

   8. lower-*.f32 N/A

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

   9. lower-+.f32 99.5

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

9. Applied rewrites 99.5%

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

Alternative 3: 43.1% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

   2. 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 9.0

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

4. Applied rewrites 9.0%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. lift-*.f32 N/A

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

   6. lower-*.f32 9.0

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 9.0

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

6. Applied rewrites 9.0%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-PI.f32 N/A

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

   4. add-log-exp N/A

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

   5. log-pow-rev N/A

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

   6. lower-log.f32 N/A

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

   7. lower-pow.f32 N/A

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

   8. lift-PI.f32 N/A

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

   9. lower-exp.f32 43.1

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

8. Applied rewrites 43.1%

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

Alternative 4: 10.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \frac{0.25}{\left(\pi \cdot s\right) \cdot r} - \frac{\frac{0.16666666666666666}{\pi} - \frac{\frac{r}{\pi} \cdot 0.06944444444444445}{s}}{s \cdot s} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (-
  (/ 0.25 (* (* PI s) r))
  (/
   (- (/ 0.16666666666666666 PI) (/ (* (/ r PI) 0.06944444444444445) s))
   (* s s))))

C

float code(float s, float r) {
	return (0.25f / ((((float) M_PI) * s) * r)) - (((0.16666666666666666f / ((float) M_PI)) - (((r / ((float) M_PI)) * 0.06944444444444445f) / s)) / (s * s));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(0.25) / Float32(Float32(Float32(pi) * s) * r)) - Float32(Float32(Float32(Float32(0.16666666666666666) / Float32(pi)) - Float32(Float32(Float32(r / Float32(pi)) * Float32(0.06944444444444445)) / s)) / Float32(s * s)))
end

MATLAB

function tmp = code(s, r)
	tmp = (single(0.25) / ((single(pi) * s) * r)) - (((single(0.16666666666666666) / single(pi)) - (((r / single(pi)) * single(0.06944444444444445)) / s)) / (s * s));
end

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in s around -inf

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

   1. lower-*.f32 N/A

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

   2. lower-/.f32 N/A

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

4. Applied rewrites 10.2%

   \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{\mathsf{fma}\left(-0.0625, \frac{r}{\pi}, -0.006944444444444444 \cdot \frac{r}{\pi}\right)}{s} - 0.16666666666666666 \cdot \frac{1}{\pi}}{s} - 0.25 \cdot \frac{1}{r \cdot \pi}}{s}} \]

6. Applied rewrites 10.2%

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

Alternative 5: 10.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.25}{\pi \cdot r} - \frac{\frac{0.16666666666666666}{\pi} - \frac{\frac{r}{\pi} \cdot 0.06944444444444445}{s}}{s}}{s} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (/
  (-
   (/ 0.25 (* PI r))
   (/ (- (/ 0.16666666666666666 PI) (/ (* (/ r PI) 0.06944444444444445) s)) s))
  s))

C

float code(float s, float r) {
	return ((0.25f / (((float) M_PI) * r)) - (((0.16666666666666666f / ((float) M_PI)) - (((r / ((float) M_PI)) * 0.06944444444444445f) / s)) / s)) / s;
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.25) / Float32(Float32(pi) * r)) - Float32(Float32(Float32(Float32(0.16666666666666666) / Float32(pi)) - Float32(Float32(Float32(r / Float32(pi)) * Float32(0.06944444444444445)) / s)) / s)) / s)
end

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.25) / (single(pi) * r)) - (((single(0.16666666666666666) / single(pi)) - (((r / single(pi)) * single(0.06944444444444445)) / s)) / s)) / s;
end

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in s around -inf

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

   1. lower-*.f32 N/A

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

   2. lower-/.f32 N/A

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

4. Applied rewrites 10.2%

   \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{\mathsf{fma}\left(-0.0625, \frac{r}{\pi}, -0.006944444444444444 \cdot \frac{r}{\pi}\right)}{s} - 0.16666666666666666 \cdot \frac{1}{\pi}}{s} - 0.25 \cdot \frac{1}{r \cdot \pi}}{s}} \]
5. Step-by-step derivation

   1. lift-*.f32 N/A

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

   2. mul-1-neg N/A

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

   3. lift-/.f32 N/A

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

   4. distribute-neg-frac N/A

      \[\leadsto \frac{\mathsf{neg}\left(\left(-1 \cdot \frac{-1 \cdot \frac{\mathsf{fma}\left(\frac{-1}{16}, \frac{r}{\pi}, \frac{-1}{144} \cdot \frac{r}{\pi}\right)}{s} - \frac{1}{6} \cdot \frac{1}{\pi}}{s} - \frac{1}{4} \cdot \frac{1}{r \cdot \pi}\right)\right)}{\color{blue}{s}} \]

   5. lower-/.f32 N/A

      \[\leadsto \frac{\mathsf{neg}\left(\left(-1 \cdot \frac{-1 \cdot \frac{\mathsf{fma}\left(\frac{-1}{16}, \frac{r}{\pi}, \frac{-1}{144} \cdot \frac{r}{\pi}\right)}{s} - \frac{1}{6} \cdot \frac{1}{\pi}}{s} - \frac{1}{4} \cdot \frac{1}{r \cdot \pi}\right)\right)}{\color{blue}{s}} \]

6. Applied rewrites 10.2%

   \[\leadsto \frac{\frac{0.25}{\pi \cdot r} - \frac{\frac{0.16666666666666666}{\pi} - \frac{\frac{r}{\pi} \cdot 0.06944444444444445}{s}}{s}}{\color{blue}{s}} \]
7. Add Preprocessing

Alternative 6: 9.5% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.6%

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

   1. lift-+.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. associate-/r* N/A

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

   5. lift-/.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. associate-/r* N/A

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

   8. div-add-rev N/A

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

   9. lower-/.f32 N/A

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

3. Applied rewrites 99.5%

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

5. Applied rewrites 99.6%

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

   1. lift-fma.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-/.f32 N/A

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

   4. frac-times N/A

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

   5. lift-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f32 N/A

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

   8. lift-/.f32 N/A

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

   9. +-commutative N/A

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

   10. lift-*.f32 N/A

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

7. Applied rewrites 99.5%

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

8. Taylor expanded in s around inf

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

10. Applied rewrites 9.5%

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

Alternative 7: 9.0% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\sqrt{\pi}} \cdot \frac{0.25}{\left(s \cdot r\right) \cdot \sqrt{\pi}} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (* (/ 1.0 (sqrt PI)) (/ 0.25 (* (* s r) (sqrt PI)))))

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

   2. 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 9.0

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

4. Applied rewrites 9.0%

   \[\leadsto \color{blue}{\frac{0.25}{r \cdot \left(s \cdot \pi\right)}} \]
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 9.0

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

6. Applied rewrites 9.0%

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

   1. lift-/.f32 N/A

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

   2. div-flip N/A

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

   3. inv-pow N/A

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

   4. lift-*.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. add-sqr-sqrt N/A

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

   7. lift-PI.f32 N/A

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

   8. lift-sqrt.f32 N/A

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

   9. lift-PI.f32 N/A

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

   10. lift-sqrt.f32 N/A

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

   11. associate-*l* N/A

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

   12. lift-*.f32 N/A

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

   13. *-commutative N/A

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

   14. associate-/l* N/A

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

   15. unpow-prod-down N/A

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

   16. inv-pow N/A

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

   17. div-flip N/A

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

8. Applied rewrites 9.0%

   \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \color{blue}{\frac{0.25}{\left(\sqrt{\pi} \cdot s\right) \cdot r}} \]
9. Step-by-step derivation

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*l* N/A

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

   4. lift-*.f32 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 9.0

      \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \frac{0.25}{\left(s \cdot r\right) \cdot \color{blue}{\sqrt{\pi}}} \]

10. Applied rewrites 9.0%

    \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \frac{0.25}{\left(s \cdot r\right) \cdot \color{blue}{\sqrt{\pi}}} \]
11. Add Preprocessing

Alternative 8: 9.0% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\pi \cdot r} \cdot \frac{0.25}{s} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 9.0

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

4. Applied rewrites 9.0%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. lift-*.f32 N/A

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

   7. associate-/l/ N/A

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

   8. mult-flip-rev N/A

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

   9. lift-/.f32 N/A

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

   10. *-commutative N/A

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

   11. associate-/l* N/A

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

   12. lower-*.f32 N/A

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

   13. lift-*.f32 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f32 N/A

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

   16. lower-/.f32 9.0

       \[\leadsto \frac{1}{\pi \cdot r} \cdot \frac{0.25}{\color{blue}{s}} \]

6. Applied rewrites 9.0%

   \[\leadsto \frac{1}{\pi \cdot r} \cdot \color{blue}{\frac{0.25}{s}} \]
7. Add Preprocessing

Alternative 9: 9.0% accurate, 6.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 9.0

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

4. Applied rewrites 9.0%

   \[\leadsto \color{blue}{\frac{0.25}{r \cdot \left(s \cdot \pi\right)}} \]
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 9.0

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

6. Applied rewrites 9.0%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*l* N/A

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

   5. associate-/r* N/A

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

   6. lower-/.f32 N/A

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

   7. lower-/.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 9.0

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

8. Applied rewrites 9.0%

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

Alternative 10: 9.0% accurate, 6.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

   2. 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 9.0

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

4. Applied rewrites 9.0%

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

Reproduce

herbie shell --seed 2025149 
(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))))