Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.6%

Time: 5.0s

Alternatives: 13

Speedup: 1.1×

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: 99.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

function code(s, r)
	return fma(Float32(exp(Float32(Float32(-r) / Float32(s * Float32(3.0)))) / Float32(Float32(Float32(Float32(6.0) * s) * Float32(pi)) * r)), Float32(0.75), Float32(Float32(exp(Float32(Float32(-r) / s)) / Float32(Float32(Float32(pi) * s) * r)) * Float32(0.125)))
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 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. mul-1-neg N/A

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

   4. distribute-frac-neg N/A

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

   5. lower-/.f32 N/A

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

   6. lift-neg.f32 N/A

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

   7. lift-/.f32 N/A

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

   8. lift-exp.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 N/A

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

   13. lift-PI.f32 99.6

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

4. Applied rewrites 99.6%

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

6. Applied rewrites 99.6%

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

   1. lift-*.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

   7. associate-*r* N/A

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

   8. lower-*.f32 N/A

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

   9. lower-*.f32 N/A

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

   10. lift-PI.f32 99.6

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

8. Applied rewrites 99.6%

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

   1. lift-+.f32 N/A

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

10. Applied rewrites 99.6%

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

Alternative 2: 99.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

function code(s, r)
	return fma(Float32(exp(Float32(Float32(-r) / Float32(Float32(3.0) * s))) / Float32(Float32(Float32(Float32(pi) * Float32(6.0)) * s) * r)), Float32(0.75), Float32(Float32(exp(Float32(Float32(-r) / s)) / Float32(Float32(Float32(pi) * s) * r)) * Float32(0.125)))
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 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. mul-1-neg N/A

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

   4. distribute-frac-neg N/A

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

   5. lower-/.f32 N/A

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

   6. lift-neg.f32 N/A

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

   7. lift-/.f32 N/A

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

   8. lift-exp.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 N/A

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

   13. lift-PI.f32 99.6

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

4. Applied rewrites 99.6%

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

6. Applied rewrites 99.6%

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

Alternative 3: 99.5% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

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

   1. lower-/.f32 N/A

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

4. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\frac{\frac{0.125 \cdot e^{\frac{-r}{s}} - -0.125 \cdot e^{-0.3333333333333333 \cdot \frac{r}{s}}}{\pi \cdot s}}{r}} \]

6. Applied rewrites 99.5%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-*l/ N/A

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

   4. lower-/.f32 N/A

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

   5. lower-*.f32 99.5

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

8. Applied rewrites 99.5%

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

Alternative 4: 99.5% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = (((single(0.125) * exp((-r / s))) - (single(-0.125) * exp(((single(-0.3333333333333333) * r) / 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. Taylor expanded in r around inf

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

   1. lower-/.f32 N/A

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

4. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\frac{\frac{0.125 \cdot e^{\frac{-r}{s}} - -0.125 \cdot e^{-0.3333333333333333 \cdot \frac{r}{s}}}{\pi \cdot s}}{r}} \]
5. Step-by-step derivation

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-*r/ N/A

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

   4. metadata-eval N/A

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

   5. lower-/.f32 N/A

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

   6. metadata-eval N/A

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

   7. lower-*.f32 99.5

      \[\leadsto \frac{\frac{0.125 \cdot e^{\frac{-r}{s}} - -0.125 \cdot e^{\frac{-0.3333333333333333 \cdot r}{s}}}{\pi \cdot s}}{r} \]

6. Applied rewrites 99.5%

   \[\leadsto \frac{\frac{0.125 \cdot e^{\frac{-r}{s}} - -0.125 \cdot e^{\frac{-0.3333333333333333 \cdot r}{s}}}{\pi \cdot s}}{r} \]
7. Add Preprocessing

Alternative 5: 99.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

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

   1. lower-/.f32 N/A

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

4. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\frac{\frac{0.125 \cdot e^{\frac{-r}{s}} - -0.125 \cdot e^{-0.3333333333333333 \cdot \frac{r}{s}}}{\pi \cdot s}}{r}} \]

6. Applied rewrites 99.5%

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

   1. lift-/.f32 N/A

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

8. Applied rewrites 99.5%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-*r/ N/A

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

   5. lower-/.f32 N/A

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

   6. lower-*.f32 99.5

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

10. Applied rewrites 99.5%

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

Alternative 6: 99.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

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

   1. lower-/.f32 N/A

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

4. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\frac{\frac{0.125 \cdot e^{\frac{-r}{s}} - -0.125 \cdot e^{-0.3333333333333333 \cdot \frac{r}{s}}}{\pi \cdot s}}{r}} \]

6. Applied rewrites 99.5%

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

   1. lift-/.f32 N/A

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

8. Applied rewrites 99.5%

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

Alternative 7: 99.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = (single(0.125) * (exp((-r / s)) + exp(((r / s) * single(-0.3333333333333333))))) / ((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. Taylor expanded in r around inf

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

   1. lower-/.f32 N/A

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

4. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\frac{\frac{0.125 \cdot e^{\frac{-r}{s}} - -0.125 \cdot e^{-0.3333333333333333 \cdot \frac{r}{s}}}{\pi \cdot s}}{r}} \]

6. Applied rewrites 99.5%

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

8. Applied rewrites 99.5%

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

Alternative 8: 9.3% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Julia

function code(s, r)
	return Float32(Float32(Float32(fma(Float32(Float32(Float32(r / Float32(s * s)) * Float32(0.06944444444444445)) - Float32(Float32(0.16666666666666666) / s)), r, Float32(0.25)) / 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. Taylor expanded in r around inf

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

   1. lower-/.f32 N/A

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

4. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\frac{\frac{0.125 \cdot e^{\frac{-r}{s}} - -0.125 \cdot e^{-0.3333333333333333 \cdot \frac{r}{s}}}{\pi \cdot s}}{r}} \]

5. Taylor expanded in r around 0

   \[\leadsto \frac{\frac{\frac{1}{4} + r \cdot \left(\frac{5}{72} \cdot \frac{r}{{s}^{2}} - \frac{1}{6} \cdot \frac{1}{s}\right)}{\pi \cdot s}}{r} \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\frac{r \cdot \left(\frac{5}{72} \cdot \frac{r}{{s}^{2}} - \frac{1}{6} \cdot \frac{1}{s}\right) + \frac{1}{4}}{\pi \cdot s}}{r} \]

   2. *-commutative N/A

      \[\leadsto \frac{\frac{\left(\frac{5}{72} \cdot \frac{r}{{s}^{2}} - \frac{1}{6} \cdot \frac{1}{s}\right) \cdot r + \frac{1}{4}}{\pi \cdot s}}{r} \]

   3. lower-fma.f32 N/A

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{5}{72} \cdot \frac{r}{{s}^{2}} - \frac{1}{6} \cdot \frac{1}{s}, r, \frac{1}{4}\right)}{\pi \cdot s}}{r} \]

   4. lower--.f32 N/A

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{5}{72} \cdot \frac{r}{{s}^{2}} - \frac{1}{6} \cdot \frac{1}{s}, r, \frac{1}{4}\right)}{\pi \cdot s}}{r} \]

   5. *-commutative N/A

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{r}{{s}^{2}} \cdot \frac{5}{72} - \frac{1}{6} \cdot \frac{1}{s}, r, \frac{1}{4}\right)}{\pi \cdot s}}{r} \]

   6. lower-*.f32 N/A

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{r}{{s}^{2}} \cdot \frac{5}{72} - \frac{1}{6} \cdot \frac{1}{s}, r, \frac{1}{4}\right)}{\pi \cdot s}}{r} \]

   7. pow2 N/A

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{5}{72} - \frac{1}{6} \cdot \frac{1}{s}, r, \frac{1}{4}\right)}{\pi \cdot s}}{r} \]

   8. lower-/.f32 N/A

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{5}{72} - \frac{1}{6} \cdot \frac{1}{s}, r, \frac{1}{4}\right)}{\pi \cdot s}}{r} \]

   9. lift-*.f32 N/A

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{5}{72} - \frac{1}{6} \cdot \frac{1}{s}, r, \frac{1}{4}\right)}{\pi \cdot s}}{r} \]

   10. associate-*r/ N/A

       \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{5}{72} - \frac{\frac{1}{6} \cdot 1}{s}, r, \frac{1}{4}\right)}{\pi \cdot s}}{r} \]

   11. metadata-eval N/A

       \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{5}{72} - \frac{\frac{1}{6}}{s}, r, \frac{1}{4}\right)}{\pi \cdot s}}{r} \]

   12. lower-/.f32 9.3

       \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot 0.06944444444444445 - \frac{0.16666666666666666}{s}, r, 0.25\right)}{\pi \cdot s}}{r} \]

7. Applied rewrites 9.3%

   \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot 0.06944444444444445 - \frac{0.16666666666666666}{s}, r, 0.25\right)}{\pi \cdot s}}{r} \]
8. Step-by-step derivation

   1. lift-/.f32 N/A

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{5}{72} - \frac{\frac{1}{6}}{s}, r, \frac{1}{4}\right)}{\pi \cdot s}}{r} \]

   2. lift-PI.f32 N/A

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{5}{72} - \frac{\frac{1}{6}}{s}, r, \frac{1}{4}\right)}{\mathsf{PI}\left(\right) \cdot s}}{r} \]

   3. lift-*.f32 N/A

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{5}{72} - \frac{\frac{1}{6}}{s}, r, \frac{1}{4}\right)}{\mathsf{PI}\left(\right) \cdot s}}{r} \]

   4. associate-/r* N/A

      \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{5}{72} - \frac{\frac{1}{6}}{s}, r, \frac{1}{4}\right)}{\mathsf{PI}\left(\right)}}{s}}{r} \]

   5. lower-/.f32 N/A

      \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{5}{72} - \frac{\frac{1}{6}}{s}, r, \frac{1}{4}\right)}{\mathsf{PI}\left(\right)}}{s}}{r} \]

9. Applied rewrites 9.3%

   \[\leadsto \frac{\frac{\frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot 0.06944444444444445 - \frac{0.16666666666666666}{s}, r, 0.25\right)}{\pi}}{s}}{r} \]
10. Add Preprocessing

Alternative 9: 9.3% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Julia

function code(s, r)
	return Float32(fma(Float32(Float32(Float32(r / Float32(s * s)) * Float32(0.06944444444444445)) - Float32(Float32(0.16666666666666666) / s)), r, Float32(0.25)) / Float32(Float32(s * r) * Float32(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 r around inf

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

   1. lower-/.f32 N/A

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

4. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\frac{\frac{0.125 \cdot e^{\frac{-r}{s}} - -0.125 \cdot e^{-0.3333333333333333 \cdot \frac{r}{s}}}{\pi \cdot s}}{r}} \]

5. Taylor expanded in r around 0

   \[\leadsto \frac{\frac{\frac{1}{4} + r \cdot \left(\frac{5}{72} \cdot \frac{r}{{s}^{2}} - \frac{1}{6} \cdot \frac{1}{s}\right)}{\pi \cdot s}}{r} \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\frac{r \cdot \left(\frac{5}{72} \cdot \frac{r}{{s}^{2}} - \frac{1}{6} \cdot \frac{1}{s}\right) + \frac{1}{4}}{\pi \cdot s}}{r} \]

   2. *-commutative N/A

      \[\leadsto \frac{\frac{\left(\frac{5}{72} \cdot \frac{r}{{s}^{2}} - \frac{1}{6} \cdot \frac{1}{s}\right) \cdot r + \frac{1}{4}}{\pi \cdot s}}{r} \]

   3. lower-fma.f32 N/A

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{5}{72} \cdot \frac{r}{{s}^{2}} - \frac{1}{6} \cdot \frac{1}{s}, r, \frac{1}{4}\right)}{\pi \cdot s}}{r} \]

   4. lower--.f32 N/A

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{5}{72} \cdot \frac{r}{{s}^{2}} - \frac{1}{6} \cdot \frac{1}{s}, r, \frac{1}{4}\right)}{\pi \cdot s}}{r} \]

   5. *-commutative N/A

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{r}{{s}^{2}} \cdot \frac{5}{72} - \frac{1}{6} \cdot \frac{1}{s}, r, \frac{1}{4}\right)}{\pi \cdot s}}{r} \]

   6. lower-*.f32 N/A

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{r}{{s}^{2}} \cdot \frac{5}{72} - \frac{1}{6} \cdot \frac{1}{s}, r, \frac{1}{4}\right)}{\pi \cdot s}}{r} \]

   7. pow2 N/A

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{5}{72} - \frac{1}{6} \cdot \frac{1}{s}, r, \frac{1}{4}\right)}{\pi \cdot s}}{r} \]

   8. lower-/.f32 N/A

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{5}{72} - \frac{1}{6} \cdot \frac{1}{s}, r, \frac{1}{4}\right)}{\pi \cdot s}}{r} \]

   9. lift-*.f32 N/A

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{5}{72} - \frac{1}{6} \cdot \frac{1}{s}, r, \frac{1}{4}\right)}{\pi \cdot s}}{r} \]

   10. associate-*r/ N/A

       \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{5}{72} - \frac{\frac{1}{6} \cdot 1}{s}, r, \frac{1}{4}\right)}{\pi \cdot s}}{r} \]

   11. metadata-eval N/A

       \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{5}{72} - \frac{\frac{1}{6}}{s}, r, \frac{1}{4}\right)}{\pi \cdot s}}{r} \]

   12. lower-/.f32 9.3

       \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot 0.06944444444444445 - \frac{0.16666666666666666}{s}, r, 0.25\right)}{\pi \cdot s}}{r} \]

7. Applied rewrites 9.3%

   \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot 0.06944444444444445 - \frac{0.16666666666666666}{s}, r, 0.25\right)}{\pi \cdot s}}{r} \]
8. Step-by-step derivation

   1. lift-/.f32 N/A

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{5}{72} - \frac{\frac{1}{6}}{s}, r, \frac{1}{4}\right)}{\pi \cdot s}}{\color{blue}{r}} \]

   2. lift-/.f32 N/A

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{5}{72} - \frac{\frac{1}{6}}{s}, r, \frac{1}{4}\right)}{\pi \cdot s}}{r} \]

   3. associate-/l/ N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{5}{72} - \frac{\frac{1}{6}}{s}, r, \frac{1}{4}\right)}{\color{blue}{\left(\pi \cdot s\right) \cdot r}} \]

   4. lift-PI.f32 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{5}{72} - \frac{\frac{1}{6}}{s}, r, \frac{1}{4}\right)}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} \]

   5. lift-*.f32 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{5}{72} - \frac{\frac{1}{6}}{s}, r, \frac{1}{4}\right)}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} \]

   6. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{5}{72} - \frac{\frac{1}{6}}{s}, r, \frac{1}{4}\right)}{\left(s \cdot \mathsf{PI}\left(\right)\right) \cdot r} \]

   7. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{5}{72} - \frac{\frac{1}{6}}{s}, r, \frac{1}{4}\right)}{r \cdot \color{blue}{\left(s \cdot \mathsf{PI}\left(\right)\right)}} \]

   8. lower-/.f32 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{5}{72} - \frac{\frac{1}{6}}{s}, r, \frac{1}{4}\right)}{\color{blue}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}} \]

9. Applied rewrites 9.3%

   \[\leadsto \frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot 0.06944444444444445 - \frac{0.16666666666666666}{s}, r, 0.25\right)}{\color{blue}{\left(\pi \cdot s\right) \cdot r}} \]
10. Step-by-step derivation

    1. lift-*.f32 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{5}{72} - \frac{\frac{1}{6}}{s}, r, \frac{1}{4}\right)}{\left(\pi \cdot s\right) \cdot \color{blue}{r}} \]

    2. lift-PI.f32 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{5}{72} - \frac{\frac{1}{6}}{s}, r, \frac{1}{4}\right)}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} \]

    3. lift-*.f32 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{5}{72} - \frac{\frac{1}{6}}{s}, r, \frac{1}{4}\right)}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} \]

    4. *-commutative N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{5}{72} - \frac{\frac{1}{6}}{s}, r, \frac{1}{4}\right)}{\left(s \cdot \mathsf{PI}\left(\right)\right) \cdot r} \]

    5. *-commutative N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{5}{72} - \frac{\frac{1}{6}}{s}, r, \frac{1}{4}\right)}{r \cdot \color{blue}{\left(s \cdot \mathsf{PI}\left(\right)\right)}} \]

    6. associate-*r* N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{5}{72} - \frac{\frac{1}{6}}{s}, r, \frac{1}{4}\right)}{\left(r \cdot s\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}} \]

    7. lower-*.f32 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{5}{72} - \frac{\frac{1}{6}}{s}, r, \frac{1}{4}\right)}{\left(r \cdot s\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}} \]

    8. *-commutative N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{5}{72} - \frac{\frac{1}{6}}{s}, r, \frac{1}{4}\right)}{\left(s \cdot r\right) \cdot \mathsf{PI}\left(\right)} \]

    9. lift-*.f32 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{5}{72} - \frac{\frac{1}{6}}{s}, r, \frac{1}{4}\right)}{\left(s \cdot r\right) \cdot \mathsf{PI}\left(\right)} \]

    10. lift-PI.f32 9.3

        \[\leadsto \frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot 0.06944444444444445 - \frac{0.16666666666666666}{s}, r, 0.25\right)}{\left(s \cdot r\right) \cdot \pi} \]

11. Applied rewrites 9.3%

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

Alternative 10: 9.3% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

function code(s, r)
	return Float32(fma(Float32(fma(Float32(0.06944444444444445), Float32(r / s), Float32(-0.16666666666666666)) / s), r, Float32(0.25)) / 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. Taylor expanded in r around inf

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

   1. lower-/.f32 N/A

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

4. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\frac{\frac{0.125 \cdot e^{\frac{-r}{s}} - -0.125 \cdot e^{-0.3333333333333333 \cdot \frac{r}{s}}}{\pi \cdot s}}{r}} \]

5. Taylor expanded in r around 0

   \[\leadsto \frac{\frac{\frac{1}{4} + r \cdot \left(\frac{5}{72} \cdot \frac{r}{{s}^{2}} - \frac{1}{6} \cdot \frac{1}{s}\right)}{\pi \cdot s}}{r} \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\frac{r \cdot \left(\frac{5}{72} \cdot \frac{r}{{s}^{2}} - \frac{1}{6} \cdot \frac{1}{s}\right) + \frac{1}{4}}{\pi \cdot s}}{r} \]

   2. *-commutative N/A

      \[\leadsto \frac{\frac{\left(\frac{5}{72} \cdot \frac{r}{{s}^{2}} - \frac{1}{6} \cdot \frac{1}{s}\right) \cdot r + \frac{1}{4}}{\pi \cdot s}}{r} \]

   3. lower-fma.f32 N/A

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{5}{72} \cdot \frac{r}{{s}^{2}} - \frac{1}{6} \cdot \frac{1}{s}, r, \frac{1}{4}\right)}{\pi \cdot s}}{r} \]

   4. lower--.f32 N/A

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{5}{72} \cdot \frac{r}{{s}^{2}} - \frac{1}{6} \cdot \frac{1}{s}, r, \frac{1}{4}\right)}{\pi \cdot s}}{r} \]

   5. *-commutative N/A

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{r}{{s}^{2}} \cdot \frac{5}{72} - \frac{1}{6} \cdot \frac{1}{s}, r, \frac{1}{4}\right)}{\pi \cdot s}}{r} \]

   6. lower-*.f32 N/A

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{r}{{s}^{2}} \cdot \frac{5}{72} - \frac{1}{6} \cdot \frac{1}{s}, r, \frac{1}{4}\right)}{\pi \cdot s}}{r} \]

   7. pow2 N/A

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{5}{72} - \frac{1}{6} \cdot \frac{1}{s}, r, \frac{1}{4}\right)}{\pi \cdot s}}{r} \]

   8. lower-/.f32 N/A

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{5}{72} - \frac{1}{6} \cdot \frac{1}{s}, r, \frac{1}{4}\right)}{\pi \cdot s}}{r} \]

   9. lift-*.f32 N/A

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{5}{72} - \frac{1}{6} \cdot \frac{1}{s}, r, \frac{1}{4}\right)}{\pi \cdot s}}{r} \]

   10. associate-*r/ N/A

       \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{5}{72} - \frac{\frac{1}{6} \cdot 1}{s}, r, \frac{1}{4}\right)}{\pi \cdot s}}{r} \]

   11. metadata-eval N/A

       \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{5}{72} - \frac{\frac{1}{6}}{s}, r, \frac{1}{4}\right)}{\pi \cdot s}}{r} \]

   12. lower-/.f32 9.3

       \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot 0.06944444444444445 - \frac{0.16666666666666666}{s}, r, 0.25\right)}{\pi \cdot s}}{r} \]

7. Applied rewrites 9.3%

   \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot 0.06944444444444445 - \frac{0.16666666666666666}{s}, r, 0.25\right)}{\pi \cdot s}}{r} \]
8. Step-by-step derivation

   1. lift-/.f32 N/A

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{5}{72} - \frac{\frac{1}{6}}{s}, r, \frac{1}{4}\right)}{\pi \cdot s}}{\color{blue}{r}} \]

   2. lift-/.f32 N/A

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{5}{72} - \frac{\frac{1}{6}}{s}, r, \frac{1}{4}\right)}{\pi \cdot s}}{r} \]

   3. associate-/l/ N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{5}{72} - \frac{\frac{1}{6}}{s}, r, \frac{1}{4}\right)}{\color{blue}{\left(\pi \cdot s\right) \cdot r}} \]

   4. lift-PI.f32 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{5}{72} - \frac{\frac{1}{6}}{s}, r, \frac{1}{4}\right)}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} \]

   5. lift-*.f32 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{5}{72} - \frac{\frac{1}{6}}{s}, r, \frac{1}{4}\right)}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} \]

   6. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{5}{72} - \frac{\frac{1}{6}}{s}, r, \frac{1}{4}\right)}{\left(s \cdot \mathsf{PI}\left(\right)\right) \cdot r} \]

   7. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{5}{72} - \frac{\frac{1}{6}}{s}, r, \frac{1}{4}\right)}{r \cdot \color{blue}{\left(s \cdot \mathsf{PI}\left(\right)\right)}} \]

   8. lower-/.f32 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{5}{72} - \frac{\frac{1}{6}}{s}, r, \frac{1}{4}\right)}{\color{blue}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}} \]

9. Applied rewrites 9.3%

   \[\leadsto \frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot 0.06944444444444445 - \frac{0.16666666666666666}{s}, r, 0.25\right)}{\color{blue}{\left(\pi \cdot s\right) \cdot r}} \]

10. Taylor expanded in s around inf

    \[\leadsto \frac{\mathsf{fma}\left(\frac{\frac{5}{72} \cdot \frac{r}{s} - \frac{1}{6}}{s}, r, \frac{1}{4}\right)}{\left(\pi \cdot s\right) \cdot r} \]
11. Step-by-step derivation

    1. lower-/.f32 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{\frac{5}{72} \cdot \frac{r}{s} - \frac{1}{6}}{s}, r, \frac{1}{4}\right)}{\left(\pi \cdot s\right) \cdot r} \]

    2. negate-sub N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{\frac{5}{72} \cdot \frac{r}{s} + \left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}{s}, r, \frac{1}{4}\right)}{\left(\pi \cdot s\right) \cdot r} \]

    3. metadata-eval N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{\frac{5}{72} \cdot \frac{r}{s} + \frac{-1}{6}}{s}, r, \frac{1}{4}\right)}{\left(\pi \cdot s\right) \cdot r} \]

    4. lower-fma.f32 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(\frac{5}{72}, \frac{r}{s}, \frac{-1}{6}\right)}{s}, r, \frac{1}{4}\right)}{\left(\pi \cdot s\right) \cdot r} \]

    5. lift-/.f32 9.3

       \[\leadsto \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.06944444444444445, \frac{r}{s}, -0.16666666666666666\right)}{s}, r, 0.25\right)}{\left(\pi \cdot s\right) \cdot r} \]

12. Applied rewrites 9.3%

    \[\leadsto \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(0.06944444444444445, \frac{r}{s}, -0.16666666666666666\right)}{s}, r, 0.25\right)}{\left(\pi \cdot s\right) \cdot r} \]
13. Add Preprocessing

Alternative 11: 8.5% accurate, 3.4× speedup

Math

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

FPCore

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

C

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

Julia

function code(s, r)
	return Float32(Float32(fma(Float32(-0.16666666666666666), Float32(r / s), Float32(0.25)) / 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. Taylor expanded in r around inf

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

   1. lower-/.f32 N/A

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

4. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\frac{\frac{0.125 \cdot e^{\frac{-r}{s}} - -0.125 \cdot e^{-0.3333333333333333 \cdot \frac{r}{s}}}{\pi \cdot s}}{r}} \]

5. Taylor expanded in r around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f32 N/A

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

   3. lift-/.f32 8.4

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

7. Applied rewrites 8.4%

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

Alternative 12: 8.4% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Julia

function code(s, r)
	return Float32(fma(Float32(-0.16666666666666666), Float32(r / s), Float32(0.25)) / 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. Taylor expanded in r around inf

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

   1. lower-/.f32 N/A

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

4. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\frac{\frac{0.125 \cdot e^{\frac{-r}{s}} - -0.125 \cdot e^{-0.3333333333333333 \cdot \frac{r}{s}}}{\pi \cdot s}}{r}} \]

5. Taylor expanded in r around 0

   \[\leadsto \frac{\frac{\frac{1}{4} + r \cdot \left(\frac{5}{72} \cdot \frac{r}{{s}^{2}} - \frac{1}{6} \cdot \frac{1}{s}\right)}{\pi \cdot s}}{r} \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\frac{r \cdot \left(\frac{5}{72} \cdot \frac{r}{{s}^{2}} - \frac{1}{6} \cdot \frac{1}{s}\right) + \frac{1}{4}}{\pi \cdot s}}{r} \]

   2. *-commutative N/A

      \[\leadsto \frac{\frac{\left(\frac{5}{72} \cdot \frac{r}{{s}^{2}} - \frac{1}{6} \cdot \frac{1}{s}\right) \cdot r + \frac{1}{4}}{\pi \cdot s}}{r} \]

   3. lower-fma.f32 N/A

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{5}{72} \cdot \frac{r}{{s}^{2}} - \frac{1}{6} \cdot \frac{1}{s}, r, \frac{1}{4}\right)}{\pi \cdot s}}{r} \]

   4. lower--.f32 N/A

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{5}{72} \cdot \frac{r}{{s}^{2}} - \frac{1}{6} \cdot \frac{1}{s}, r, \frac{1}{4}\right)}{\pi \cdot s}}{r} \]

   5. *-commutative N/A

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{r}{{s}^{2}} \cdot \frac{5}{72} - \frac{1}{6} \cdot \frac{1}{s}, r, \frac{1}{4}\right)}{\pi \cdot s}}{r} \]

   6. lower-*.f32 N/A

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{r}{{s}^{2}} \cdot \frac{5}{72} - \frac{1}{6} \cdot \frac{1}{s}, r, \frac{1}{4}\right)}{\pi \cdot s}}{r} \]

   7. pow2 N/A

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{5}{72} - \frac{1}{6} \cdot \frac{1}{s}, r, \frac{1}{4}\right)}{\pi \cdot s}}{r} \]

   8. lower-/.f32 N/A

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{5}{72} - \frac{1}{6} \cdot \frac{1}{s}, r, \frac{1}{4}\right)}{\pi \cdot s}}{r} \]

   9. lift-*.f32 N/A

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{5}{72} - \frac{1}{6} \cdot \frac{1}{s}, r, \frac{1}{4}\right)}{\pi \cdot s}}{r} \]

   10. associate-*r/ N/A

       \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{5}{72} - \frac{\frac{1}{6} \cdot 1}{s}, r, \frac{1}{4}\right)}{\pi \cdot s}}{r} \]

   11. metadata-eval N/A

       \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{5}{72} - \frac{\frac{1}{6}}{s}, r, \frac{1}{4}\right)}{\pi \cdot s}}{r} \]

   12. lower-/.f32 9.3

       \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot 0.06944444444444445 - \frac{0.16666666666666666}{s}, r, 0.25\right)}{\pi \cdot s}}{r} \]

7. Applied rewrites 9.3%

   \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot 0.06944444444444445 - \frac{0.16666666666666666}{s}, r, 0.25\right)}{\pi \cdot s}}{r} \]
8. Step-by-step derivation

   1. lift-/.f32 N/A

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{5}{72} - \frac{\frac{1}{6}}{s}, r, \frac{1}{4}\right)}{\pi \cdot s}}{\color{blue}{r}} \]

   2. lift-/.f32 N/A

      \[\leadsto \frac{\frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{5}{72} - \frac{\frac{1}{6}}{s}, r, \frac{1}{4}\right)}{\pi \cdot s}}{r} \]

   3. associate-/l/ N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{5}{72} - \frac{\frac{1}{6}}{s}, r, \frac{1}{4}\right)}{\color{blue}{\left(\pi \cdot s\right) \cdot r}} \]

   4. lift-PI.f32 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{5}{72} - \frac{\frac{1}{6}}{s}, r, \frac{1}{4}\right)}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} \]

   5. lift-*.f32 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{5}{72} - \frac{\frac{1}{6}}{s}, r, \frac{1}{4}\right)}{\left(\mathsf{PI}\left(\right) \cdot s\right) \cdot r} \]

   6. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{5}{72} - \frac{\frac{1}{6}}{s}, r, \frac{1}{4}\right)}{\left(s \cdot \mathsf{PI}\left(\right)\right) \cdot r} \]

   7. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{5}{72} - \frac{\frac{1}{6}}{s}, r, \frac{1}{4}\right)}{r \cdot \color{blue}{\left(s \cdot \mathsf{PI}\left(\right)\right)}} \]

   8. lower-/.f32 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot \frac{5}{72} - \frac{\frac{1}{6}}{s}, r, \frac{1}{4}\right)}{\color{blue}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}} \]

9. Applied rewrites 9.3%

   \[\leadsto \frac{\mathsf{fma}\left(\frac{r}{s \cdot s} \cdot 0.06944444444444445 - \frac{0.16666666666666666}{s}, r, 0.25\right)}{\color{blue}{\left(\pi \cdot s\right) \cdot r}} \]

10. Taylor expanded in s around inf

    \[\leadsto \frac{\frac{1}{4} + \left(\frac{-1}{8} \cdot \frac{r}{s} + \frac{-1}{24} \cdot \frac{r}{s}\right)}{\color{blue}{\left(\pi \cdot s\right)} \cdot r} \]
11. Step-by-step derivation

    1. fp-cancel-sign-sub-inv N/A

       \[\leadsto \frac{\frac{1}{4} + \left(\frac{-1}{8} \cdot \frac{r}{s} + \frac{-1}{24} \cdot \frac{r}{s}\right)}{\left(\color{blue}{\pi} \cdot s\right) \cdot r} \]

    2. metadata-eval N/A

       \[\leadsto \frac{\frac{1}{4} + \left(\frac{-1}{8} \cdot \frac{r}{s} + \frac{-1}{24} \cdot \frac{r}{s}\right)}{\left(\pi \cdot s\right) \cdot r} \]

    3. *-commutative N/A

       \[\leadsto \frac{\frac{1}{4} + \left(\frac{-1}{8} \cdot \frac{r}{s} + \frac{-1}{24} \cdot \frac{r}{s}\right)}{\left(\pi \cdot s\right) \cdot r} \]

    4. +-commutative N/A

       \[\leadsto \frac{\left(\frac{-1}{8} \cdot \frac{r}{s} + \frac{-1}{24} \cdot \frac{r}{s}\right) + \frac{1}{4}}{\left(\pi \cdot \color{blue}{s}\right) \cdot r} \]

    5. distribute-rgt-out N/A

       \[\leadsto \frac{\frac{r}{s} \cdot \left(\frac{-1}{8} + \frac{-1}{24}\right) + \frac{1}{4}}{\left(\pi \cdot s\right) \cdot r} \]

    6. metadata-eval N/A

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

    7. *-commutative N/A

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

    8. lower-fma.f32 N/A

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

    9. lift-/.f32 8.4

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

12. Applied rewrites 8.4%

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

Alternative 13: 8.4% 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 99.6%

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

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. *-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 8.5

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

4. Applied rewrites 8.5%

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

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

   10. lift-PI.f32 8.5

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

6. Applied rewrites 8.5%

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

Reproduce

herbie shell --seed 2025110 
(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))))