Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.6%

Time: 5.7s

Alternatives: 20

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: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = (single(0.75) * (exp((-r / (single(3.0) * s))) / (((single(6.0) * single(pi)) * s) * r))) + ((exp((-r / s)) / ((single(pi) * s) * r)) * single(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. Step-by-step derivation

   1. lift-*.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. lower-*.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. lift-PI.f32 99.5

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

3. Applied rewrites 99.5%

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

4. Taylor expanded in s around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. mul-1-neg N/A

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

   4. distribute-frac-neg N/A

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

   5. lower-/.f32 N/A

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

   6. lift-neg.f32 N/A

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

   7. lift-/.f32 N/A

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

   8. lift-exp.f32 N/A

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

   9. *-commutative N/A

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

   10. *-commutative N/A

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

   11. lift-*.f32 N/A

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

   12. lift-PI.f32 N/A

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

   13. lift-*.f32 99.5

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

6. Applied rewrites 99.5%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f32 N/A

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

   5. lower-/.f32 99.5

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

8. Applied rewrites 99.5%

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

   1. lift-+.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-exp.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. lift-/.f32 N/A

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

   7. lift-neg.f32 N/A

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

   8. lift-*.f32 N/A

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

   9. lift-PI.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. lift-*.f32 N/A

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

10. Applied rewrites 99.6%

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

Alternative 2: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = ((exp((-r / s)) / ((single(pi) * s) * r)) * single(0.125)) + ((single(0.75) * exp((-r / (single(3.0) * s)))) / (((single(6.0) * 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. Step-by-step derivation

   1. lift-*.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. lower-*.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. lift-PI.f32 99.5

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

3. Applied rewrites 99.5%

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

4. Taylor expanded in s around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. mul-1-neg N/A

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

   4. distribute-frac-neg N/A

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

   5. lower-/.f32 N/A

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

   6. lift-neg.f32 N/A

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

   7. lift-/.f32 N/A

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

   8. lift-exp.f32 N/A

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

   9. *-commutative N/A

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

   10. *-commutative N/A

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

   11. lift-*.f32 N/A

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

   12. lift-PI.f32 N/A

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

   13. lift-*.f32 99.5

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

6. Applied rewrites 99.5%

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

Alternative 3: 99.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{e^{\frac{-r}{s}}}{\left(\pi \cdot s\right) \cdot r} \cdot 0.125 + \frac{0.75 \cdot e^{\frac{-0.3333333333333333 \cdot r}{s}}}{\left(\left(6 \cdot s\right) \cdot \pi\right) \cdot r} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (* (/ (exp (/ (- r) s)) (* (* PI s) r)) 0.125)
  (/ (* 0.75 (exp (/ (* -0.3333333333333333 r) s))) (* (* (* 6.0 s) PI) r))))

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = ((exp((-r / s)) / ((single(pi) * s) * r)) * single(0.125)) + ((single(0.75) * exp(((single(-0.3333333333333333) * r) / s))) / (((single(6.0) * 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. Step-by-step derivation

   1. lift-*.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. lower-*.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. lift-PI.f32 99.5

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

3. Applied rewrites 99.5%

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

4. Taylor expanded in s around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. mul-1-neg N/A

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

   4. distribute-frac-neg N/A

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

   5. lower-/.f32 N/A

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

   6. lift-neg.f32 N/A

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

   7. lift-/.f32 N/A

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

   8. lift-exp.f32 N/A

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

   9. *-commutative N/A

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

   10. *-commutative N/A

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

   11. lift-*.f32 N/A

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

   12. lift-PI.f32 N/A

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

   13. lift-*.f32 99.5

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

6. Applied rewrites 99.5%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f32 N/A

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

   5. lower-/.f32 99.5

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

8. Applied rewrites 99.5%

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

9. Taylor expanded in r around 0

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

    1. lower-*.f32 99.5

       \[\leadsto \frac{e^{\frac{-r}{s}}}{\left(\pi \cdot s\right) \cdot r} \cdot 0.125 + \frac{0.75 \cdot e^{\frac{-0.3333333333333333 \cdot \color{blue}{r}}{s}}}{\left(\left(6 \cdot s\right) \cdot \pi\right) \cdot r} \]

11. Applied rewrites 99.5%

    \[\leadsto \frac{e^{\frac{-r}{s}}}{\left(\pi \cdot s\right) \cdot r} \cdot 0.125 + \frac{0.75 \cdot e^{\frac{\color{blue}{-0.3333333333333333 \cdot r}}{s}}}{\left(\left(6 \cdot s\right) \cdot \pi\right) \cdot r} \]
12. Add Preprocessing

Alternative 4: 99.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.125 \cdot e^{\frac{-0.3333333333333333 \cdot r}{s}}}{\left(\pi \cdot s\right) \cdot r} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/ (* 0.25 (exp (/ (- r) s))) (* (* (* 2.0 PI) s) r))
  (/ (* 0.125 (exp (/ (* -0.3333333333333333 r) s))) (* (* PI s) r))))

C

float code(float s, float r) {
	return ((0.25f * expf((-r / s))) / (((2.0f * ((float) M_PI)) * s) * r)) + ((0.125f * expf(((-0.3333333333333333f * r) / s))) / ((((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(0.125) * exp(Float32(Float32(Float32(-0.3333333333333333) * r) / s))) / Float32(Float32(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(0.125) * 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 s around 0

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

   1. associate-*r/ N/A

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

   2. lower-/.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-exp.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-/.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f32 N/A

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

   11. lift-PI.f32 99.5

       \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.125 \cdot e^{-0.3333333333333333 \cdot \frac{r}{s}}}{\left(\pi \cdot s\right) \cdot r} \]

4. Applied rewrites 99.5%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \color{blue}{\frac{0.125 \cdot e^{-0.3333333333333333 \cdot \frac{r}{s}}}{\left(\pi \cdot s\right) \cdot r}} \]
5. Step-by-step derivation

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-*r/ N/A

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

   4. metadata-eval N/A

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

   5. lower-/.f32 N/A

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

   6. metadata-eval N/A

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

   7. lower-*.f32 99.5

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.125 \cdot e^{\frac{-0.3333333333333333 \cdot r}{s}}}{\left(\pi \cdot s\right) \cdot r} \]

6. Applied rewrites 99.5%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.125 \cdot e^{\frac{-0.3333333333333333 \cdot r}{s}}}{\left(\pi \cdot s\right) \cdot r} \]
7. Add Preprocessing

Alternative 5: 99.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\pi + \pi\right) \cdot \left(s \cdot r\right)} + \frac{0.125 \cdot e^{-0.3333333333333333 \cdot \frac{r}{s}}}{\left(\pi \cdot s\right) \cdot r} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/ (* 0.25 (exp (/ (- r) s))) (* (+ PI PI) (* s r)))
  (/ (* 0.125 (exp (* -0.3333333333333333 (/ r s)))) (* (* PI s) r))))

C

float code(float s, float r) {
	return ((0.25f * expf((-r / s))) / ((((float) M_PI) + ((float) M_PI)) * (s * r))) + ((0.125f * expf((-0.3333333333333333f * (r / s)))) / ((((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(pi) + Float32(pi)) * Float32(s * r))) + Float32(Float32(Float32(0.125) * exp(Float32(Float32(-0.3333333333333333) * Float32(r / s)))) / Float32(Float32(Float32(pi) * s) * r)))
end

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.25) * exp((-r / s))) / ((single(pi) + single(pi)) * (s * r))) + ((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 s around 0

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

   1. associate-*r/ N/A

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

   2. lower-/.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-exp.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-/.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f32 N/A

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

   11. lift-PI.f32 99.5

       \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.125 \cdot e^{-0.3333333333333333 \cdot \frac{r}{s}}}{\left(\pi \cdot s\right) \cdot r} \]

4. Applied rewrites 99.5%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \color{blue}{\frac{0.125 \cdot e^{-0.3333333333333333 \cdot \frac{r}{s}}}{\left(\pi \cdot s\right) \cdot r}} \]
5. Step-by-step derivation

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lift-PI.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. count-2-rev N/A

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

   9. lower-+.f32 N/A

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

   10. lift-PI.f32 N/A

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

   11. lift-PI.f32 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f32 99.5

       \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\pi + \pi\right) \cdot \color{blue}{\left(s \cdot r\right)}} + \frac{0.125 \cdot e^{-0.3333333333333333 \cdot \frac{r}{s}}}{\left(\pi \cdot s\right) \cdot r} \]

6. Applied rewrites 99.5%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\color{blue}{\left(\pi + \pi\right) \cdot \left(s \cdot r\right)}} + \frac{0.125 \cdot e^{-0.3333333333333333 \cdot \frac{r}{s}}}{\left(\pi \cdot s\right) \cdot r} \]
7. Add Preprocessing

Alternative 6: 99.5% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = (single(0.125) * ((exp((-r / s)) / (single(pi) * r)) + (exp((single(-0.3333333333333333) * (r / s))) / (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 0

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

   1. lower-/.f32 N/A

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

4. Applied rewrites 99.5%

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

Alternative 7: 41.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 3.200000047683716:\\ \;\;\;\;-\frac{\left(-\frac{\left(-\frac{\left(\frac{r}{\pi \cdot s} \cdot 0.021604938271604937 - \frac{0.06944444444444445}{\pi}\right) \cdot r}{s}\right) - \frac{0.16666666666666666}{\pi}}{s}\right) - \frac{\frac{0.25}{r}}{\pi}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{\log \left({\left(e^{r}\right)}^{\left(\pi \cdot s\right)}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (if (<= r 3.200000047683716)
   (-
    (/
     (-
      (-
       (/
        (-
         (-
          (/
           (*
            (-
             (* (/ r (* PI s)) 0.021604938271604937)
             (/ 0.06944444444444445 PI))
            r)
           s))
         (/ 0.16666666666666666 PI))
        s))
      (/ (/ 0.25 r) PI))
     s))
   (/ 0.25 (log (pow (exp r) (* PI s))))))

C

float code(float s, float r) {
	float tmp;
	if (r <= 3.200000047683716f) {
		tmp = -((-((-(((((r / (((float) M_PI) * s)) * 0.021604938271604937f) - (0.06944444444444445f / ((float) M_PI))) * r) / s) - (0.16666666666666666f / ((float) M_PI))) / s) - ((0.25f / r) / ((float) M_PI))) / s);
	} else {
		tmp = 0.25f / logf(powf(expf(r), (((float) M_PI) * s)));
	}
	return tmp;
}

Julia

function code(s, r)
	tmp = Float32(0.0)
	if (r <= Float32(3.200000047683716))
		tmp = Float32(-Float32(Float32(Float32(-Float32(Float32(Float32(-Float32(Float32(Float32(Float32(Float32(r / Float32(Float32(pi) * s)) * Float32(0.021604938271604937)) - Float32(Float32(0.06944444444444445) / Float32(pi))) * r) / s)) - Float32(Float32(0.16666666666666666) / Float32(pi))) / s)) - Float32(Float32(Float32(0.25) / r) / Float32(pi))) / s));
	else
		tmp = Float32(Float32(0.25) / log((exp(r) ^ Float32(Float32(pi) * s))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(s, r)
	tmp = single(0.0);
	if (r <= single(3.200000047683716))
		tmp = -((-((-(((((r / (single(pi) * s)) * single(0.021604938271604937)) - (single(0.06944444444444445) / single(pi))) * r) / s) - (single(0.16666666666666666) / single(pi))) / s) - ((single(0.25) / r) / single(pi))) / s);
	else
		tmp = single(0.25) / log((exp(r) ^ (single(pi) * s)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if r < 3.20000005

   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

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{\frac{-1}{48} \cdot \frac{{r}^{2}}{\mathsf{PI}\left(\right)} + \frac{-1}{1296} \cdot \frac{{r}^{2}}{\mathsf{PI}\left(\right)}}{s} + \left(\frac{-1}{16} \cdot \frac{r}{\mathsf{PI}\left(\right)} + \frac{-1}{144} \cdot \frac{r}{\mathsf{PI}\left(\right)}\right)}{s} - \frac{1}{6} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{s} - \frac{1}{4} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s}} \]

   4. Applied rewrites 9.8%

      \[\leadsto \color{blue}{-\frac{\left(-\frac{\left(-\frac{\mathsf{fma}\left(\frac{r}{\pi}, -0.06944444444444445, -\frac{\frac{r \cdot r}{\pi} \cdot -0.021604938271604937}{s}\right)}{s}\right) - \frac{0.16666666666666666}{\pi}}{s}\right) - \frac{0.25}{\pi \cdot r}}{s}} \]
   5. Step-by-step derivation

      1. lift-/.f32 N/A

         \[\leadsto -\frac{\left(-\frac{\left(-\frac{\mathsf{fma}\left(\frac{r}{\pi}, \frac{-5}{72}, -\frac{\frac{r \cdot r}{\pi} \cdot \frac{-7}{324}}{s}\right)}{s}\right) - \frac{\frac{1}{6}}{\pi}}{s}\right) - \frac{\frac{1}{4}}{\pi \cdot r}}{s} \]

      2. lift-PI.f32 N/A

         \[\leadsto -\frac{\left(-\frac{\left(-\frac{\mathsf{fma}\left(\frac{r}{\pi}, \frac{-5}{72}, -\frac{\frac{r \cdot r}{\pi} \cdot \frac{-7}{324}}{s}\right)}{s}\right) - \frac{\frac{1}{6}}{\pi}}{s}\right) - \frac{\frac{1}{4}}{\mathsf{PI}\left(\right) \cdot r}}{s} \]

      3. lift-*.f32 N/A

         \[\leadsto -\frac{\left(-\frac{\left(-\frac{\mathsf{fma}\left(\frac{r}{\pi}, \frac{-5}{72}, -\frac{\frac{r \cdot r}{\pi} \cdot \frac{-7}{324}}{s}\right)}{s}\right) - \frac{\frac{1}{6}}{\pi}}{s}\right) - \frac{\frac{1}{4}}{\mathsf{PI}\left(\right) \cdot r}}{s} \]

      4. *-commutative N/A

         \[\leadsto -\frac{\left(-\frac{\left(-\frac{\mathsf{fma}\left(\frac{r}{\pi}, \frac{-5}{72}, -\frac{\frac{r \cdot r}{\pi} \cdot \frac{-7}{324}}{s}\right)}{s}\right) - \frac{\frac{1}{6}}{\pi}}{s}\right) - \frac{\frac{1}{4}}{r \cdot \mathsf{PI}\left(\right)}}{s} \]

      5. associate-/r* N/A

         \[\leadsto -\frac{\left(-\frac{\left(-\frac{\mathsf{fma}\left(\frac{r}{\pi}, \frac{-5}{72}, -\frac{\frac{r \cdot r}{\pi} \cdot \frac{-7}{324}}{s}\right)}{s}\right) - \frac{\frac{1}{6}}{\pi}}{s}\right) - \frac{\frac{\frac{1}{4}}{r}}{\mathsf{PI}\left(\right)}}{s} \]

      6. lower-/.f32 N/A

         \[\leadsto -\frac{\left(-\frac{\left(-\frac{\mathsf{fma}\left(\frac{r}{\pi}, \frac{-5}{72}, -\frac{\frac{r \cdot r}{\pi} \cdot \frac{-7}{324}}{s}\right)}{s}\right) - \frac{\frac{1}{6}}{\pi}}{s}\right) - \frac{\frac{\frac{1}{4}}{r}}{\mathsf{PI}\left(\right)}}{s} \]

      7. lower-/.f32 N/A

         \[\leadsto -\frac{\left(-\frac{\left(-\frac{\mathsf{fma}\left(\frac{r}{\pi}, \frac{-5}{72}, -\frac{\frac{r \cdot r}{\pi} \cdot \frac{-7}{324}}{s}\right)}{s}\right) - \frac{\frac{1}{6}}{\pi}}{s}\right) - \frac{\frac{\frac{1}{4}}{r}}{\mathsf{PI}\left(\right)}}{s} \]

      8. lift-PI.f32 9.8

         \[\leadsto -\frac{\left(-\frac{\left(-\frac{\mathsf{fma}\left(\frac{r}{\pi}, -0.06944444444444445, -\frac{\frac{r \cdot r}{\pi} \cdot -0.021604938271604937}{s}\right)}{s}\right) - \frac{0.16666666666666666}{\pi}}{s}\right) - \frac{\frac{0.25}{r}}{\pi}}{s} \]

   6. Applied rewrites 9.8%

      \[\leadsto -\frac{\left(-\frac{\left(-\frac{\mathsf{fma}\left(\frac{r}{\pi}, -0.06944444444444445, -\frac{\frac{r \cdot r}{\pi} \cdot -0.021604938271604937}{s}\right)}{s}\right) - \frac{0.16666666666666666}{\pi}}{s}\right) - \frac{\frac{0.25}{r}}{\pi}}{s} \]

   7. Taylor expanded in r around 0

      \[\leadsto -\frac{\left(-\frac{\left(-\frac{r \cdot \left(\frac{7}{324} \cdot \frac{r}{s \cdot \mathsf{PI}\left(\right)} - \frac{5}{72} \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{s}\right) - \frac{\frac{1}{6}}{\pi}}{s}\right) - \frac{\frac{\frac{1}{4}}{r}}{\pi}}{s} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto -\frac{\left(-\frac{\left(-\frac{\left(\frac{7}{324} \cdot \frac{r}{s \cdot \mathsf{PI}\left(\right)} - \frac{5}{72} \cdot \frac{1}{\mathsf{PI}\left(\right)}\right) \cdot r}{s}\right) - \frac{\frac{1}{6}}{\pi}}{s}\right) - \frac{\frac{\frac{1}{4}}{r}}{\pi}}{s} \]

      2. lower-*.f32 N/A

         \[\leadsto -\frac{\left(-\frac{\left(-\frac{\left(\frac{7}{324} \cdot \frac{r}{s \cdot \mathsf{PI}\left(\right)} - \frac{5}{72} \cdot \frac{1}{\mathsf{PI}\left(\right)}\right) \cdot r}{s}\right) - \frac{\frac{1}{6}}{\pi}}{s}\right) - \frac{\frac{\frac{1}{4}}{r}}{\pi}}{s} \]

      3. lower--.f32 N/A

         \[\leadsto -\frac{\left(-\frac{\left(-\frac{\left(\frac{7}{324} \cdot \frac{r}{s \cdot \mathsf{PI}\left(\right)} - \frac{5}{72} \cdot \frac{1}{\mathsf{PI}\left(\right)}\right) \cdot r}{s}\right) - \frac{\frac{1}{6}}{\pi}}{s}\right) - \frac{\frac{\frac{1}{4}}{r}}{\pi}}{s} \]

      4. *-commutative N/A

         \[\leadsto -\frac{\left(-\frac{\left(-\frac{\left(\frac{r}{s \cdot \mathsf{PI}\left(\right)} \cdot \frac{7}{324} - \frac{5}{72} \cdot \frac{1}{\mathsf{PI}\left(\right)}\right) \cdot r}{s}\right) - \frac{\frac{1}{6}}{\pi}}{s}\right) - \frac{\frac{\frac{1}{4}}{r}}{\pi}}{s} \]

      5. lower-*.f32 N/A

         \[\leadsto -\frac{\left(-\frac{\left(-\frac{\left(\frac{r}{s \cdot \mathsf{PI}\left(\right)} \cdot \frac{7}{324} - \frac{5}{72} \cdot \frac{1}{\mathsf{PI}\left(\right)}\right) \cdot r}{s}\right) - \frac{\frac{1}{6}}{\pi}}{s}\right) - \frac{\frac{\frac{1}{4}}{r}}{\pi}}{s} \]

      6. lower-/.f32 N/A

         \[\leadsto -\frac{\left(-\frac{\left(-\frac{\left(\frac{r}{s \cdot \mathsf{PI}\left(\right)} \cdot \frac{7}{324} - \frac{5}{72} \cdot \frac{1}{\mathsf{PI}\left(\right)}\right) \cdot r}{s}\right) - \frac{\frac{1}{6}}{\pi}}{s}\right) - \frac{\frac{\frac{1}{4}}{r}}{\pi}}{s} \]

      7. *-commutative N/A

         \[\leadsto -\frac{\left(-\frac{\left(-\frac{\left(\frac{r}{\mathsf{PI}\left(\right) \cdot s} \cdot \frac{7}{324} - \frac{5}{72} \cdot \frac{1}{\mathsf{PI}\left(\right)}\right) \cdot r}{s}\right) - \frac{\frac{1}{6}}{\pi}}{s}\right) - \frac{\frac{\frac{1}{4}}{r}}{\pi}}{s} \]

      8. lift-*.f32 N/A

         \[\leadsto -\frac{\left(-\frac{\left(-\frac{\left(\frac{r}{\mathsf{PI}\left(\right) \cdot s} \cdot \frac{7}{324} - \frac{5}{72} \cdot \frac{1}{\mathsf{PI}\left(\right)}\right) \cdot r}{s}\right) - \frac{\frac{1}{6}}{\pi}}{s}\right) - \frac{\frac{\frac{1}{4}}{r}}{\pi}}{s} \]

      9. lift-PI.f32 N/A

         \[\leadsto -\frac{\left(-\frac{\left(-\frac{\left(\frac{r}{\pi \cdot s} \cdot \frac{7}{324} - \frac{5}{72} \cdot \frac{1}{\mathsf{PI}\left(\right)}\right) \cdot r}{s}\right) - \frac{\frac{1}{6}}{\pi}}{s}\right) - \frac{\frac{\frac{1}{4}}{r}}{\pi}}{s} \]

      10. associate-*r/ N/A

          \[\leadsto -\frac{\left(-\frac{\left(-\frac{\left(\frac{r}{\pi \cdot s} \cdot \frac{7}{324} - \frac{\frac{5}{72} \cdot 1}{\mathsf{PI}\left(\right)}\right) \cdot r}{s}\right) - \frac{\frac{1}{6}}{\pi}}{s}\right) - \frac{\frac{\frac{1}{4}}{r}}{\pi}}{s} \]

      11. metadata-eval N/A

          \[\leadsto -\frac{\left(-\frac{\left(-\frac{\left(\frac{r}{\pi \cdot s} \cdot \frac{7}{324} - \frac{\frac{5}{72}}{\mathsf{PI}\left(\right)}\right) \cdot r}{s}\right) - \frac{\frac{1}{6}}{\pi}}{s}\right) - \frac{\frac{\frac{1}{4}}{r}}{\pi}}{s} \]

      12. lower-/.f32 N/A

          \[\leadsto -\frac{\left(-\frac{\left(-\frac{\left(\frac{r}{\pi \cdot s} \cdot \frac{7}{324} - \frac{\frac{5}{72}}{\mathsf{PI}\left(\right)}\right) \cdot r}{s}\right) - \frac{\frac{1}{6}}{\pi}}{s}\right) - \frac{\frac{\frac{1}{4}}{r}}{\pi}}{s} \]

      13. lift-PI.f32 9.8

          \[\leadsto -\frac{\left(-\frac{\left(-\frac{\left(\frac{r}{\pi \cdot s} \cdot 0.021604938271604937 - \frac{0.06944444444444445}{\pi}\right) \cdot r}{s}\right) - \frac{0.16666666666666666}{\pi}}{s}\right) - \frac{\frac{0.25}{r}}{\pi}}{s} \]

   9. Applied rewrites 9.8%

      \[\leadsto -\frac{\left(-\frac{\left(-\frac{\left(\frac{r}{\pi \cdot s} \cdot 0.021604938271604937 - \frac{0.06944444444444445}{\pi}\right) \cdot r}{s}\right) - \frac{0.16666666666666666}{\pi}}{s}\right) - \frac{\frac{0.25}{r}}{\pi}}{s} \]

   if 3.20000005 < r

   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 9.1

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

   4. Applied rewrites 9.1%

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

      1. lift-*.f32 N/A

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

      2. lift-PI.f32 N/A

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

      3. lift-*.f32 N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. add-log-exp N/A

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

      8. log-pow-rev N/A

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

      9. lower-log.f32 N/A

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

      10. lower-pow.f32 N/A

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

      11. lower-exp.f32 N/A

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

      12. lift-PI.f32 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f32 10.2

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

   6. Applied rewrites 10.2%

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

      1. lift-*.f32 N/A

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

      2. lift-pow.f32 N/A

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

      3. lift-PI.f32 N/A

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

      4. lift-exp.f32 N/A

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

      5. pow-exp N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. exp-prod N/A

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

      10. lower-pow.f32 N/A

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

      11. lower-exp.f32 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f32 N/A

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

      14. lift-PI.f32 38.2

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

   8. Applied rewrites 38.2%

      \[\leadsto \frac{0.25}{\log \left({\left(e^{r}\right)}^{\left(\pi \cdot s\right)}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 41.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 3.200000047683716:\\ \;\;\;\;-\frac{\left(-\frac{\left(-\frac{\left(\frac{r}{\pi \cdot s} \cdot 0.021604938271604937 - \frac{0.06944444444444445}{\pi}\right) \cdot r}{s}\right) - \frac{0.16666666666666666}{\pi}}{s}\right) - \frac{0.25}{\pi \cdot r}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{\log \left({\left(e^{r}\right)}^{\left(\pi \cdot s\right)}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (if (<= r 3.200000047683716)
   (-
    (/
     (-
      (-
       (/
        (-
         (-
          (/
           (*
            (-
             (* (/ r (* PI s)) 0.021604938271604937)
             (/ 0.06944444444444445 PI))
            r)
           s))
         (/ 0.16666666666666666 PI))
        s))
      (/ 0.25 (* PI r)))
     s))
   (/ 0.25 (log (pow (exp r) (* PI s))))))

C

float code(float s, float r) {
	float tmp;
	if (r <= 3.200000047683716f) {
		tmp = -((-((-(((((r / (((float) M_PI) * s)) * 0.021604938271604937f) - (0.06944444444444445f / ((float) M_PI))) * r) / s) - (0.16666666666666666f / ((float) M_PI))) / s) - (0.25f / (((float) M_PI) * r))) / s);
	} else {
		tmp = 0.25f / logf(powf(expf(r), (((float) M_PI) * s)));
	}
	return tmp;
}

Julia

function code(s, r)
	tmp = Float32(0.0)
	if (r <= Float32(3.200000047683716))
		tmp = Float32(-Float32(Float32(Float32(-Float32(Float32(Float32(-Float32(Float32(Float32(Float32(Float32(r / Float32(Float32(pi) * s)) * Float32(0.021604938271604937)) - Float32(Float32(0.06944444444444445) / Float32(pi))) * r) / s)) - Float32(Float32(0.16666666666666666) / Float32(pi))) / s)) - Float32(Float32(0.25) / Float32(Float32(pi) * r))) / s));
	else
		tmp = Float32(Float32(0.25) / log((exp(r) ^ Float32(Float32(pi) * s))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(s, r)
	tmp = single(0.0);
	if (r <= single(3.200000047683716))
		tmp = -((-((-(((((r / (single(pi) * s)) * single(0.021604938271604937)) - (single(0.06944444444444445) / single(pi))) * r) / s) - (single(0.16666666666666666) / single(pi))) / s) - (single(0.25) / (single(pi) * r))) / s);
	else
		tmp = single(0.25) / log((exp(r) ^ (single(pi) * s)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if r < 3.20000005

   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

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{\frac{-1}{48} \cdot \frac{{r}^{2}}{\mathsf{PI}\left(\right)} + \frac{-1}{1296} \cdot \frac{{r}^{2}}{\mathsf{PI}\left(\right)}}{s} + \left(\frac{-1}{16} \cdot \frac{r}{\mathsf{PI}\left(\right)} + \frac{-1}{144} \cdot \frac{r}{\mathsf{PI}\left(\right)}\right)}{s} - \frac{1}{6} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{s} - \frac{1}{4} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s}} \]

   4. Applied rewrites 9.8%

      \[\leadsto \color{blue}{-\frac{\left(-\frac{\left(-\frac{\mathsf{fma}\left(\frac{r}{\pi}, -0.06944444444444445, -\frac{\frac{r \cdot r}{\pi} \cdot -0.021604938271604937}{s}\right)}{s}\right) - \frac{0.16666666666666666}{\pi}}{s}\right) - \frac{0.25}{\pi \cdot r}}{s}} \]

   5. Taylor expanded in r around 0

      \[\leadsto -\frac{\left(-\frac{\left(-\frac{r \cdot \left(\frac{7}{324} \cdot \frac{r}{s \cdot \mathsf{PI}\left(\right)} - \frac{5}{72} \cdot \frac{1}{\mathsf{PI}\left(\right)}\right)}{s}\right) - \frac{\frac{1}{6}}{\pi}}{s}\right) - \frac{\frac{1}{4}}{\pi \cdot r}}{s} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto -\frac{\left(-\frac{\left(-\frac{\left(\frac{7}{324} \cdot \frac{r}{s \cdot \mathsf{PI}\left(\right)} - \frac{5}{72} \cdot \frac{1}{\mathsf{PI}\left(\right)}\right) \cdot r}{s}\right) - \frac{\frac{1}{6}}{\pi}}{s}\right) - \frac{\frac{1}{4}}{\pi \cdot r}}{s} \]

      2. lower-*.f32 N/A

         \[\leadsto -\frac{\left(-\frac{\left(-\frac{\left(\frac{7}{324} \cdot \frac{r}{s \cdot \mathsf{PI}\left(\right)} - \frac{5}{72} \cdot \frac{1}{\mathsf{PI}\left(\right)}\right) \cdot r}{s}\right) - \frac{\frac{1}{6}}{\pi}}{s}\right) - \frac{\frac{1}{4}}{\pi \cdot r}}{s} \]

      3. lower--.f32 N/A

         \[\leadsto -\frac{\left(-\frac{\left(-\frac{\left(\frac{7}{324} \cdot \frac{r}{s \cdot \mathsf{PI}\left(\right)} - \frac{5}{72} \cdot \frac{1}{\mathsf{PI}\left(\right)}\right) \cdot r}{s}\right) - \frac{\frac{1}{6}}{\pi}}{s}\right) - \frac{\frac{1}{4}}{\pi \cdot r}}{s} \]

      4. *-commutative N/A

         \[\leadsto -\frac{\left(-\frac{\left(-\frac{\left(\frac{r}{s \cdot \mathsf{PI}\left(\right)} \cdot \frac{7}{324} - \frac{5}{72} \cdot \frac{1}{\mathsf{PI}\left(\right)}\right) \cdot r}{s}\right) - \frac{\frac{1}{6}}{\pi}}{s}\right) - \frac{\frac{1}{4}}{\pi \cdot r}}{s} \]

      5. lower-*.f32 N/A

         \[\leadsto -\frac{\left(-\frac{\left(-\frac{\left(\frac{r}{s \cdot \mathsf{PI}\left(\right)} \cdot \frac{7}{324} - \frac{5}{72} \cdot \frac{1}{\mathsf{PI}\left(\right)}\right) \cdot r}{s}\right) - \frac{\frac{1}{6}}{\pi}}{s}\right) - \frac{\frac{1}{4}}{\pi \cdot r}}{s} \]

      6. lower-/.f32 N/A

         \[\leadsto -\frac{\left(-\frac{\left(-\frac{\left(\frac{r}{s \cdot \mathsf{PI}\left(\right)} \cdot \frac{7}{324} - \frac{5}{72} \cdot \frac{1}{\mathsf{PI}\left(\right)}\right) \cdot r}{s}\right) - \frac{\frac{1}{6}}{\pi}}{s}\right) - \frac{\frac{1}{4}}{\pi \cdot r}}{s} \]

      7. *-commutative N/A

         \[\leadsto -\frac{\left(-\frac{\left(-\frac{\left(\frac{r}{\mathsf{PI}\left(\right) \cdot s} \cdot \frac{7}{324} - \frac{5}{72} \cdot \frac{1}{\mathsf{PI}\left(\right)}\right) \cdot r}{s}\right) - \frac{\frac{1}{6}}{\pi}}{s}\right) - \frac{\frac{1}{4}}{\pi \cdot r}}{s} \]

      8. lift-*.f32 N/A

         \[\leadsto -\frac{\left(-\frac{\left(-\frac{\left(\frac{r}{\mathsf{PI}\left(\right) \cdot s} \cdot \frac{7}{324} - \frac{5}{72} \cdot \frac{1}{\mathsf{PI}\left(\right)}\right) \cdot r}{s}\right) - \frac{\frac{1}{6}}{\pi}}{s}\right) - \frac{\frac{1}{4}}{\pi \cdot r}}{s} \]

      9. lift-PI.f32 N/A

         \[\leadsto -\frac{\left(-\frac{\left(-\frac{\left(\frac{r}{\pi \cdot s} \cdot \frac{7}{324} - \frac{5}{72} \cdot \frac{1}{\mathsf{PI}\left(\right)}\right) \cdot r}{s}\right) - \frac{\frac{1}{6}}{\pi}}{s}\right) - \frac{\frac{1}{4}}{\pi \cdot r}}{s} \]

      10. associate-*r/ N/A

          \[\leadsto -\frac{\left(-\frac{\left(-\frac{\left(\frac{r}{\pi \cdot s} \cdot \frac{7}{324} - \frac{\frac{5}{72} \cdot 1}{\mathsf{PI}\left(\right)}\right) \cdot r}{s}\right) - \frac{\frac{1}{6}}{\pi}}{s}\right) - \frac{\frac{1}{4}}{\pi \cdot r}}{s} \]

      11. metadata-eval N/A

          \[\leadsto -\frac{\left(-\frac{\left(-\frac{\left(\frac{r}{\pi \cdot s} \cdot \frac{7}{324} - \frac{\frac{5}{72}}{\mathsf{PI}\left(\right)}\right) \cdot r}{s}\right) - \frac{\frac{1}{6}}{\pi}}{s}\right) - \frac{\frac{1}{4}}{\pi \cdot r}}{s} \]

      12. lower-/.f32 N/A

          \[\leadsto -\frac{\left(-\frac{\left(-\frac{\left(\frac{r}{\pi \cdot s} \cdot \frac{7}{324} - \frac{\frac{5}{72}}{\mathsf{PI}\left(\right)}\right) \cdot r}{s}\right) - \frac{\frac{1}{6}}{\pi}}{s}\right) - \frac{\frac{1}{4}}{\pi \cdot r}}{s} \]

      13. lift-PI.f32 9.8

          \[\leadsto -\frac{\left(-\frac{\left(-\frac{\left(\frac{r}{\pi \cdot s} \cdot 0.021604938271604937 - \frac{0.06944444444444445}{\pi}\right) \cdot r}{s}\right) - \frac{0.16666666666666666}{\pi}}{s}\right) - \frac{0.25}{\pi \cdot r}}{s} \]

   7. Applied rewrites 9.8%

      \[\leadsto -\frac{\left(-\frac{\left(-\frac{\left(\frac{r}{\pi \cdot s} \cdot 0.021604938271604937 - \frac{0.06944444444444445}{\pi}\right) \cdot r}{s}\right) - \frac{0.16666666666666666}{\pi}}{s}\right) - \frac{0.25}{\pi \cdot r}}{s} \]

   if 3.20000005 < r

   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 9.1

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

   4. Applied rewrites 9.1%

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

      1. lift-*.f32 N/A

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

      2. lift-PI.f32 N/A

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

      3. lift-*.f32 N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. add-log-exp N/A

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

      8. log-pow-rev N/A

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

      9. lower-log.f32 N/A

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

      10. lower-pow.f32 N/A

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

      11. lower-exp.f32 N/A

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

      12. lift-PI.f32 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f32 10.2

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

   6. Applied rewrites 10.2%

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

      1. lift-*.f32 N/A

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

      2. lift-pow.f32 N/A

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

      3. lift-PI.f32 N/A

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

      4. lift-exp.f32 N/A

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

      5. pow-exp N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. exp-prod N/A

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

      10. lower-pow.f32 N/A

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

      11. lower-exp.f32 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f32 N/A

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

      14. lift-PI.f32 38.2

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

   8. Applied rewrites 38.2%

      \[\leadsto \frac{0.25}{\log \left({\left(e^{r}\right)}^{\left(\pi \cdot s\right)}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 41.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 82:\\ \;\;\;\;-\frac{\frac{\mathsf{fma}\left(\frac{r}{\left(s \cdot s\right) \cdot \pi}, -0.06944444444444445, \frac{0.16666666666666666}{\pi \cdot s}\right) \cdot r - \frac{0.25}{\pi}}{r}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{\log \left({\left(e^{r}\right)}^{\left(\pi \cdot s\right)}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (if (<= r 82.0)
   (-
    (/
     (/
      (-
       (*
        (fma
         (/ r (* (* s s) PI))
         -0.06944444444444445
         (/ 0.16666666666666666 (* PI s)))
        r)
       (/ 0.25 PI))
      r)
     s))
   (/ 0.25 (log (pow (exp r) (* PI s))))))

C

float code(float s, float r) {
	float tmp;
	if (r <= 82.0f) {
		tmp = -((((fmaf((r / ((s * s) * ((float) M_PI))), -0.06944444444444445f, (0.16666666666666666f / (((float) M_PI) * s))) * r) - (0.25f / ((float) M_PI))) / r) / s);
	} else {
		tmp = 0.25f / logf(powf(expf(r), (((float) M_PI) * s)));
	}
	return tmp;
}

Julia

function code(s, r)
	tmp = Float32(0.0)
	if (r <= Float32(82.0))
		tmp = Float32(-Float32(Float32(Float32(Float32(fma(Float32(r / Float32(Float32(s * s) * Float32(pi))), Float32(-0.06944444444444445), Float32(Float32(0.16666666666666666) / Float32(Float32(pi) * s))) * r) - Float32(Float32(0.25) / Float32(pi))) / r) / s));
	else
		tmp = Float32(Float32(0.25) / log((exp(r) ^ Float32(Float32(pi) * s))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if r < 82

   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

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{\frac{-1}{48} \cdot \frac{{r}^{2}}{\mathsf{PI}\left(\right)} + \frac{-1}{1296} \cdot \frac{{r}^{2}}{\mathsf{PI}\left(\right)}}{s} + \left(\frac{-1}{16} \cdot \frac{r}{\mathsf{PI}\left(\right)} + \frac{-1}{144} \cdot \frac{r}{\mathsf{PI}\left(\right)}\right)}{s} - \frac{1}{6} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{s} - \frac{1}{4} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s}} \]

   4. Applied rewrites 9.8%

      \[\leadsto \color{blue}{-\frac{\left(-\frac{\left(-\frac{\mathsf{fma}\left(\frac{r}{\pi}, -0.06944444444444445, -\frac{\frac{r \cdot r}{\pi} \cdot -0.021604938271604937}{s}\right)}{s}\right) - \frac{0.16666666666666666}{\pi}}{s}\right) - \frac{0.25}{\pi \cdot r}}{s}} \]

   5. Taylor expanded in r around 0

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

      1. lower-/.f32 N/A

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

   7. Applied rewrites 10.0%

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

   if 82 < r

   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 9.1

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

   4. Applied rewrites 9.1%

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

      1. lift-*.f32 N/A

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

      2. lift-PI.f32 N/A

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

      3. lift-*.f32 N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. add-log-exp N/A

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

      8. log-pow-rev N/A

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

      9. lower-log.f32 N/A

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

      10. lower-pow.f32 N/A

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

      11. lower-exp.f32 N/A

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

      12. lift-PI.f32 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f32 10.2

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

   6. Applied rewrites 10.2%

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

      1. lift-*.f32 N/A

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

      2. lift-pow.f32 N/A

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

      3. lift-PI.f32 N/A

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

      4. lift-exp.f32 N/A

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

      5. pow-exp N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. exp-prod N/A

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

      10. lower-pow.f32 N/A

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

      11. lower-exp.f32 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f32 N/A

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

      14. lift-PI.f32 38.2

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

   8. Applied rewrites 38.2%

      \[\leadsto \frac{0.25}{\log \left({\left(e^{r}\right)}^{\left(\pi \cdot s\right)}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 13.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 125:\\ \;\;\;\;-\frac{\frac{\mathsf{fma}\left(\frac{r}{\left(s \cdot s\right) \cdot \pi}, -0.06944444444444445, \frac{0.16666666666666666}{\pi \cdot s}\right) \cdot r - \frac{0.25}{\pi}}{r}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{\log \left(e^{\left(\pi \cdot s\right) \cdot r}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (if (<= r 125.0)
   (-
    (/
     (/
      (-
       (*
        (fma
         (/ r (* (* s s) PI))
         -0.06944444444444445
         (/ 0.16666666666666666 (* PI s)))
        r)
       (/ 0.25 PI))
      r)
     s))
   (/ 0.25 (log (exp (* (* PI s) r))))))

C

float code(float s, float r) {
	float tmp;
	if (r <= 125.0f) {
		tmp = -((((fmaf((r / ((s * s) * ((float) M_PI))), -0.06944444444444445f, (0.16666666666666666f / (((float) M_PI) * s))) * r) - (0.25f / ((float) M_PI))) / r) / s);
	} else {
		tmp = 0.25f / logf(expf(((((float) M_PI) * s) * r)));
	}
	return tmp;
}

Julia

function code(s, r)
	tmp = Float32(0.0)
	if (r <= Float32(125.0))
		tmp = Float32(-Float32(Float32(Float32(Float32(fma(Float32(r / Float32(Float32(s * s) * Float32(pi))), Float32(-0.06944444444444445), Float32(Float32(0.16666666666666666) / Float32(Float32(pi) * s))) * r) - Float32(Float32(0.25) / Float32(pi))) / r) / s));
	else
		tmp = Float32(Float32(0.25) / log(exp(Float32(Float32(Float32(pi) * s) * r))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if r < 125

   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

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{\frac{-1}{48} \cdot \frac{{r}^{2}}{\mathsf{PI}\left(\right)} + \frac{-1}{1296} \cdot \frac{{r}^{2}}{\mathsf{PI}\left(\right)}}{s} + \left(\frac{-1}{16} \cdot \frac{r}{\mathsf{PI}\left(\right)} + \frac{-1}{144} \cdot \frac{r}{\mathsf{PI}\left(\right)}\right)}{s} - \frac{1}{6} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{s} - \frac{1}{4} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s}} \]

   4. Applied rewrites 9.8%

      \[\leadsto \color{blue}{-\frac{\left(-\frac{\left(-\frac{\mathsf{fma}\left(\frac{r}{\pi}, -0.06944444444444445, -\frac{\frac{r \cdot r}{\pi} \cdot -0.021604938271604937}{s}\right)}{s}\right) - \frac{0.16666666666666666}{\pi}}{s}\right) - \frac{0.25}{\pi \cdot r}}{s}} \]

   5. Taylor expanded in r around 0

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

      1. lower-/.f32 N/A

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

   7. Applied rewrites 10.0%

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

   if 125 < r

   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 9.1

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

   4. Applied rewrites 9.1%

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

      1. lift-*.f32 N/A

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

      2. lift-PI.f32 N/A

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

      3. lift-*.f32 N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. add-log-exp N/A

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

      8. log-pow-rev N/A

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

      9. lower-log.f32 N/A

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

      10. lower-pow.f32 N/A

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

      11. lower-exp.f32 N/A

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

      12. lift-PI.f32 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f32 10.2

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

   6. Applied rewrites 10.2%

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

      1. lift-*.f32 N/A

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

      2. lift-pow.f32 N/A

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

      3. lift-PI.f32 N/A

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

      4. lift-exp.f32 N/A

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

      5. pow-exp N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-exp.f32 N/A

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

      10. *-commutative N/A

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

      11. *-commutative N/A

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

      12. lift-*.f32 N/A

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

      13. lift-PI.f32 N/A

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

      14. lift-*.f32 10.2

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

   8. Applied rewrites 10.2%

      \[\leadsto \frac{0.25}{\log \left(e^{\left(\pi \cdot s\right) \cdot r}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 13.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 125:\\ \;\;\;\;-\frac{\frac{\mathsf{fma}\left(\frac{r}{\pi \cdot s}, -0.06944444444444445, \frac{0.16666666666666666}{\pi}\right)}{s} - \frac{\frac{0.25}{\pi}}{r}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{\log \left(e^{\left(\pi \cdot s\right) \cdot r}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (if (<= r 125.0)
   (-
    (/
     (-
      (/
       (fma (/ r (* PI s)) -0.06944444444444445 (/ 0.16666666666666666 PI))
       s)
      (/ (/ 0.25 PI) r))
     s))
   (/ 0.25 (log (exp (* (* PI s) r))))))

C

float code(float s, float r) {
	float tmp;
	if (r <= 125.0f) {
		tmp = -(((fmaf((r / (((float) M_PI) * s)), -0.06944444444444445f, (0.16666666666666666f / ((float) M_PI))) / s) - ((0.25f / ((float) M_PI)) / r)) / s);
	} else {
		tmp = 0.25f / logf(expf(((((float) M_PI) * s) * r)));
	}
	return tmp;
}

Julia

function code(s, r)
	tmp = Float32(0.0)
	if (r <= Float32(125.0))
		tmp = Float32(-Float32(Float32(Float32(fma(Float32(r / Float32(Float32(pi) * s)), Float32(-0.06944444444444445), Float32(Float32(0.16666666666666666) / Float32(pi))) / s) - Float32(Float32(Float32(0.25) / Float32(pi)) / r)) / s));
	else
		tmp = Float32(Float32(0.25) / log(exp(Float32(Float32(Float32(pi) * s) * r))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if r < 125

   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

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{\frac{-1}{48} \cdot \frac{{r}^{2}}{\mathsf{PI}\left(\right)} + \frac{-1}{1296} \cdot \frac{{r}^{2}}{\mathsf{PI}\left(\right)}}{s} + \left(\frac{-1}{16} \cdot \frac{r}{\mathsf{PI}\left(\right)} + \frac{-1}{144} \cdot \frac{r}{\mathsf{PI}\left(\right)}\right)}{s} - \frac{1}{6} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{s} - \frac{1}{4} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s}} \]

   4. Applied rewrites 9.8%

      \[\leadsto \color{blue}{-\frac{\left(-\frac{\left(-\frac{\mathsf{fma}\left(\frac{r}{\pi}, -0.06944444444444445, -\frac{\frac{r \cdot r}{\pi} \cdot -0.021604938271604937}{s}\right)}{s}\right) - \frac{0.16666666666666666}{\pi}}{s}\right) - \frac{0.25}{\pi \cdot r}}{s}} \]

   5. Taylor expanded in s around inf

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

      1. lower-/.f32 N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

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

      4. lower-/.f32 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f32 N/A

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

      7. lift-PI.f32 N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. lift-/.f32 N/A

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

      11. lift-PI.f32 10.0

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

   7. Applied rewrites 10.0%

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

      1. lift-/.f32 N/A

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

      2. lift-PI.f32 N/A

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

      3. lift-*.f32 N/A

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

      4. associate-/r* N/A

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

      5. metadata-eval N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f32 N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. lift-/.f32 N/A

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

      11. lift-PI.f32 10.0

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

   9. Applied rewrites 10.0%

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

   if 125 < r

   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 9.1

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

   4. Applied rewrites 9.1%

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

      1. lift-*.f32 N/A

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

      2. lift-PI.f32 N/A

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

      3. lift-*.f32 N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. add-log-exp N/A

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

      8. log-pow-rev N/A

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

      9. lower-log.f32 N/A

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

      10. lower-pow.f32 N/A

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

      11. lower-exp.f32 N/A

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

      12. lift-PI.f32 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f32 10.2

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

   6. Applied rewrites 10.2%

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

      1. lift-*.f32 N/A

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

      2. lift-pow.f32 N/A

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

      3. lift-PI.f32 N/A

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

      4. lift-exp.f32 N/A

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

      5. pow-exp N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-exp.f32 N/A

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

      10. *-commutative N/A

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

      11. *-commutative N/A

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

      12. lift-*.f32 N/A

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

      13. lift-PI.f32 N/A

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

      14. lift-*.f32 10.2

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

   8. Applied rewrites 10.2%

      \[\leadsto \frac{0.25}{\log \left(e^{\left(\pi \cdot s\right) \cdot r}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 13.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 125:\\ \;\;\;\;-\frac{\frac{\frac{0.16666666666666666}{\pi} + \frac{r}{\pi \cdot s} \cdot -0.06944444444444445}{s} - \frac{0.25}{\pi \cdot r}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{\log \left(e^{\left(\pi \cdot s\right) \cdot r}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (if (<= r 125.0)
   (-
    (/
     (-
      (/
       (+ (/ 0.16666666666666666 PI) (* (/ r (* PI s)) -0.06944444444444445))
       s)
      (/ 0.25 (* PI r)))
     s))
   (/ 0.25 (log (exp (* (* PI s) r))))))

C

float code(float s, float r) {
	float tmp;
	if (r <= 125.0f) {
		tmp = -(((((0.16666666666666666f / ((float) M_PI)) + ((r / (((float) M_PI) * s)) * -0.06944444444444445f)) / s) - (0.25f / (((float) M_PI) * r))) / s);
	} else {
		tmp = 0.25f / logf(expf(((((float) M_PI) * s) * r)));
	}
	return tmp;
}

Julia

function code(s, r)
	tmp = Float32(0.0)
	if (r <= Float32(125.0))
		tmp = Float32(-Float32(Float32(Float32(Float32(Float32(Float32(0.16666666666666666) / Float32(pi)) + Float32(Float32(r / Float32(Float32(pi) * s)) * Float32(-0.06944444444444445))) / s) - Float32(Float32(0.25) / Float32(Float32(pi) * r))) / s));
	else
		tmp = Float32(Float32(0.25) / log(exp(Float32(Float32(Float32(pi) * s) * r))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(s, r)
	tmp = single(0.0);
	if (r <= single(125.0))
		tmp = -(((((single(0.16666666666666666) / single(pi)) + ((r / (single(pi) * s)) * single(-0.06944444444444445))) / s) - (single(0.25) / (single(pi) * r))) / s);
	else
		tmp = single(0.25) / log(exp(((single(pi) * s) * r)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if r < 125

   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

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{\frac{-1}{48} \cdot \frac{{r}^{2}}{\mathsf{PI}\left(\right)} + \frac{-1}{1296} \cdot \frac{{r}^{2}}{\mathsf{PI}\left(\right)}}{s} + \left(\frac{-1}{16} \cdot \frac{r}{\mathsf{PI}\left(\right)} + \frac{-1}{144} \cdot \frac{r}{\mathsf{PI}\left(\right)}\right)}{s} - \frac{1}{6} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{s} - \frac{1}{4} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s}} \]

   4. Applied rewrites 9.8%

      \[\leadsto \color{blue}{-\frac{\left(-\frac{\left(-\frac{\mathsf{fma}\left(\frac{r}{\pi}, -0.06944444444444445, -\frac{\frac{r \cdot r}{\pi} \cdot -0.021604938271604937}{s}\right)}{s}\right) - \frac{0.16666666666666666}{\pi}}{s}\right) - \frac{0.25}{\pi \cdot r}}{s}} \]

   5. Taylor expanded in s around inf

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

      1. lower-/.f32 N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

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

      4. lower-/.f32 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f32 N/A

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

      7. lift-PI.f32 N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. lift-/.f32 N/A

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

      11. lift-PI.f32 10.0

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

   7. Applied rewrites 10.0%

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

      1. lift-fma.f32 N/A

         \[\leadsto -\frac{\frac{\frac{r}{\pi \cdot s} \cdot \frac{-5}{72} + \frac{\frac{1}{6}}{\pi}}{s} - \frac{\frac{1}{4}}{\pi \cdot r}}{s} \]

      2. *-commutative N/A

         \[\leadsto -\frac{\frac{\frac{-5}{72} \cdot \frac{r}{\pi \cdot s} + \frac{\frac{1}{6}}{\pi}}{s} - \frac{\frac{1}{4}}{\pi \cdot r}}{s} \]

      3. lift-/.f32 N/A

         \[\leadsto -\frac{\frac{\frac{-5}{72} \cdot \frac{r}{\pi \cdot s} + \frac{\frac{1}{6}}{\pi}}{s} - \frac{\frac{1}{4}}{\pi \cdot r}}{s} \]

      4. lift-PI.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. *-commutative N/A

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

      7. lift-PI.f32 N/A

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

      8. lift-/.f32 N/A

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

      9. metadata-eval N/A

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

      10. associate-*r/ N/A

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

      11. +-commutative N/A

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

      12. lower-+.f32 N/A

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

      13. associate-*r/ N/A

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

      14. metadata-eval N/A

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

      15. lift-/.f32 N/A

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

      16. lift-PI.f32 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f32 N/A

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

      19. lower-/.f32 N/A

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

      20. *-commutative N/A

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

      21. lift-*.f32 N/A

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

      22. lift-PI.f32 10.0

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

   9. Applied rewrites 10.0%

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

   if 125 < r

   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 9.1

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

   4. Applied rewrites 9.1%

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

      1. lift-*.f32 N/A

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

      2. lift-PI.f32 N/A

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

      3. lift-*.f32 N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. add-log-exp N/A

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

      8. log-pow-rev N/A

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

      9. lower-log.f32 N/A

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

      10. lower-pow.f32 N/A

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

      11. lower-exp.f32 N/A

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

      12. lift-PI.f32 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f32 10.2

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

   6. Applied rewrites 10.2%

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

      1. lift-*.f32 N/A

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

      2. lift-pow.f32 N/A

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

      3. lift-PI.f32 N/A

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

      4. lift-exp.f32 N/A

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

      5. pow-exp N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-exp.f32 N/A

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

      10. *-commutative N/A

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

      11. *-commutative N/A

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

      12. lift-*.f32 N/A

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

      13. lift-PI.f32 N/A

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

      14. lift-*.f32 10.2

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

   8. Applied rewrites 10.2%

      \[\leadsto \frac{0.25}{\log \left(e^{\left(\pi \cdot s\right) \cdot r}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 13.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 125:\\ \;\;\;\;-\frac{\frac{\mathsf{fma}\left(\frac{r}{\pi \cdot s}, -0.06944444444444445, \frac{0.16666666666666666}{\pi}\right)}{s} - \frac{0.25}{\pi \cdot r}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{\log \left(e^{\left(\pi \cdot s\right) \cdot r}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (if (<= r 125.0)
   (-
    (/
     (-
      (/
       (fma (/ r (* PI s)) -0.06944444444444445 (/ 0.16666666666666666 PI))
       s)
      (/ 0.25 (* PI r)))
     s))
   (/ 0.25 (log (exp (* (* PI s) r))))))

C

float code(float s, float r) {
	float tmp;
	if (r <= 125.0f) {
		tmp = -(((fmaf((r / (((float) M_PI) * s)), -0.06944444444444445f, (0.16666666666666666f / ((float) M_PI))) / s) - (0.25f / (((float) M_PI) * r))) / s);
	} else {
		tmp = 0.25f / logf(expf(((((float) M_PI) * s) * r)));
	}
	return tmp;
}

Julia

function code(s, r)
	tmp = Float32(0.0)
	if (r <= Float32(125.0))
		tmp = Float32(-Float32(Float32(Float32(fma(Float32(r / Float32(Float32(pi) * s)), Float32(-0.06944444444444445), Float32(Float32(0.16666666666666666) / Float32(pi))) / s) - Float32(Float32(0.25) / Float32(Float32(pi) * r))) / s));
	else
		tmp = Float32(Float32(0.25) / log(exp(Float32(Float32(Float32(pi) * s) * r))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if r < 125

   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

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{\frac{-1}{48} \cdot \frac{{r}^{2}}{\mathsf{PI}\left(\right)} + \frac{-1}{1296} \cdot \frac{{r}^{2}}{\mathsf{PI}\left(\right)}}{s} + \left(\frac{-1}{16} \cdot \frac{r}{\mathsf{PI}\left(\right)} + \frac{-1}{144} \cdot \frac{r}{\mathsf{PI}\left(\right)}\right)}{s} - \frac{1}{6} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{s} - \frac{1}{4} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s}} \]

   4. Applied rewrites 9.8%

      \[\leadsto \color{blue}{-\frac{\left(-\frac{\left(-\frac{\mathsf{fma}\left(\frac{r}{\pi}, -0.06944444444444445, -\frac{\frac{r \cdot r}{\pi} \cdot -0.021604938271604937}{s}\right)}{s}\right) - \frac{0.16666666666666666}{\pi}}{s}\right) - \frac{0.25}{\pi \cdot r}}{s}} \]

   5. Taylor expanded in s around inf

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

      1. lower-/.f32 N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

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

      4. lower-/.f32 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f32 N/A

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

      7. lift-PI.f32 N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. lift-/.f32 N/A

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

      11. lift-PI.f32 10.0

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

   7. Applied rewrites 10.0%

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

   if 125 < r

   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 9.1

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

   4. Applied rewrites 9.1%

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

      1. lift-*.f32 N/A

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

      2. lift-PI.f32 N/A

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

      3. lift-*.f32 N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. add-log-exp N/A

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

      8. log-pow-rev N/A

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

      9. lower-log.f32 N/A

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

      10. lower-pow.f32 N/A

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

      11. lower-exp.f32 N/A

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

      12. lift-PI.f32 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f32 10.2

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

   6. Applied rewrites 10.2%

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

      1. lift-*.f32 N/A

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

      2. lift-pow.f32 N/A

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

      3. lift-PI.f32 N/A

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

      4. lift-exp.f32 N/A

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

      5. pow-exp N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-exp.f32 N/A

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

      10. *-commutative N/A

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

      11. *-commutative N/A

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

      12. lift-*.f32 N/A

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

      13. lift-PI.f32 N/A

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

      14. lift-*.f32 10.2

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

   8. Applied rewrites 10.2%

      \[\leadsto \frac{0.25}{\log \left(e^{\left(\pi \cdot s\right) \cdot r}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 10.2% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

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

   1. lower-/.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lift-PI.f32 9.1

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

4. Applied rewrites 9.1%

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

   1. lift-*.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. add-log-exp N/A

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

   8. log-pow-rev N/A

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

   9. lower-log.f32 N/A

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

   10. lower-pow.f32 N/A

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

   11. lower-exp.f32 N/A

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

   12. lift-PI.f32 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f32 10.2

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

6. Applied rewrites 10.2%

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

   1. lift-*.f32 N/A

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

   2. lift-pow.f32 N/A

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

   3. lift-PI.f32 N/A

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

   4. lift-exp.f32 N/A

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

   5. pow-exp N/A

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

   6. associate-*l* N/A

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

   7. *-commutative N/A

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

   8. *-commutative N/A

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

   9. lower-exp.f32 N/A

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

   10. *-commutative N/A

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

   11. *-commutative N/A

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

   12. lift-*.f32 N/A

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

   13. lift-PI.f32 N/A

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

   14. lift-*.f32 10.2

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

8. Applied rewrites 10.2%

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

Alternative 15: 9.1% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

   \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{\frac{-1}{48} \cdot \frac{{r}^{2}}{\mathsf{PI}\left(\right)} + \frac{-1}{1296} \cdot \frac{{r}^{2}}{\mathsf{PI}\left(\right)}}{s} + \left(\frac{-1}{16} \cdot \frac{r}{\mathsf{PI}\left(\right)} + \frac{-1}{144} \cdot \frac{r}{\mathsf{PI}\left(\right)}\right)}{s} - \frac{1}{6} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{s} - \frac{1}{4} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s}} \]

4. Applied rewrites 9.8%

   \[\leadsto \color{blue}{-\frac{\left(-\frac{\left(-\frac{\mathsf{fma}\left(\frac{r}{\pi}, -0.06944444444444445, -\frac{\frac{r \cdot r}{\pi} \cdot -0.021604938271604937}{s}\right)}{s}\right) - \frac{0.16666666666666666}{\pi}}{s}\right) - \frac{0.25}{\pi \cdot r}}{s}} \]

5. Taylor expanded in r around 0

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

   1. lower-/.f32 N/A

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

   2. lower--.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lower-/.f32 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f32 N/A

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

   8. lift-PI.f32 N/A

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

   9. associate-*r/ N/A

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

   10. metadata-eval N/A

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

   11. lower-/.f32 N/A

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

   12. lift-PI.f32 9.0

       \[\leadsto -\frac{\frac{\frac{r}{\pi \cdot s} \cdot 0.16666666666666666 - \frac{0.25}{\pi}}{r}}{s} \]

7. Applied rewrites 9.0%

   \[\leadsto -\frac{\frac{\frac{r}{\pi \cdot s} \cdot 0.16666666666666666 - \frac{0.25}{\pi}}{r}}{s} \]
8. Add Preprocessing

Alternative 16: 9.1% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

function code(s, r)
	return Float32(Float32(fma(Float32(r / Float32(Float32(pi) * s)), Float32(-0.16666666666666666), Float32(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 0

   \[\leadsto \color{blue}{\frac{r \cdot \left(\frac{5}{72} \cdot \frac{r}{{s}^{3} \cdot \mathsf{PI}\left(\right)} - \frac{1}{6} \cdot \frac{1}{{s}^{2} \cdot \mathsf{PI}\left(\right)}\right) + \frac{1}{4} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}}{r}} \]
3. Step-by-step derivation

   1. lower-/.f32 N/A

      \[\leadsto \frac{r \cdot \left(\frac{5}{72} \cdot \frac{r}{{s}^{3} \cdot \mathsf{PI}\left(\right)} - \frac{1}{6} \cdot \frac{1}{{s}^{2} \cdot \mathsf{PI}\left(\right)}\right) + \frac{1}{4} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}}{\color{blue}{r}} \]

4. Applied rewrites 8.6%

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

5. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

   4. lower-/.f32 N/A

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

   5. *-commutative N/A

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

   6. lift-*.f32 N/A

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

   7. lift-PI.f32 N/A

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

   8. associate-*r/ N/A

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

   9. metadata-eval N/A

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

   10. lower-/.f32 N/A

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

   11. lift-PI.f32 9.0

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

7. Applied rewrites 9.0%

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

Alternative 17: 9.0% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

   \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{\frac{-1}{48} \cdot \frac{{r}^{2}}{\mathsf{PI}\left(\right)} + \frac{-1}{1296} \cdot \frac{{r}^{2}}{\mathsf{PI}\left(\right)}}{s} + \left(\frac{-1}{16} \cdot \frac{r}{\mathsf{PI}\left(\right)} + \frac{-1}{144} \cdot \frac{r}{\mathsf{PI}\left(\right)}\right)}{s} - \frac{1}{6} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{s} - \frac{1}{4} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s}} \]

4. Applied rewrites 9.8%

   \[\leadsto \color{blue}{-\frac{\left(-\frac{\left(-\frac{\mathsf{fma}\left(\frac{r}{\pi}, -0.06944444444444445, -\frac{\frac{r \cdot r}{\pi} \cdot -0.021604938271604937}{s}\right)}{s}\right) - \frac{0.16666666666666666}{\pi}}{s}\right) - \frac{0.25}{\pi \cdot r}}{s}} \]
5. Step-by-step derivation

   1. lift-/.f32 N/A

      \[\leadsto -\frac{\left(-\frac{\left(-\frac{\mathsf{fma}\left(\frac{r}{\pi}, \frac{-5}{72}, -\frac{\frac{r \cdot r}{\pi} \cdot \frac{-7}{324}}{s}\right)}{s}\right) - \frac{\frac{1}{6}}{\pi}}{s}\right) - \frac{\frac{1}{4}}{\pi \cdot r}}{s} \]

   2. lift-PI.f32 N/A

      \[\leadsto -\frac{\left(-\frac{\left(-\frac{\mathsf{fma}\left(\frac{r}{\pi}, \frac{-5}{72}, -\frac{\frac{r \cdot r}{\pi} \cdot \frac{-7}{324}}{s}\right)}{s}\right) - \frac{\frac{1}{6}}{\pi}}{s}\right) - \frac{\frac{1}{4}}{\mathsf{PI}\left(\right) \cdot r}}{s} \]

   3. lift-*.f32 N/A

      \[\leadsto -\frac{\left(-\frac{\left(-\frac{\mathsf{fma}\left(\frac{r}{\pi}, \frac{-5}{72}, -\frac{\frac{r \cdot r}{\pi} \cdot \frac{-7}{324}}{s}\right)}{s}\right) - \frac{\frac{1}{6}}{\pi}}{s}\right) - \frac{\frac{1}{4}}{\mathsf{PI}\left(\right) \cdot r}}{s} \]

   4. *-commutative N/A

      \[\leadsto -\frac{\left(-\frac{\left(-\frac{\mathsf{fma}\left(\frac{r}{\pi}, \frac{-5}{72}, -\frac{\frac{r \cdot r}{\pi} \cdot \frac{-7}{324}}{s}\right)}{s}\right) - \frac{\frac{1}{6}}{\pi}}{s}\right) - \frac{\frac{1}{4}}{r \cdot \mathsf{PI}\left(\right)}}{s} \]

   5. associate-/r* N/A

      \[\leadsto -\frac{\left(-\frac{\left(-\frac{\mathsf{fma}\left(\frac{r}{\pi}, \frac{-5}{72}, -\frac{\frac{r \cdot r}{\pi} \cdot \frac{-7}{324}}{s}\right)}{s}\right) - \frac{\frac{1}{6}}{\pi}}{s}\right) - \frac{\frac{\frac{1}{4}}{r}}{\mathsf{PI}\left(\right)}}{s} \]

   6. lower-/.f32 N/A

      \[\leadsto -\frac{\left(-\frac{\left(-\frac{\mathsf{fma}\left(\frac{r}{\pi}, \frac{-5}{72}, -\frac{\frac{r \cdot r}{\pi} \cdot \frac{-7}{324}}{s}\right)}{s}\right) - \frac{\frac{1}{6}}{\pi}}{s}\right) - \frac{\frac{\frac{1}{4}}{r}}{\mathsf{PI}\left(\right)}}{s} \]

   7. lower-/.f32 N/A

      \[\leadsto -\frac{\left(-\frac{\left(-\frac{\mathsf{fma}\left(\frac{r}{\pi}, \frac{-5}{72}, -\frac{\frac{r \cdot r}{\pi} \cdot \frac{-7}{324}}{s}\right)}{s}\right) - \frac{\frac{1}{6}}{\pi}}{s}\right) - \frac{\frac{\frac{1}{4}}{r}}{\mathsf{PI}\left(\right)}}{s} \]

   8. lift-PI.f32 9.8

      \[\leadsto -\frac{\left(-\frac{\left(-\frac{\mathsf{fma}\left(\frac{r}{\pi}, -0.06944444444444445, -\frac{\frac{r \cdot r}{\pi} \cdot -0.021604938271604937}{s}\right)}{s}\right) - \frac{0.16666666666666666}{\pi}}{s}\right) - \frac{\frac{0.25}{r}}{\pi}}{s} \]

6. Applied rewrites 9.8%

   \[\leadsto -\frac{\left(-\frac{\left(-\frac{\mathsf{fma}\left(\frac{r}{\pi}, -0.06944444444444445, -\frac{\frac{r \cdot r}{\pi} \cdot -0.021604938271604937}{s}\right)}{s}\right) - \frac{0.16666666666666666}{\pi}}{s}\right) - \frac{\frac{0.25}{r}}{\pi}}{s} \]

7. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. lift-PI.f32 9.0

      \[\leadsto -\frac{\frac{0.16666666666666666}{\pi \cdot s} - \frac{\frac{0.25}{r}}{\pi}}{s} \]

9. Applied rewrites 9.0%

   \[\leadsto -\frac{\frac{0.16666666666666666}{\pi \cdot s} - \frac{\frac{0.25}{r}}{\pi}}{s} \]
10. Add Preprocessing

Alternative 18: 9.0% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

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

   1. lower-/.f32 N/A

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

4. Applied rewrites 9.0%

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

Alternative 19: 9.0% accurate, 6.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

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 9.1

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

4. Applied rewrites 9.1%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-/r* N/A

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

   4. lift-PI.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. metadata-eval N/A

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

   7. *-commutative N/A

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

   8. associate-*r/ N/A

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

   9. lower-/.f32 N/A

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

   10. associate-*r/ N/A

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

   11. metadata-eval N/A

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

   12. lower-/.f32 N/A

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

   13. *-commutative N/A

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

   14. lift-*.f32 N/A

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

   15. lift-PI.f32 9.1

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

6. Applied rewrites 9.1%

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

Alternative 20: 9.0% 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 9.1

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

4. Applied rewrites 9.1%

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

   1. lift-*.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. 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 9.1

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

6. Applied rewrites 9.1%

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

Reproduce

herbie shell --seed 2025142 
(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))))