Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.5% → 99.5%

Time: 14.8s

Alternatives: 19

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Final simplification 99.5%

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

Alternative 2: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

   1. lift-/.f32 N/A

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

   2. clear-num N/A

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

   3. lift-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. *-commutative N/A

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

   6. times-frac N/A

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

   7. associate-/r* N/A

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

   8. clear-num N/A

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

   9. lower-/.f32 N/A

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

4. Applied rewrites 99.5%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \color{blue}{\frac{\frac{e^{\frac{r}{s \cdot -3}}}{s \cdot \left(\pi \cdot 6\right)}}{r \cdot 1.3333333333333333}} \]

5. Taylor expanded in s around 0

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

   1. lower-/.f32 N/A

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

7. Applied rewrites 99.5%

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

Alternative 3: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

4. Applied rewrites 99.5%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-/r* N/A

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

   4. div-inv N/A

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

   5. metadata-eval N/A

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

   6. lower-*.f32 N/A

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

   7. lower-/.f32 99.5

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

6. Applied rewrites 99.5%

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

Alternative 4: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

4. Applied rewrites 99.5%

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

5. Taylor expanded in r around inf

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

   1. associate-*r/ N/A

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

   2. metadata-eval N/A

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

   3. associate-*r* N/A

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

   4. distribute-lft-out N/A

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

   5. lower-/.f32 N/A

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

7. Applied rewrites 99.4%

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

9. Applied rewrites 99.5%

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

10. Final simplification 99.5%

    \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{r}{-s}} + e^{\frac{r}{s \cdot -3}}}{r} \]
11. Add Preprocessing

Alternative 5: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

4. Applied rewrites 99.5%

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

5. Taylor expanded in r around inf

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

   1. associate-*r/ N/A

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

   2. metadata-eval N/A

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

   3. associate-*r* N/A

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

   4. distribute-lft-out N/A

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

   5. lower-/.f32 N/A

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

7. Applied rewrites 99.4%

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

9. Applied rewrites 99.5%

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

10. Final simplification 99.5%

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

Alternative 6: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

4. Applied rewrites 99.5%

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

5. Taylor expanded in r around inf

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

   1. associate-*r/ N/A

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

   2. metadata-eval N/A

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

   3. associate-*r* N/A

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

   4. distribute-lft-out N/A

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

   5. lower-/.f32 N/A

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

7. Applied rewrites 99.4%

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

Alternative 7: 99.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

4. Applied rewrites 99.5%

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

5. Taylor expanded in r around inf

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

   1. associate-*r/ N/A

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

   2. metadata-eval N/A

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

   3. associate-*r* N/A

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

   4. distribute-lft-out N/A

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

   5. lower-/.f32 N/A

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

7. Applied rewrites 99.4%

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

9. Applied rewrites 99.4%

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

10. Final simplification 99.4%

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

Alternative 8: 97.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

4. Applied rewrites 99.5%

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

5. Taylor expanded in r around inf

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

   1. associate-*r/ N/A

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

   2. metadata-eval N/A

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

   3. associate-*r* N/A

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

   4. distribute-lft-out N/A

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

   5. lower-/.f32 N/A

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

7. Applied rewrites 99.4%

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

8. Taylor expanded in r around inf

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

   1. associate-*r/ N/A

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

   2. *-commutative N/A

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

   3. associate-/l* N/A

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

   4. lower-*.f32 N/A

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

   5. lower-+.f32 N/A

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

   6. lower-exp.f32 N/A

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

   7. mul-1-neg N/A

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

   8. lower-neg.f32 N/A

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

   9. lower-/.f32 N/A

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

   10. lower-exp.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 N/A

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

   13. lower-/.f32 N/A

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

   14. lower-/.f32 N/A

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

   15. *-commutative N/A

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

   16. associate-*r* N/A

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

   17. lower-*.f32 N/A

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

10. Applied rewrites 98.2%

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

11. Final simplification 98.2%

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

Alternative 9: 10.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \frac{0.25 \cdot e^{\frac{r}{-s}}}{r \cdot \left(s \cdot \left(2 \cdot \pi\right)\right)} + \frac{\frac{0.125}{r \cdot \pi} + \mathsf{fma}\left(r, \frac{0.006944444444444444}{s \cdot \left(s \cdot \pi\right)}, \frac{-0.041666666666666664}{s \cdot \pi}\right)}{s} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/ (* 0.25 (exp (/ r (- s)))) (* r (* s (* 2.0 PI))))
  (/
   (+
    (/ 0.125 (* r PI))
    (fma
     r
     (/ 0.006944444444444444 (* s (* s PI)))
     (/ -0.041666666666666664 (* s PI))))
   s)))

C

float code(float s, float r) {
	return ((0.25f * expf((r / -s))) / (r * (s * (2.0f * ((float) M_PI))))) + (((0.125f / (r * ((float) M_PI))) + fmaf(r, (0.006944444444444444f / (s * (s * ((float) M_PI)))), (-0.041666666666666664f / (s * ((float) M_PI))))) / s);
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.25) * exp(Float32(r / Float32(-s)))) / Float32(r * Float32(s * Float32(Float32(2.0) * Float32(pi))))) + Float32(Float32(Float32(Float32(0.125) / Float32(r * Float32(pi))) + fma(r, Float32(Float32(0.006944444444444444) / Float32(s * Float32(s * Float32(pi)))), Float32(Float32(-0.041666666666666664) / Float32(s * Float32(pi))))) / s))
end

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in s around inf

   \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \color{blue}{\frac{\left(\frac{1}{144} \cdot \frac{r}{{s}^{2} \cdot \mathsf{PI}\left(\right)} + \frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}\right) - \frac{\frac{1}{24}}{s \cdot \mathsf{PI}\left(\right)}}{s}} \]

5. Applied rewrites 10.6%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \color{blue}{\frac{\frac{0.125}{r \cdot \pi} + \mathsf{fma}\left(r, \frac{0.006944444444444444}{s \cdot \left(s \cdot \pi\right)}, \frac{-0.041666666666666664}{s \cdot \pi}\right)}{s}} \]

6. Final simplification 10.6%

   \[\leadsto \frac{0.25 \cdot e^{\frac{r}{-s}}}{r \cdot \left(s \cdot \left(2 \cdot \pi\right)\right)} + \frac{\frac{0.125}{r \cdot \pi} + \mathsf{fma}\left(r, \frac{0.006944444444444444}{s \cdot \left(s \cdot \pi\right)}, \frac{-0.041666666666666664}{s \cdot \pi}\right)}{s} \]
7. Add Preprocessing

Alternative 10: 10.2% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \frac{0.125}{s \cdot \pi} \cdot \frac{\mathsf{fma}\left(r, \mathsf{fma}\left(r, \frac{0.5555555555555556}{s \cdot s}, \frac{-1.3333333333333333}{s}\right), 2\right)}{r} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (*
  (/ 0.125 (* s PI))
  (/
   (fma r (fma r (/ 0.5555555555555556 (* s s)) (/ -1.3333333333333333 s)) 2.0)
   r)))

C

float code(float s, float r) {
	return (0.125f / (s * ((float) M_PI))) * (fmaf(r, fmaf(r, (0.5555555555555556f / (s * s)), (-1.3333333333333333f / s)), 2.0f) / r);
}

Julia

function code(s, r)
	return Float32(Float32(Float32(0.125) / Float32(s * Float32(pi))) * Float32(fma(r, fma(r, Float32(Float32(0.5555555555555556) / Float32(s * s)), Float32(Float32(-1.3333333333333333) / s)), Float32(2.0)) / r))
end

Derivation

1. Initial program 99.5%

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

4. Applied rewrites 99.5%

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

5. Taylor expanded in r around inf

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

   1. associate-*r/ N/A

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

   2. metadata-eval N/A

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

   3. associate-*r* N/A

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

   4. distribute-lft-out N/A

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

   5. lower-/.f32 N/A

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

7. Applied rewrites 99.4%

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

9. Applied rewrites 99.5%

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

10. Taylor expanded in r around 0

    \[\leadsto \frac{2 + r \cdot \left(\frac{5}{9} \cdot \frac{r}{{s}^{2}} - \frac{4}{3} \cdot \frac{1}{s}\right)}{r} \cdot \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \]
11. Step-by-step derivation

12. Applied rewrites 10.1%

    \[\leadsto \frac{\mathsf{fma}\left(r, \mathsf{fma}\left(r, \frac{0.5555555555555556}{s \cdot s}, \frac{-1.3333333333333333}{s}\right), 2\right)}{r} \cdot \frac{0.125}{s \cdot \pi} \]

13. Final simplification 10.1%

    \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \frac{\mathsf{fma}\left(r, \mathsf{fma}\left(r, \frac{0.5555555555555556}{s \cdot s}, \frac{-1.3333333333333333}{s}\right), 2\right)}{r} \]
14. Add Preprocessing

Alternative 11: 10.2% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \frac{0.125}{s \cdot \pi} \cdot \left(\frac{2}{r} - \frac{\mathsf{fma}\left(\frac{r}{s}, -0.5555555555555556, 1.3333333333333333\right)}{s}\right) \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (*
  (/ 0.125 (* s PI))
  (- (/ 2.0 r) (/ (fma (/ r s) -0.5555555555555556 1.3333333333333333) s))))

C

float code(float s, float r) {
	return (0.125f / (s * ((float) M_PI))) * ((2.0f / r) - (fmaf((r / s), -0.5555555555555556f, 1.3333333333333333f) / s));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(0.125) / Float32(s * Float32(pi))) * Float32(Float32(Float32(2.0) / r) - Float32(fma(Float32(r / s), Float32(-0.5555555555555556), Float32(1.3333333333333333)) / s)))
end

Derivation

1. Initial program 99.5%

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

4. Applied rewrites 99.5%

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

5. Taylor expanded in r around inf

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

   1. associate-*r/ N/A

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

   2. metadata-eval N/A

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

   3. associate-*r* N/A

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

   4. distribute-lft-out N/A

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

   5. lower-/.f32 N/A

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

7. Applied rewrites 99.4%

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

9. Applied rewrites 99.5%

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

10. Taylor expanded in s around -inf

    \[\leadsto \left(-1 \cdot \frac{\frac{4}{3} + -1 \cdot \frac{\frac{1}{18} \cdot r + \frac{1}{2} \cdot r}{s}}{s} + 2 \cdot \frac{1}{r}\right) \cdot \frac{\color{blue}{\frac{1}{8}}}{s \cdot \mathsf{PI}\left(\right)} \]
11. Step-by-step derivation

12. Applied rewrites 10.1%

    \[\leadsto \left(\frac{2}{r} - \frac{\mathsf{fma}\left(\frac{r}{s}, -0.5555555555555556, 1.3333333333333333\right)}{s}\right) \cdot \frac{\color{blue}{0.125}}{s \cdot \pi} \]

13. Final simplification 10.1%

    \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{2}{r} - \frac{\mathsf{fma}\left(\frac{r}{s}, -0.5555555555555556, 1.3333333333333333\right)}{s}\right) \]
14. Add Preprocessing

Alternative 12: 10.2% accurate, 4.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.5%

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

4. Applied rewrites 99.5%

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

5. Taylor expanded in r around inf

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

   1. associate-*r/ N/A

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

   2. metadata-eval N/A

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

   3. associate-*r* N/A

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

   4. distribute-lft-out N/A

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

   5. lower-/.f32 N/A

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

7. Applied rewrites 99.4%

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

8. Taylor expanded in r around 0

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

10. Applied rewrites 10.1%

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

Alternative 13: 9.1% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in s around inf

   \[\leadsto \color{blue}{\frac{\left(\frac{-1}{48} \cdot \frac{{r}^{2}}{{s}^{3} \cdot \mathsf{PI}\left(\right)} + \left(\frac{-1}{1296} \cdot \frac{{r}^{2}}{{s}^{3} \cdot \mathsf{PI}\left(\right)} + \left(\frac{1}{144} \cdot \frac{r}{{s}^{2} \cdot \mathsf{PI}\left(\right)} + \left(\frac{1}{16} \cdot \frac{r}{{s}^{2} \cdot \mathsf{PI}\left(\right)} + \frac{1}{4} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}\right)\right)\right)\right) - \frac{\frac{1}{6}}{s \cdot \mathsf{PI}\left(\right)}}{s}} \]

5. Applied rewrites 8.5%

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

6. Taylor expanded in s around inf

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

8. Applied rewrites 9.0%

   \[\leadsto \frac{\frac{0.25}{r \cdot \pi} + \frac{-0.16666666666666666}{s \cdot \pi}}{s} \]
9. Step-by-step derivation

10. Applied rewrites 9.0%

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

11. Final simplification 9.0%

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

Alternative 14: 9.1% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \frac{0.125}{s \cdot \pi} \cdot \frac{\mathsf{fma}\left(\frac{r}{s}, -1.3333333333333333, 2\right)}{r} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (* (/ 0.125 (* s PI)) (/ (fma (/ r s) -1.3333333333333333 2.0) r)))

C

float code(float s, float r) {
	return (0.125f / (s * ((float) M_PI))) * (fmaf((r / s), -1.3333333333333333f, 2.0f) / r);
}

Julia

function code(s, r)
	return Float32(Float32(Float32(0.125) / Float32(s * Float32(pi))) * Float32(fma(Float32(r / s), Float32(-1.3333333333333333), Float32(2.0)) / r))
end

Derivation

1. Initial program 99.5%

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

4. Applied rewrites 99.5%

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

5. Taylor expanded in r around inf

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

   1. associate-*r/ N/A

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

   2. metadata-eval N/A

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

   3. associate-*r* N/A

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

   4. distribute-lft-out N/A

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

   5. lower-/.f32 N/A

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

7. Applied rewrites 99.4%

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

9. Applied rewrites 99.5%

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

10. Taylor expanded in r around 0

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

12. Applied rewrites 9.0%

    \[\leadsto \frac{\mathsf{fma}\left(\frac{r}{s}, -1.3333333333333333, 2\right)}{r} \cdot \frac{0.125}{s \cdot \pi} \]

13. Final simplification 9.0%

    \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \frac{\mathsf{fma}\left(\frac{r}{s}, -1.3333333333333333, 2\right)}{r} \]
14. Add Preprocessing

Alternative 15: 9.1% accurate, 7.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

4. Applied rewrites 99.5%

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

5. Taylor expanded in r around inf

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

   1. associate-*r/ N/A

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

   2. metadata-eval N/A

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

   3. associate-*r* N/A

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

   4. distribute-lft-out N/A

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

   5. lower-/.f32 N/A

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

7. Applied rewrites 99.4%

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

8. Taylor expanded in s around inf

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

10. Applied rewrites 9.0%

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

11. Final simplification 9.0%

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

Alternative 16: 9.1% accurate, 7.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.5%

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

4. Applied rewrites 99.5%

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

5. Taylor expanded in r around inf

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

   1. associate-*r/ N/A

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

   2. metadata-eval N/A

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

   3. associate-*r* N/A

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

   4. distribute-lft-out N/A

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

   5. lower-/.f32 N/A

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

7. Applied rewrites 99.4%

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

8. Taylor expanded in r around 0

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

10. Applied rewrites 9.0%

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

Alternative 17: 9.0% accurate, 9.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

   1. lift-/.f32 N/A

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

   2. clear-num N/A

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

   3. lift-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. *-commutative N/A

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

   6. times-frac N/A

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

   7. associate-/r* N/A

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

   8. clear-num N/A

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

   9. lower-/.f32 N/A

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

4. Applied rewrites 99.5%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \color{blue}{\frac{\frac{e^{\frac{r}{s \cdot -3}}}{s \cdot \left(\pi \cdot 6\right)}}{r \cdot 1.3333333333333333}} \]

5. Taylor expanded in s around 0

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

   1. lower-/.f32 N/A

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

7. Applied rewrites 99.5%

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

8. Taylor expanded in r around 0

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

   1. lower-/.f32 N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-PI.f32 8.7

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

10. Applied rewrites 8.7%

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

12. Applied rewrites 8.7%

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

Alternative 18: 9.0% accurate, 10.6× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in r around 0

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 8.7

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

5. Applied rewrites 8.7%

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

7. Applied rewrites 8.7%

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

Alternative 19: 9.0% accurate, 13.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in r around 0

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 8.7

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

5. Applied rewrites 8.7%

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

Reproduce

herbie shell --seed 2024226 
(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))))