Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.6%

Time: 19.2s

Alternatives: 16

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: 99.6% 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, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.7%

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

   1. add-cube-cbrt 99.7%

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

   2. pow3 99.7%

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

   3. *-commutative 99.7%

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

   4. *-commutative 99.7%

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

   5. *-commutative 99.7%

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

4. Applied egg-rr 99.7%

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

5. Final simplification 99.7%

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

Alternative 2: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Final simplification 99.7%

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

Alternative 3: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

   1. times-frac 99.8%

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

   2. *-commutative 99.8%

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

   3. distribute-frac-neg 99.8%

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

   4. associate-/l* 99.7%

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

   5. *-commutative 99.7%

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

   6. *-commutative 99.7%

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

   7. associate-*l* 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in s around 0 99.7%

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

6. Taylor expanded in s around 0 99.7%

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

   1. *-commutative 99.7%

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

   2. associate-*r* 99.7%

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

8. Simplified 99.7%

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

9. Final simplification 99.7%

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

Alternative 4: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Simplified 99.6%

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

5. Taylor expanded in s around 0 99.7%

   \[\leadsto \color{blue}{0.125 \cdot \frac{\frac{e^{-1 \cdot \frac{r}{s}}}{r} + \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r}}{s \cdot \pi}} \]
6. Step-by-step derivation

   1. associate-*r/ 99.7%

      \[\leadsto \color{blue}{\frac{0.125 \cdot \left(\frac{e^{-1 \cdot \frac{r}{s}}}{r} + \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r}\right)}{s \cdot \pi}} \]

   2. times-frac 99.7%

      \[\leadsto \color{blue}{\frac{0.125}{s} \cdot \frac{\frac{e^{-1 \cdot \frac{r}{s}}}{r} + \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r}}{\pi}} \]

   3. mul-1-neg 99.7%

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

   4. distribute-neg-frac2 99.7%

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

   5. *-commutative 99.7%

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

   6. associate-*l/ 99.7%

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

7. Simplified 99.7%

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

8. Taylor expanded in r around 0 99.7%

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

   1. associate-*r/ 99.7%

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

   2. *-commutative 99.7%

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

   3. associate-*r/ 99.7%

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

10. Simplified 99.7%

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

11. Final simplification 99.7%

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

Alternative 5: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Simplified 99.6%

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

5. Taylor expanded in r around inf 99.7%

   \[\leadsto \color{blue}{0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}} + e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r \cdot \left(s \cdot \pi\right)}} \]
6. Step-by-step derivation

   1. associate-*r/ 99.7%

      \[\leadsto \color{blue}{\frac{0.125 \cdot \left(e^{-1 \cdot \frac{r}{s}} + e^{-0.3333333333333333 \cdot \frac{r}{s}}\right)}{r \cdot \left(s \cdot \pi\right)}} \]

   2. times-frac 99.7%

      \[\leadsto \color{blue}{\frac{0.125}{r} \cdot \frac{e^{-1 \cdot \frac{r}{s}} + e^{-0.3333333333333333 \cdot \frac{r}{s}}}{s \cdot \pi}} \]

   3. mul-1-neg 99.7%

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

   4. distribute-neg-frac2 99.7%

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

   5. *-commutative 99.7%

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

   6. associate-*l/ 99.7%

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

7. Simplified 99.7%

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

8. Final simplification 99.7%

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

Alternative 6: 99.5% accurate, 1.1× speedup

Math

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

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(0.125) * Float32(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 99.7%

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

3. Simplified 99.6%

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

5. Taylor expanded in r around inf 99.7%

   \[\leadsto \color{blue}{0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}} + e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r \cdot \left(s \cdot \pi\right)}} \]

6. Final simplification 99.7%

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

Alternative 7: 11.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{0.25}{\mathsf{log1p}\left(\mathsf{expm1}\left(r \cdot \left(s \cdot \pi\right)\right)\right)} \end{array} \]

FPCore

(FPCore (s r) :precision binary32 (/ 0.25 (log1p (expm1 (* r (* s PI))))))

C

float code(float s, float r) {
	return 0.25f / log1pf(expm1f((r * (s * ((float) M_PI)))));
}

Julia

function code(s, r)
	return Float32(Float32(0.25) / log1p(expm1(Float32(r * Float32(s * Float32(pi))))))
end

Derivation

1. Initial program 99.7%

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

3. Simplified 99.6%

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

5. Taylor expanded in s around inf 8.7%

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

   1. log1p-expm1-u 9.9%

      \[\leadsto \frac{0.25}{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(r \cdot \left(s \cdot \pi\right)\right)\right)}} \]

7. Applied egg-rr 9.9%

   \[\leadsto \frac{0.25}{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(r \cdot \left(s \cdot \pi\right)\right)\right)}} \]

8. Final simplification 9.9%

   \[\leadsto \frac{0.25}{\mathsf{log1p}\left(\mathsf{expm1}\left(r \cdot \left(s \cdot \pi\right)\right)\right)} \]
9. Add Preprocessing

Alternative 8: 11.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{0.25}{\pi \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(r \cdot s\right)\right)} \end{array} \]

FPCore

(FPCore (s r) :precision binary32 (/ 0.25 (* PI (log1p (expm1 (* r s))))))

C

float code(float s, float r) {
	return 0.25f / (((float) M_PI) * log1pf(expm1f((r * s))));
}

Julia

function code(s, r)
	return Float32(Float32(0.25) / Float32(Float32(pi) * log1p(expm1(Float32(r * s)))))
end

Derivation

1. Initial program 99.7%

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

3. Simplified 99.6%

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

5. Taylor expanded in s around inf 9.3%

   \[\leadsto \color{blue}{0.25 \cdot \frac{1}{r \cdot \left(s \cdot \pi\right)} - 0.16666666666666666 \cdot \frac{1}{{s}^{2} \cdot \pi}} \]
6. Step-by-step derivation

   1. associate-*r/ 9.3%

      \[\leadsto \color{blue}{\frac{0.25 \cdot 1}{r \cdot \left(s \cdot \pi\right)}} - 0.16666666666666666 \cdot \frac{1}{{s}^{2} \cdot \pi} \]

   2. metadata-eval 9.3%

      \[\leadsto \frac{\color{blue}{0.25}}{r \cdot \left(s \cdot \pi\right)} - 0.16666666666666666 \cdot \frac{1}{{s}^{2} \cdot \pi} \]

   3. associate-/r* 9.3%

      \[\leadsto \color{blue}{\frac{\frac{0.25}{r}}{s \cdot \pi}} - 0.16666666666666666 \cdot \frac{1}{{s}^{2} \cdot \pi} \]

   4. associate-*r/ 9.3%

      \[\leadsto \frac{\frac{0.25}{r}}{s \cdot \pi} - \color{blue}{\frac{0.16666666666666666 \cdot 1}{{s}^{2} \cdot \pi}} \]

   5. metadata-eval 9.3%

      \[\leadsto \frac{\frac{0.25}{r}}{s \cdot \pi} - \frac{\color{blue}{0.16666666666666666}}{{s}^{2} \cdot \pi} \]

   6. *-commutative 9.3%

      \[\leadsto \frac{\frac{0.25}{r}}{s \cdot \pi} - \frac{0.16666666666666666}{\color{blue}{\pi \cdot {s}^{2}}} \]

7. Simplified 9.3%

   \[\leadsto \color{blue}{\frac{\frac{0.25}{r}}{s \cdot \pi} - \frac{0.16666666666666666}{\pi \cdot {s}^{2}}} \]

8. Taylor expanded in r around 0 8.7%

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

   1. associate-/r* 8.7%

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

   2. associate-/r* 8.7%

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

   3. associate-/l/ 8.7%

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

   4. *-commutative 8.7%

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

   5. associate-/l/ 8.7%

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

   6. *-commutative 8.7%

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

10. Simplified 8.7%

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

    1. log1p-expm1-u 9.9%

       \[\leadsto \frac{0.25}{\pi \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(s \cdot r\right)\right)}} \]

    2. *-commutative 9.9%

       \[\leadsto \frac{0.25}{\pi \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{r \cdot s}\right)\right)} \]

12. Applied egg-rr 9.9%

    \[\leadsto \frac{0.25}{\pi \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(r \cdot s\right)\right)}} \]

13. Final simplification 9.9%

    \[\leadsto \frac{0.25}{\pi \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(r \cdot s\right)\right)} \]
14. Add Preprocessing

Alternative 9: 9.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \frac{0.125}{s} \cdot \frac{\frac{1 - \frac{r}{s}}{r} + \frac{e^{\frac{r \cdot -0.3333333333333333}{s}}}{r}}{\pi} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (*
  (/ 0.125 s)
  (/
   (+ (/ (- 1.0 (/ r s)) r) (/ (exp (/ (* r -0.3333333333333333) s)) r))
   PI)))

C

float code(float s, float r) {
	return (0.125f / s) * ((((1.0f - (r / s)) / r) + (expf(((r * -0.3333333333333333f) / s)) / r)) / ((float) M_PI));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(0.125) / s) * Float32(Float32(Float32(Float32(Float32(1.0) - Float32(r / s)) / r) + Float32(exp(Float32(Float32(r * Float32(-0.3333333333333333)) / s)) / r)) / Float32(pi)))
end

MATLAB

function tmp = code(s, r)
	tmp = (single(0.125) / s) * ((((single(1.0) - (r / s)) / r) + (exp(((r * single(-0.3333333333333333)) / s)) / r)) / single(pi));
end

Derivation

1. Initial program 99.7%

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

3. Simplified 99.6%

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

5. Taylor expanded in s around 0 99.7%

   \[\leadsto \color{blue}{0.125 \cdot \frac{\frac{e^{-1 \cdot \frac{r}{s}}}{r} + \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r}}{s \cdot \pi}} \]
6. Step-by-step derivation

   1. associate-*r/ 99.7%

      \[\leadsto \color{blue}{\frac{0.125 \cdot \left(\frac{e^{-1 \cdot \frac{r}{s}}}{r} + \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r}\right)}{s \cdot \pi}} \]

   2. times-frac 99.7%

      \[\leadsto \color{blue}{\frac{0.125}{s} \cdot \frac{\frac{e^{-1 \cdot \frac{r}{s}}}{r} + \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r}}{\pi}} \]

   3. mul-1-neg 99.7%

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

   4. distribute-neg-frac2 99.7%

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

   5. *-commutative 99.7%

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

   6. associate-*l/ 99.7%

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

7. Simplified 99.7%

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

8. Taylor expanded in r around 0 9.4%

   \[\leadsto \frac{0.125}{s} \cdot \frac{\frac{\color{blue}{1 + -1 \cdot \frac{r}{s}}}{r} + \frac{e^{\frac{r \cdot -0.3333333333333333}{s}}}{r}}{\pi} \]
9. Step-by-step derivation

   1. mul-1-neg 9.4%

      \[\leadsto \frac{0.125}{s} \cdot \frac{\frac{1 + \color{blue}{\left(-\frac{r}{s}\right)}}{r} + \frac{e^{\frac{r \cdot -0.3333333333333333}{s}}}{r}}{\pi} \]

   2. unsub-neg 9.4%

      \[\leadsto \frac{0.125}{s} \cdot \frac{\frac{\color{blue}{1 - \frac{r}{s}}}{r} + \frac{e^{\frac{r \cdot -0.3333333333333333}{s}}}{r}}{\pi} \]

10. Simplified 9.4%

    \[\leadsto \frac{0.125}{s} \cdot \frac{\frac{\color{blue}{1 - \frac{r}{s}}}{r} + \frac{e^{\frac{r \cdot -0.3333333333333333}{s}}}{r}}{\pi} \]

11. Final simplification 9.4%

    \[\leadsto \frac{0.125}{s} \cdot \frac{\frac{1 - \frac{r}{s}}{r} + \frac{e^{\frac{r \cdot -0.3333333333333333}{s}}}{r}}{\pi} \]
12. Add Preprocessing

Alternative 10: 9.3% accurate, 15.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = (single(0.125) / s) * (((single(2.0) / r) / single(pi)) + (single(-1.3333333333333333) / (s * single(pi))));
end

Derivation

1. Initial program 99.7%

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

3. Simplified 99.6%

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

5. Taylor expanded in s around 0 99.7%

   \[\leadsto \color{blue}{0.125 \cdot \frac{\frac{e^{-1 \cdot \frac{r}{s}}}{r} + \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r}}{s \cdot \pi}} \]
6. Step-by-step derivation

   1. associate-*r/ 99.7%

      \[\leadsto \color{blue}{\frac{0.125 \cdot \left(\frac{e^{-1 \cdot \frac{r}{s}}}{r} + \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r}\right)}{s \cdot \pi}} \]

   2. times-frac 99.7%

      \[\leadsto \color{blue}{\frac{0.125}{s} \cdot \frac{\frac{e^{-1 \cdot \frac{r}{s}}}{r} + \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r}}{\pi}} \]

   3. mul-1-neg 99.7%

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

   4. distribute-neg-frac2 99.7%

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

   5. *-commutative 99.7%

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

   6. associate-*l/ 99.7%

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

7. Simplified 99.7%

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

8. Taylor expanded in r around 0 9.4%

   \[\leadsto \frac{0.125}{s} \cdot \frac{\frac{\color{blue}{1 + -1 \cdot \frac{r}{s}}}{r} + \frac{e^{\frac{r \cdot -0.3333333333333333}{s}}}{r}}{\pi} \]
9. Step-by-step derivation

   1. mul-1-neg 9.4%

      \[\leadsto \frac{0.125}{s} \cdot \frac{\frac{1 + \color{blue}{\left(-\frac{r}{s}\right)}}{r} + \frac{e^{\frac{r \cdot -0.3333333333333333}{s}}}{r}}{\pi} \]

   2. unsub-neg 9.4%

      \[\leadsto \frac{0.125}{s} \cdot \frac{\frac{\color{blue}{1 - \frac{r}{s}}}{r} + \frac{e^{\frac{r \cdot -0.3333333333333333}{s}}}{r}}{\pi} \]

10. Simplified 9.4%

    \[\leadsto \frac{0.125}{s} \cdot \frac{\frac{\color{blue}{1 - \frac{r}{s}}}{r} + \frac{e^{\frac{r \cdot -0.3333333333333333}{s}}}{r}}{\pi} \]

11. Taylor expanded in r around 0 9.4%

    \[\leadsto \frac{0.125}{s} \cdot \color{blue}{\left(2 \cdot \frac{1}{r \cdot \pi} - 1.3333333333333333 \cdot \frac{1}{s \cdot \pi}\right)} \]
12. Step-by-step derivation

    1. sub-neg 9.4%

       \[\leadsto \frac{0.125}{s} \cdot \color{blue}{\left(2 \cdot \frac{1}{r \cdot \pi} + \left(-1.3333333333333333 \cdot \frac{1}{s \cdot \pi}\right)\right)} \]

    2. associate-*r/ 9.4%

       \[\leadsto \frac{0.125}{s} \cdot \left(\color{blue}{\frac{2 \cdot 1}{r \cdot \pi}} + \left(-1.3333333333333333 \cdot \frac{1}{s \cdot \pi}\right)\right) \]

    3. metadata-eval 9.4%

       \[\leadsto \frac{0.125}{s} \cdot \left(\frac{\color{blue}{2}}{r \cdot \pi} + \left(-1.3333333333333333 \cdot \frac{1}{s \cdot \pi}\right)\right) \]

    4. associate-/r* 9.3%

       \[\leadsto \frac{0.125}{s} \cdot \left(\color{blue}{\frac{\frac{2}{r}}{\pi}} + \left(-1.3333333333333333 \cdot \frac{1}{s \cdot \pi}\right)\right) \]

    5. associate-*r/ 9.3%

       \[\leadsto \frac{0.125}{s} \cdot \left(\frac{\frac{2}{r}}{\pi} + \left(-\color{blue}{\frac{1.3333333333333333 \cdot 1}{s \cdot \pi}}\right)\right) \]

    6. metadata-eval 9.3%

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

    7. distribute-neg-frac 9.3%

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

    8. metadata-eval 9.3%

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

13. Simplified 9.3%

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

14. Final simplification 9.3%

    \[\leadsto \frac{0.125}{s} \cdot \left(\frac{\frac{2}{r}}{\pi} + \frac{-1.3333333333333333}{s \cdot \pi}\right) \]
15. Add Preprocessing

Alternative 11: 9.3% accurate, 15.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = (single(0.125) / s) * ((single(2.0) / (r * single(pi))) - (single(1.3333333333333333) / (s * single(pi))));
end

Derivation

1. Initial program 99.7%

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

3. Simplified 99.6%

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

5. Taylor expanded in s around 0 99.7%

   \[\leadsto \color{blue}{0.125 \cdot \frac{\frac{e^{-1 \cdot \frac{r}{s}}}{r} + \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r}}{s \cdot \pi}} \]
6. Step-by-step derivation

   1. associate-*r/ 99.7%

      \[\leadsto \color{blue}{\frac{0.125 \cdot \left(\frac{e^{-1 \cdot \frac{r}{s}}}{r} + \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r}\right)}{s \cdot \pi}} \]

   2. times-frac 99.7%

      \[\leadsto \color{blue}{\frac{0.125}{s} \cdot \frac{\frac{e^{-1 \cdot \frac{r}{s}}}{r} + \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r}}{\pi}} \]

   3. mul-1-neg 99.7%

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

   4. distribute-neg-frac2 99.7%

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

   5. *-commutative 99.7%

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

   6. associate-*l/ 99.7%

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

7. Simplified 99.7%

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

8. Taylor expanded in r around 0 9.4%

   \[\leadsto \frac{0.125}{s} \cdot \frac{\frac{\color{blue}{1 + -1 \cdot \frac{r}{s}}}{r} + \frac{e^{\frac{r \cdot -0.3333333333333333}{s}}}{r}}{\pi} \]
9. Step-by-step derivation

   1. mul-1-neg 9.4%

      \[\leadsto \frac{0.125}{s} \cdot \frac{\frac{1 + \color{blue}{\left(-\frac{r}{s}\right)}}{r} + \frac{e^{\frac{r \cdot -0.3333333333333333}{s}}}{r}}{\pi} \]

   2. unsub-neg 9.4%

      \[\leadsto \frac{0.125}{s} \cdot \frac{\frac{\color{blue}{1 - \frac{r}{s}}}{r} + \frac{e^{\frac{r \cdot -0.3333333333333333}{s}}}{r}}{\pi} \]

10. Simplified 9.4%

    \[\leadsto \frac{0.125}{s} \cdot \frac{\frac{\color{blue}{1 - \frac{r}{s}}}{r} + \frac{e^{\frac{r \cdot -0.3333333333333333}{s}}}{r}}{\pi} \]

11. Taylor expanded in r around 0 9.4%

    \[\leadsto \frac{0.125}{s} \cdot \color{blue}{\left(2 \cdot \frac{1}{r \cdot \pi} - 1.3333333333333333 \cdot \frac{1}{s \cdot \pi}\right)} \]
12. Step-by-step derivation

    1. associate-*r/ 9.4%

       \[\leadsto \frac{0.125}{s} \cdot \left(\color{blue}{\frac{2 \cdot 1}{r \cdot \pi}} - 1.3333333333333333 \cdot \frac{1}{s \cdot \pi}\right) \]

    2. metadata-eval 9.4%

       \[\leadsto \frac{0.125}{s} \cdot \left(\frac{\color{blue}{2}}{r \cdot \pi} - 1.3333333333333333 \cdot \frac{1}{s \cdot \pi}\right) \]

    3. associate-*r/ 9.4%

       \[\leadsto \frac{0.125}{s} \cdot \left(\frac{2}{r \cdot \pi} - \color{blue}{\frac{1.3333333333333333 \cdot 1}{s \cdot \pi}}\right) \]

    4. metadata-eval 9.4%

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

13. Simplified 9.4%

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

14. Final simplification 9.4%

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

Alternative 12: 9.2% accurate, 25.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Simplified 99.6%

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

5. Taylor expanded in s around inf 8.7%

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

   1. associate-/r* 8.7%

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

   2. div-inv 8.7%

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

7. Applied egg-rr 8.7%

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

8. Final simplification 8.7%

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

Alternative 13: 9.2% accurate, 33.0× 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 99.7%

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

3. Simplified 99.6%

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

5. Taylor expanded in s around inf 8.7%

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

6. Final simplification 8.7%

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

Alternative 14: 9.2% accurate, 33.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Simplified 99.6%

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

5. Taylor expanded in s around inf 9.3%

   \[\leadsto \color{blue}{0.25 \cdot \frac{1}{r \cdot \left(s \cdot \pi\right)} - 0.16666666666666666 \cdot \frac{1}{{s}^{2} \cdot \pi}} \]
6. Step-by-step derivation

   1. associate-*r/ 9.3%

      \[\leadsto \color{blue}{\frac{0.25 \cdot 1}{r \cdot \left(s \cdot \pi\right)}} - 0.16666666666666666 \cdot \frac{1}{{s}^{2} \cdot \pi} \]

   2. metadata-eval 9.3%

      \[\leadsto \frac{\color{blue}{0.25}}{r \cdot \left(s \cdot \pi\right)} - 0.16666666666666666 \cdot \frac{1}{{s}^{2} \cdot \pi} \]

   3. associate-/r* 9.3%

      \[\leadsto \color{blue}{\frac{\frac{0.25}{r}}{s \cdot \pi}} - 0.16666666666666666 \cdot \frac{1}{{s}^{2} \cdot \pi} \]

   4. associate-*r/ 9.3%

      \[\leadsto \frac{\frac{0.25}{r}}{s \cdot \pi} - \color{blue}{\frac{0.16666666666666666 \cdot 1}{{s}^{2} \cdot \pi}} \]

   5. metadata-eval 9.3%

      \[\leadsto \frac{\frac{0.25}{r}}{s \cdot \pi} - \frac{\color{blue}{0.16666666666666666}}{{s}^{2} \cdot \pi} \]

   6. *-commutative 9.3%

      \[\leadsto \frac{\frac{0.25}{r}}{s \cdot \pi} - \frac{0.16666666666666666}{\color{blue}{\pi \cdot {s}^{2}}} \]

7. Simplified 9.3%

   \[\leadsto \color{blue}{\frac{\frac{0.25}{r}}{s \cdot \pi} - \frac{0.16666666666666666}{\pi \cdot {s}^{2}}} \]

8. Taylor expanded in r around 0 8.7%

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

   1. associate-/r* 8.7%

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

   2. associate-/r* 8.7%

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

   3. associate-/l/ 8.7%

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

   4. *-commutative 8.7%

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

   5. associate-/l/ 8.7%

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

   6. *-commutative 8.7%

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

10. Simplified 8.7%

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

11. Final simplification 8.7%

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

Alternative 15: 9.2% accurate, 33.0× speedup

Math

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

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(Float32(0.25) / 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 99.7%

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

3. Simplified 99.6%

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

5. Taylor expanded in s around inf 8.7%

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

   1. associate-/r* 8.7%

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

7. Simplified 8.7%

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

8. Final simplification 8.7%

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

Alternative 16: 9.1% accurate, 33.0× speedup

Math

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

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(Float32(0.25) / r) / Float32(pi)) / s)
end

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Simplified 99.6%

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

5. Taylor expanded in s around inf 8.7%

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

   1. associate-/r* 8.7%

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

   2. *-un-lft-identity 8.7%

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

   3. *-commutative 8.7%

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

   4. times-frac 8.7%

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

7. Applied egg-rr 8.7%

   \[\leadsto \color{blue}{\frac{1}{\pi} \cdot \frac{\frac{0.25}{r}}{s}} \]
8. Step-by-step derivation

   1. frac-times 8.7%

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

   2. *-un-lft-identity 8.7%

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

9. Applied egg-rr 8.7%

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

10. Taylor expanded in r around 0 8.7%

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

    1. associate-/r* 8.7%

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

    2. associate-/l/ 8.7%

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

12. Simplified 8.7%

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

13. Final simplification 8.7%

    \[\leadsto \frac{\frac{\frac{0.25}{r}}{\pi}}{s} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024046 
(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))))