Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.5% → 99.1%

Time: 18.8s

Alternatives: 14

Speedup: N/A×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 99.5% 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: 99.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\sqrt{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s}\right)}}}{r}\right) \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (*
  (/ (/ 0.125 PI) s)
  (+
   (/ (exp (/ r (- s))) r)
   (/ (sqrt (pow (exp -0.6666666666666666) (/ r s))) r))))

C

float code(float s, float r) {
	return ((0.125f / ((float) M_PI)) / s) * ((expf((r / -s)) / r) + (sqrtf(powf(expf(-0.6666666666666666f), (r / s))) / r));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.125) / Float32(pi)) / s) * Float32(Float32(exp(Float32(r / Float32(-s))) / r) + Float32(sqrt((exp(Float32(-0.6666666666666666)) ^ Float32(r / s))) / r)))
end

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.125) / single(pi)) / s) * ((exp((r / -s)) / r) + (sqrt((exp(single(-0.6666666666666666)) ^ (r / s))) / r));
end

Derivation

1. Initial program 99.3%

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

3. Simplified 98.9%

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

   1. add-sqr-sqrt 98.9%

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

   2. sqrt-unprod 98.9%

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

   3. pow-prod-down 98.9%

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

   4. prod-exp 99.3%

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

   5. metadata-eval 99.3%

      \[\leadsto \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\sqrt{{\left(e^{\color{blue}{-0.6666666666666666}}\right)}^{\left(\frac{r}{s}\right)}}}{r}\right) \]

5. Applied egg-rr 99.3%

   \[\leadsto \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\color{blue}{\sqrt{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s}\right)}}}}{r}\right) \]

6. Final simplification 99.3%

   \[\leadsto \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\sqrt{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s}\right)}}}{r}\right) \]

Alternative 2: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{0.25 \cdot e^{\frac{-r}{s}}}{r \cdot \left(s \cdot \left(\pi \cdot 2\right)\right)} + 0.125 \cdot \frac{e^{\frac{r}{s \cdot -3}}}{r \cdot \left(\pi \cdot s\right)} \end{array} \]

FPCore

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

C

float code(float s, float r) {
	return ((0.25f * expf((-r / s))) / (r * (s * (((float) M_PI) * 2.0f)))) + (0.125f * (expf((r / (s * -3.0f))) / (r * (((float) M_PI) * s))));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.25) * exp(Float32(Float32(-r) / s))) / Float32(r * Float32(s * Float32(Float32(pi) * Float32(2.0))))) + Float32(Float32(0.125) * Float32(exp(Float32(r / Float32(s * Float32(-3.0)))) / Float32(r * Float32(Float32(pi) * s)))))
end

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.25) * exp((-r / s))) / (r * (s * (single(pi) * single(2.0))))) + (single(0.125) * (exp((r / (s * single(-3.0)))) / (r * (single(pi) * s))));
end

Derivation

1. Initial program 99.3%

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

2. Taylor expanded in r around inf 99.2%

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

   1. metadata-eval 99.2%

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + 0.125 \cdot \frac{e^{\color{blue}{\frac{-1}{3}} \cdot \frac{r}{s}}}{r \cdot \left(s \cdot \pi\right)} \]

   2. times-frac 99.3%

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + 0.125 \cdot \frac{e^{\color{blue}{\frac{-1 \cdot r}{3 \cdot s}}}}{r \cdot \left(s \cdot \pi\right)} \]

   3. neg-mul-1 99.3%

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + 0.125 \cdot \frac{e^{\frac{\color{blue}{-r}}{3 \cdot s}}}{r \cdot \left(s \cdot \pi\right)} \]

   4. frac-2neg 99.3%

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + 0.125 \cdot \frac{e^{\color{blue}{\frac{-\left(-r\right)}{-3 \cdot s}}}}{r \cdot \left(s \cdot \pi\right)} \]

   5. remove-double-neg 99.3%

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + 0.125 \cdot \frac{e^{\frac{\color{blue}{r}}{-3 \cdot s}}}{r \cdot \left(s \cdot \pi\right)} \]

   6. *-commutative 99.3%

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + 0.125 \cdot \frac{e^{\frac{r}{-\color{blue}{s \cdot 3}}}}{r \cdot \left(s \cdot \pi\right)} \]

   7. distribute-rgt-neg-in 99.3%

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + 0.125 \cdot \frac{e^{\frac{r}{\color{blue}{s \cdot \left(-3\right)}}}}{r \cdot \left(s \cdot \pi\right)} \]

   8. metadata-eval 99.3%

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + 0.125 \cdot \frac{e^{\frac{r}{s \cdot \color{blue}{-3}}}}{r \cdot \left(s \cdot \pi\right)} \]

4. Applied egg-rr 99.3%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + 0.125 \cdot \frac{e^{\color{blue}{\frac{r}{s \cdot -3}}}}{r \cdot \left(s \cdot \pi\right)} \]

5. Final simplification 99.3%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{r \cdot \left(s \cdot \left(\pi \cdot 2\right)\right)} + 0.125 \cdot \frac{e^{\frac{r}{s \cdot -3}}}{r \cdot \left(\pi \cdot s\right)} \]

Alternative 3: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{0.25 \cdot e^{\frac{-r}{s}}}{r \cdot \left(s \cdot \left(\pi \cdot 2\right)\right)} + 0.125 \cdot \frac{e^{\frac{r}{s \cdot -3}}}{s \cdot \left(\pi \cdot r\right)} \end{array} \]

FPCore

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

C

float code(float s, float r) {
	return ((0.25f * expf((-r / s))) / (r * (s * (((float) M_PI) * 2.0f)))) + (0.125f * (expf((r / (s * -3.0f))) / (s * (((float) M_PI) * r))));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.25) * exp(Float32(Float32(-r) / s))) / Float32(r * Float32(s * Float32(Float32(pi) * Float32(2.0))))) + Float32(Float32(0.125) * Float32(exp(Float32(r / Float32(s * Float32(-3.0)))) / Float32(s * Float32(Float32(pi) * r)))))
end

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.25) * exp((-r / s))) / (r * (s * (single(pi) * single(2.0))))) + (single(0.125) * (exp((r / (s * single(-3.0)))) / (s * (single(pi) * r))));
end

Derivation

1. Initial program 99.3%

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

2. Taylor expanded in r around inf 99.2%

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

   1. metadata-eval 99.2%

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + 0.125 \cdot \frac{e^{\color{blue}{\frac{-1}{3}} \cdot \frac{r}{s}}}{r \cdot \left(s \cdot \pi\right)} \]

   2. times-frac 99.3%

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + 0.125 \cdot \frac{e^{\color{blue}{\frac{-1 \cdot r}{3 \cdot s}}}}{r \cdot \left(s \cdot \pi\right)} \]

   3. neg-mul-1 99.3%

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + 0.125 \cdot \frac{e^{\frac{\color{blue}{-r}}{3 \cdot s}}}{r \cdot \left(s \cdot \pi\right)} \]

   4. frac-2neg 99.3%

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + 0.125 \cdot \frac{e^{\color{blue}{\frac{-\left(-r\right)}{-3 \cdot s}}}}{r \cdot \left(s \cdot \pi\right)} \]

   5. remove-double-neg 99.3%

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + 0.125 \cdot \frac{e^{\frac{\color{blue}{r}}{-3 \cdot s}}}{r \cdot \left(s \cdot \pi\right)} \]

   6. *-commutative 99.3%

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + 0.125 \cdot \frac{e^{\frac{r}{-\color{blue}{s \cdot 3}}}}{r \cdot \left(s \cdot \pi\right)} \]

   7. distribute-rgt-neg-in 99.3%

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + 0.125 \cdot \frac{e^{\frac{r}{\color{blue}{s \cdot \left(-3\right)}}}}{r \cdot \left(s \cdot \pi\right)} \]

   8. metadata-eval 99.3%

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + 0.125 \cdot \frac{e^{\frac{r}{s \cdot \color{blue}{-3}}}}{r \cdot \left(s \cdot \pi\right)} \]

4. Applied egg-rr 99.3%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + 0.125 \cdot \frac{e^{\color{blue}{\frac{r}{s \cdot -3}}}}{r \cdot \left(s \cdot \pi\right)} \]
5. Step-by-step derivation

   1. expm1-log1p-u 99.2%

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + 0.125 \cdot \frac{e^{\frac{r}{s \cdot -3}}}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(r \cdot \left(s \cdot \pi\right)\right)\right)}} \]

   2. expm1-udef 27.2%

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + 0.125 \cdot \frac{e^{\frac{r}{s \cdot -3}}}{\color{blue}{e^{\mathsf{log1p}\left(r \cdot \left(s \cdot \pi\right)\right)} - 1}} \]

   3. *-commutative 27.2%

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + 0.125 \cdot \frac{e^{\frac{r}{s \cdot -3}}}{e^{\mathsf{log1p}\left(\color{blue}{\left(s \cdot \pi\right) \cdot r}\right)} - 1} \]

   4. *-commutative 27.2%

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + 0.125 \cdot \frac{e^{\frac{r}{s \cdot -3}}}{e^{\mathsf{log1p}\left(\color{blue}{\left(\pi \cdot s\right)} \cdot r\right)} - 1} \]

   5. associate-*l* 27.2%

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + 0.125 \cdot \frac{e^{\frac{r}{s \cdot -3}}}{e^{\mathsf{log1p}\left(\color{blue}{\pi \cdot \left(s \cdot r\right)}\right)} - 1} \]

6. Applied egg-rr 27.2%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + 0.125 \cdot \frac{e^{\frac{r}{s \cdot -3}}}{\color{blue}{e^{\mathsf{log1p}\left(\pi \cdot \left(s \cdot r\right)\right)} - 1}} \]
7. Step-by-step derivation

   1. expm1-def 99.2%

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + 0.125 \cdot \frac{e^{\frac{r}{s \cdot -3}}}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(s \cdot r\right)\right)\right)}} \]

   2. expm1-log1p 99.3%

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + 0.125 \cdot \frac{e^{\frac{r}{s \cdot -3}}}{\color{blue}{\pi \cdot \left(s \cdot r\right)}} \]

   3. associate-*r* 99.3%

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + 0.125 \cdot \frac{e^{\frac{r}{s \cdot -3}}}{\color{blue}{\left(\pi \cdot s\right) \cdot r}} \]

   4. *-commutative 99.3%

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + 0.125 \cdot \frac{e^{\frac{r}{s \cdot -3}}}{\color{blue}{\left(s \cdot \pi\right)} \cdot r} \]

   5. associate-*l* 99.3%

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + 0.125 \cdot \frac{e^{\frac{r}{s \cdot -3}}}{\color{blue}{s \cdot \left(\pi \cdot r\right)}} \]

8. Simplified 99.3%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + 0.125 \cdot \frac{e^{\frac{r}{s \cdot -3}}}{\color{blue}{s \cdot \left(\pi \cdot r\right)}} \]

9. Final simplification 99.3%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{r \cdot \left(s \cdot \left(\pi \cdot 2\right)\right)} + 0.125 \cdot \frac{e^{\frac{r}{s \cdot -3}}}{s \cdot \left(\pi \cdot r\right)} \]

Alternative 4: 99.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{e^{\frac{1}{s \cdot \frac{-3}{r}}}}{r}\right) \cdot \frac{\frac{0.125}{s}}{\pi} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (*
  (+ (/ (exp (/ r (- s))) r) (/ (exp (/ 1.0 (* s (/ -3.0 r)))) r))
  (/ (/ 0.125 s) PI)))

C

float code(float s, float r) {
	return ((expf((r / -s)) / r) + (expf((1.0f / (s * (-3.0f / r)))) / r)) * ((0.125f / s) / ((float) M_PI));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(exp(Float32(r / Float32(-s))) / r) + Float32(exp(Float32(Float32(1.0) / Float32(s * Float32(Float32(-3.0) / r)))) / r)) * Float32(Float32(Float32(0.125) / s) / Float32(pi)))
end

MATLAB

function tmp = code(s, r)
	tmp = ((exp((r / -s)) / r) + (exp((single(1.0) / (s * (single(-3.0) / r)))) / r)) * ((single(0.125) / s) / single(pi));
end

Derivation

1. Initial program 99.3%

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

3. Simplified 98.9%

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

   1. pow-to-exp 98.9%

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

   2. clear-num 99.0%

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

   3. un-div-inv 98.9%

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

   4. rem-log-exp 99.2%

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

5. Applied egg-rr 99.2%

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

   1. div-inv 99.1%

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

   2. metadata-eval 99.1%

      \[\leadsto \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{e^{\color{blue}{\frac{1}{-3}} \cdot \frac{1}{\frac{s}{r}}}}{r}\right) \]

   3. clear-num 99.1%

      \[\leadsto \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{e^{\frac{1}{-3} \cdot \color{blue}{\frac{r}{s}}}}{r}\right) \]

   4. times-frac 99.2%

      \[\leadsto \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{e^{\color{blue}{\frac{1 \cdot r}{-3 \cdot s}}}}{r}\right) \]

   5. *-un-lft-identity 99.2%

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

   6. *-commutative 99.2%

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

   7. clear-num 99.2%

      \[\leadsto \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{e^{\color{blue}{\frac{1}{\frac{s \cdot -3}{r}}}}}{r}\right) \]

   8. inv-pow 99.2%

      \[\leadsto \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{e^{\color{blue}{{\left(\frac{s \cdot -3}{r}\right)}^{-1}}}}{r}\right) \]

   9. *-un-lft-identity 99.2%

      \[\leadsto \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{e^{{\left(\frac{s \cdot -3}{\color{blue}{1 \cdot r}}\right)}^{-1}}}{r}\right) \]

   10. times-frac 99.3%

       \[\leadsto \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{e^{{\color{blue}{\left(\frac{s}{1} \cdot \frac{-3}{r}\right)}}^{-1}}}{r}\right) \]

   11. /-rgt-identity 99.3%

       \[\leadsto \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{e^{{\left(\color{blue}{s} \cdot \frac{-3}{r}\right)}^{-1}}}{r}\right) \]

7. Applied egg-rr 99.3%

   \[\leadsto \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{e^{\color{blue}{{\left(s \cdot \frac{-3}{r}\right)}^{-1}}}}{r}\right) \]
8. Step-by-step derivation

   1. unpow-1 99.3%

      \[\leadsto \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{e^{\color{blue}{\frac{1}{s \cdot \frac{-3}{r}}}}}{r}\right) \]

9. Simplified 99.3%

   \[\leadsto \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{e^{\color{blue}{\frac{1}{s \cdot \frac{-3}{r}}}}}{r}\right) \]
10. Step-by-step derivation

    1. div-inv 99.2%

       \[\leadsto \color{blue}{\left(\frac{0.125}{\pi} \cdot \frac{1}{s}\right)} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{e^{\frac{1}{s \cdot \frac{-3}{r}}}}{r}\right) \]

11. Applied egg-rr 99.2%

    \[\leadsto \color{blue}{\left(\frac{0.125}{\pi} \cdot \frac{1}{s}\right)} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{e^{\frac{1}{s \cdot \frac{-3}{r}}}}{r}\right) \]
12. Step-by-step derivation

    1. associate-*l/ 99.3%

       \[\leadsto \color{blue}{\frac{0.125 \cdot \frac{1}{s}}{\pi}} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{e^{\frac{1}{s \cdot \frac{-3}{r}}}}{r}\right) \]

    2. div-inv 99.3%

       \[\leadsto \frac{\color{blue}{\frac{0.125}{s}}}{\pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{e^{\frac{1}{s \cdot \frac{-3}{r}}}}{r}\right) \]

13. Applied egg-rr 99.3%

    \[\leadsto \color{blue}{\frac{\frac{0.125}{s}}{\pi}} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{e^{\frac{1}{s \cdot \frac{-3}{r}}}}{r}\right) \]

14. Final simplification 99.3%

    \[\leadsto \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{e^{\frac{1}{s \cdot \frac{-3}{r}}}}{r}\right) \cdot \frac{\frac{0.125}{s}}{\pi} \]

Alternative 5: 99.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{e^{\frac{1}{s \cdot \frac{-3}{r}}}}{r}\right) \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (*
  (/ (/ 0.125 PI) s)
  (+ (/ (exp (/ r (- s))) r) (/ (exp (/ 1.0 (* s (/ -3.0 r)))) r))))

C

float code(float s, float r) {
	return ((0.125f / ((float) M_PI)) / s) * ((expf((r / -s)) / r) + (expf((1.0f / (s * (-3.0f / r)))) / r));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.125) / Float32(pi)) / s) * Float32(Float32(exp(Float32(r / Float32(-s))) / r) + Float32(exp(Float32(Float32(1.0) / Float32(s * Float32(Float32(-3.0) / r)))) / r)))
end

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.125) / single(pi)) / s) * ((exp((r / -s)) / r) + (exp((single(1.0) / (s * (single(-3.0) / r)))) / r));
end

Derivation

1. Initial program 99.3%

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

3. Simplified 98.9%

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

   1. pow-to-exp 98.9%

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

   2. clear-num 99.0%

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

   3. un-div-inv 98.9%

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

   4. rem-log-exp 99.2%

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

5. Applied egg-rr 99.2%

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

   1. div-inv 99.1%

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

   2. metadata-eval 99.1%

      \[\leadsto \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{e^{\color{blue}{\frac{1}{-3}} \cdot \frac{1}{\frac{s}{r}}}}{r}\right) \]

   3. clear-num 99.1%

      \[\leadsto \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{e^{\frac{1}{-3} \cdot \color{blue}{\frac{r}{s}}}}{r}\right) \]

   4. times-frac 99.2%

      \[\leadsto \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{e^{\color{blue}{\frac{1 \cdot r}{-3 \cdot s}}}}{r}\right) \]

   5. *-un-lft-identity 99.2%

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

   6. *-commutative 99.2%

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

   7. clear-num 99.2%

      \[\leadsto \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{e^{\color{blue}{\frac{1}{\frac{s \cdot -3}{r}}}}}{r}\right) \]

   8. inv-pow 99.2%

      \[\leadsto \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{e^{\color{blue}{{\left(\frac{s \cdot -3}{r}\right)}^{-1}}}}{r}\right) \]

   9. *-un-lft-identity 99.2%

      \[\leadsto \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{e^{{\left(\frac{s \cdot -3}{\color{blue}{1 \cdot r}}\right)}^{-1}}}{r}\right) \]

   10. times-frac 99.3%

       \[\leadsto \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{e^{{\color{blue}{\left(\frac{s}{1} \cdot \frac{-3}{r}\right)}}^{-1}}}{r}\right) \]

   11. /-rgt-identity 99.3%

       \[\leadsto \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{e^{{\left(\color{blue}{s} \cdot \frac{-3}{r}\right)}^{-1}}}{r}\right) \]

7. Applied egg-rr 99.3%

   \[\leadsto \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{e^{\color{blue}{{\left(s \cdot \frac{-3}{r}\right)}^{-1}}}}{r}\right) \]
8. Step-by-step derivation

   1. unpow-1 99.3%

      \[\leadsto \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{e^{\color{blue}{\frac{1}{s \cdot \frac{-3}{r}}}}}{r}\right) \]

9. Simplified 99.3%

   \[\leadsto \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{e^{\color{blue}{\frac{1}{s \cdot \frac{-3}{r}}}}}{r}\right) \]

10. Final simplification 99.3%

    \[\leadsto \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{e^{\frac{1}{s \cdot \frac{-3}{r}}}}{r}\right) \]

Alternative 6: 99.5% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.3%

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

3. Simplified 98.9%

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

   1. pow-to-exp 98.9%

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

   2. clear-num 99.0%

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

   3. un-div-inv 98.9%

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

   4. rem-log-exp 99.2%

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

5. Applied egg-rr 99.2%

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

6. Taylor expanded in s around 0 99.2%

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

7. Final simplification 99.2%

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

Alternative 7: 99.5% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.3%

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

3. Simplified 98.9%

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

   1. pow-to-exp 98.9%

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

   2. clear-num 99.0%

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

   3. un-div-inv 98.9%

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

   4. rem-log-exp 99.2%

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

5. Applied egg-rr 99.2%

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

6. Final simplification 99.2%

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

Alternative 8: 12.2% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.3%

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

3. Simplified 98.9%

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

4. Taylor expanded in r around 0 10.1%

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

5. Taylor expanded in s around inf 9.5%

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

   1. add-sqr-sqrt 9.5%

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

   2. sqrt-unprod 9.5%

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

   3. sqr-neg 9.5%

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

   4. sqrt-unprod -0.0%

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

   5. add-sqr-sqrt 4.3%

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

   6. distribute-lft-neg-in 4.3%

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

   7. log1p-expm1-u 8.0%

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

   8. distribute-lft-neg-in 8.0%

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

   9. add-sqr-sqrt -0.0%

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

   10. sqrt-unprod 12.7%

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

   11. sqr-neg 12.7%

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

   12. sqrt-unprod 12.7%

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

   13. add-sqr-sqrt 12.7%

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

   14. *-commutative 12.7%

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

   15. *-commutative 12.7%

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

   16. associate-*l* 12.7%

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

7. Applied egg-rr 12.7%

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

8. Final simplification 12.7%

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

Alternative 9: 43.6% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.3%

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

3. Simplified 98.9%

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

4. Taylor expanded in r around 0 10.1%

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

5. Taylor expanded in r around inf 10.1%

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

   1. associate-*r/ 10.1%

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

   2. neg-mul-1 10.1%

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

7. Simplified 10.1%

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

8. Taylor expanded in s around inf 9.5%

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

   1. *-commutative 9.5%

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

   2. associate-*l* 9.5%

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

10. Simplified 9.5%

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

    1. log1p-expm1-u 39.8%

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

12. Applied egg-rr 39.8%

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

13. Final simplification 39.8%

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

Alternative 10: 9.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.3%

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

3. Simplified 98.9%

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

4. Taylor expanded in r around 0 10.1%

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

5. Taylor expanded in s around 0 10.1%

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

   1. associate-*r/ 10.1%

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

   2. associate-*r/ 10.1%

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

   3. neg-mul-1 10.1%

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

7. Simplified 10.1%

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

8. Final simplification 10.1%

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

Alternative 11: 9.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.3%

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

3. Simplified 98.9%

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

4. Taylor expanded in r around 0 10.1%

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

5. Taylor expanded in r around inf 10.1%

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

   1. associate-*r/ 10.1%

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

   2. neg-mul-1 10.1%

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

7. Simplified 10.1%

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

8. Final simplification 10.1%

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

Alternative 12: 9.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.3%

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

3. Simplified 98.9%

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

4. Taylor expanded in r around 0 10.1%

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

5. Taylor expanded in r around inf 10.1%

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

   1. associate-*r/ 10.1%

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

   2. times-frac 10.1%

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

   3. associate-*r/ 10.1%

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

   4. neg-mul-1 10.1%

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

7. Simplified 10.1%

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

8. Final simplification 10.1%

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

Alternative 13: 9.0% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.3%

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

3. Simplified 98.9%

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

4. Taylor expanded in r around 0 10.1%

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

5. Taylor expanded in s around inf 9.5%

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

   1. associate-/r* 9.5%

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

   2. div-inv 9.5%

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

7. Applied egg-rr 9.5%

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

8. Final simplification 9.5%

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

Alternative 14: 9.0% accurate, 4.0× speedup

Math

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

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

MATLAB

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

Derivation

1. Initial program 99.3%

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

3. Simplified 98.9%

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

4. Taylor expanded in r around 0 10.1%

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

5. Taylor expanded in s around inf 9.5%

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

6. Final simplification 9.5%

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

Reproduce

herbie shell --seed 2023321 
(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))))