Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.6%

Time: 14.1s

Alternatives: 19

Speedup: 0.7×

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, 0.7× speedup

Math

\[\begin{array}{l} \\ \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s \cdot 2}\right)}}{r}\right) \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (*
  (/ 0.125 (* s PI))
  (+
   (/ (exp (/ r (- s))) r)
   (/ (pow (exp -0.6666666666666666) (/ r (* s 2.0))) r))))

C

float code(float s, float r) {
	return (0.125f / (s * ((float) M_PI))) * ((expf((r / -s)) / r) + (powf(expf(-0.6666666666666666f), (r / (s * 2.0f))) / r));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(0.125) / Float32(s * Float32(pi))) * Float32(Float32(exp(Float32(r / Float32(-s))) / r) + Float32((exp(Float32(-0.6666666666666666)) ^ Float32(r / Float32(s * Float32(2.0)))) / r)))
end

MATLAB

function tmp = code(s, r)
	tmp = (single(0.125) / (s * single(pi))) * ((exp((r / -s)) / r) + ((exp(single(-0.6666666666666666)) ^ (r / (s * single(2.0)))) / r));
end

Derivation

1. Initial program 99.4%

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

3. Simplified 99.1%

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

   1. add-sqr-sqrt 99.1%

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

   2. sqrt-unprod 97.9%

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

   3. pow-prod-down 97.9%

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

   4. prod-exp 98.1%

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

   5. metadata-eval 98.1%

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

6. Applied egg-rr 98.1%

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

   1. sqrt-pow1 99.5%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\color{blue}{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{\frac{r}{s}}{2}\right)}}}{r}\right) \]

   2. associate-/l/ 99.5%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{{\left(e^{-0.6666666666666666}\right)}^{\color{blue}{\left(\frac{r}{2 \cdot s}\right)}}}{r}\right) \]

8. Applied egg-rr 99.5%

   \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\color{blue}{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{2 \cdot s}\right)}}}{r}\right) \]

9. Final simplification 99.5%

   \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s \cdot 2}\right)}}{r}\right) \]
10. Add Preprocessing

Alternative 2: 99.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ 0.125 \cdot \frac{e^{\frac{r}{-s}} + {\left(e^{-0.6666666666666666}\right)}^{\left(\frac{\frac{r}{s}}{2}\right)}}{\left(s \cdot \pi\right) \cdot r} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (*
  0.125
  (/
   (+ (exp (/ r (- s))) (pow (exp -0.6666666666666666) (/ (/ r s) 2.0)))
   (* (* s PI) r))))

C

float code(float s, float r) {
	return 0.125f * ((expf((r / -s)) + powf(expf(-0.6666666666666666f), ((r / s) / 2.0f))) / ((s * ((float) M_PI)) * r));
}

Julia

function code(s, r)
	return Float32(Float32(0.125) * Float32(Float32(exp(Float32(r / Float32(-s))) + (exp(Float32(-0.6666666666666666)) ^ Float32(Float32(r / s) / Float32(2.0)))) / Float32(Float32(s * Float32(pi)) * r)))
end

MATLAB

function tmp = code(s, r)
	tmp = single(0.125) * ((exp((r / -s)) + (exp(single(-0.6666666666666666)) ^ ((r / s) / single(2.0)))) / ((s * single(pi)) * r));
end

Derivation

1. Initial program 99.4%

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

3. Simplified 99.1%

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

5. Taylor expanded in r around inf 99.3%

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

   1. mul-1-neg 99.3%

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

   2. distribute-frac-neg2 99.3%

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

   3. expm1-log1p-u 99.3%

      \[\leadsto 0.125 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{r}{-s}}\right)\right)} + e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r \cdot \left(s \cdot \pi\right)} \]

   4. expm1-undefine 99.3%

      \[\leadsto 0.125 \cdot \frac{\color{blue}{\left(e^{\mathsf{log1p}\left(e^{\frac{r}{-s}}\right)} - 1\right)} + e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r \cdot \left(s \cdot \pi\right)} \]

7. Applied egg-rr 99.3%

   \[\leadsto 0.125 \cdot \frac{\color{blue}{\left(e^{\mathsf{log1p}\left(e^{\frac{r}{-s}}\right)} - 1\right)} + e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r \cdot \left(s \cdot \pi\right)} \]
8. Step-by-step derivation

   1. log1p-undefine 99.3%

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

   2. rem-exp-log 99.3%

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

   3. associate-+r- 99.3%

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

   4. expm1-undefine 99.3%

      \[\leadsto 0.125 \cdot \frac{\left(1 + \color{blue}{\mathsf{expm1}\left(\frac{r}{-s}\right)}\right) + e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r \cdot \left(s \cdot \pi\right)} \]

   5. rem-exp-log 99.3%

      \[\leadsto 0.125 \cdot \frac{\color{blue}{e^{\log \left(1 + \mathsf{expm1}\left(\frac{r}{-s}\right)\right)}} + e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r \cdot \left(s \cdot \pi\right)} \]

   6. log1p-define 99.3%

      \[\leadsto 0.125 \cdot \frac{e^{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{r}{-s}\right)\right)}} + e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r \cdot \left(s \cdot \pi\right)} \]

   7. log1p-expm1 99.3%

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

9. Simplified 99.3%

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

    1. pow-exp 99.1%

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

    2. sqr-pow 99.1%

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

    3. pow-prod-down 99.0%

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

    4. prod-exp 99.5%

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

    5. metadata-eval 99.5%

       \[\leadsto 0.125 \cdot \frac{e^{\frac{r}{-s}} + {\left(e^{\color{blue}{-0.6666666666666666}}\right)}^{\left(\frac{\frac{r}{s}}{2}\right)}}{r \cdot \left(s \cdot \pi\right)} \]

11. Applied egg-rr 99.5%

    \[\leadsto 0.125 \cdot \frac{e^{\frac{r}{-s}} + \color{blue}{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{\frac{r}{s}}{2}\right)}}}{r \cdot \left(s \cdot \pi\right)} \]

12. Final simplification 99.5%

    \[\leadsto 0.125 \cdot \frac{e^{\frac{r}{-s}} + {\left(e^{-0.6666666666666666}\right)}^{\left(\frac{\frac{r}{s}}{2}\right)}}{\left(s \cdot \pi\right) \cdot r} \]
13. Add Preprocessing

Alternative 3: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

float code(float s, float r) {
	return ((0.125f / expf((r / s))) / (s * (((float) M_PI) * 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) / exp(Float32(r / s))) / Float32(s * Float32(Float32(pi) * r))) + Float32(Float32(0.75) * Float32(exp(Float32(r / Float32(s * Float32(-Float32(3.0))))) / Float32(r * Float32(Float32(s * Float32(pi)) * Float32(6.0))))))
end

MATLAB

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

Derivation

1. Initial program 99.4%

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

   1. times-frac 99.4%

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

   2. *-commutative 99.4%

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

   3. distribute-frac-neg 99.4%

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

   4. associate-/l* 99.4%

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

   5. *-commutative 99.4%

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

   6. *-commutative 99.4%

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

   7. associate-*l* 99.4%

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

3. Simplified 99.4%

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

5. Taylor expanded in s around 0 99.4%

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

   1. associate-*r/ 99.4%

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

   2. rec-exp 99.5%

      \[\leadsto \frac{0.125 \cdot \color{blue}{\frac{1}{e^{\frac{r}{s}}}}}{r \cdot \left(s \cdot \pi\right)} + 0.75 \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r \cdot \left(6 \cdot \left(\pi \cdot s\right)\right)} \]

   3. associate-*r/ 99.5%

      \[\leadsto \frac{\color{blue}{\frac{0.125 \cdot 1}{e^{\frac{r}{s}}}}}{r \cdot \left(s \cdot \pi\right)} + 0.75 \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r \cdot \left(6 \cdot \left(\pi \cdot s\right)\right)} \]

   4. metadata-eval 99.5%

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

   5. *-commutative 99.5%

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

   6. associate-*l* 99.4%

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

   7. *-commutative 99.4%

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

7. Simplified 99.4%

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

8. Final simplification 99.4%

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

Alternative 4: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(s, r)
	return Float32(Float32(Float32(0.125) / Float32(s * Float32(pi))) * Float32(Float32(exp(Float32(r / Float32(-s))) / r) + Float32((Float32(exp(1)) ^ Float32(r * Float32(Float32(-0.3333333333333333) / s))) / r)))
end

MATLAB

function tmp = code(s, r)
	tmp = (single(0.125) / (s * single(pi))) * ((exp((r / -s)) / r) + ((single(2.71828182845904523536) ^ (r * (single(-0.3333333333333333) / s))) / r));
end

Derivation

1. Initial program 99.4%

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

3. Simplified 99.1%

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

   1. pow-exp 99.3%

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

   2. *-commutative 99.3%

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

6. Applied egg-rr 99.3%

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

   1. *-commutative 99.3%

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

   2. *-un-lft-identity 99.3%

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

   3. exp-prod 99.4%

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

   4. clear-num 99.4%

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

   5. un-div-inv 99.4%

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

8. Applied egg-rr 99.4%

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

   1. exp-1-e 99.4%

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

   2. associate-/r/ 99.4%

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

10. Simplified 99.4%

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

11. Final simplification 99.4%

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

Alternative 5: 99.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

3. Simplified 99.1%

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

5. Taylor expanded in s around 0 99.4%

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

6. Final simplification 99.4%

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

Alternative 6: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

3. Simplified 99.1%

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

   1. pow-exp 99.3%

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

   2. *-commutative 99.3%

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

6. Applied egg-rr 99.3%

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

   1. add-cbrt-cube 35.3%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{0.125}{s \cdot \pi} \cdot \frac{0.125}{s \cdot \pi}\right) \cdot \frac{0.125}{s \cdot \pi}}} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{e^{\frac{r}{s} \cdot -0.3333333333333333}}{r}\right) \]

   2. pow1/3 35.0%

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

   3. pow3 35.0%

      \[\leadsto {\color{blue}{\left({\left(\frac{0.125}{s \cdot \pi}\right)}^{3}\right)}}^{0.3333333333333333} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{e^{\frac{r}{s} \cdot -0.3333333333333333}}{r}\right) \]

8. Applied egg-rr 35.0%

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

   1. unpow1/3 35.3%

      \[\leadsto \color{blue}{\sqrt[3]{{\left(\frac{0.125}{s \cdot \pi}\right)}^{3}}} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{e^{\frac{r}{s} \cdot -0.3333333333333333}}{r}\right) \]

   2. rem-cbrt-cube 99.3%

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

   3. associate-/r* 99.4%

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

10. Applied egg-rr 99.4%

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

Alternative 7: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

3. Simplified 99.1%

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

   1. pow-exp 99.3%

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

   2. *-commutative 99.3%

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

6. Applied egg-rr 99.3%

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

   1. associate-*l/ 99.4%

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

8. Applied egg-rr 99.4%

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

Alternative 8: 99.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

3. Simplified 99.1%

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

5. Taylor expanded in r around inf 99.3%

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

   1. mul-1-neg 99.3%

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

   2. exp-neg 99.3%

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

7. Applied egg-rr 99.3%

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

8. Final simplification 99.3%

   \[\leadsto 0.125 \cdot \frac{e^{\frac{r}{s} \cdot -0.3333333333333333} + \frac{1}{e^{\frac{r}{s}}}}{\left(s \cdot \pi\right) \cdot r} \]
9. Add Preprocessing

Alternative 9: 99.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

3. Simplified 99.1%

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

5. Taylor expanded in r around inf 99.3%

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

   1. pow-exp 99.1%

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

   2. sqr-pow 99.1%

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

   3. pow-prod-down 99.0%

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

   4. prod-exp 99.5%

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

   5. metadata-eval 99.5%

      \[\leadsto 0.125 \cdot \frac{e^{\frac{r}{-s}} + {\left(e^{\color{blue}{-0.6666666666666666}}\right)}^{\left(\frac{\frac{r}{s}}{2}\right)}}{r \cdot \left(s \cdot \pi\right)} \]

7. Applied egg-rr 99.5%

   \[\leadsto 0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}} + \color{blue}{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{\frac{r}{s}}{2}\right)}}}{r \cdot \left(s \cdot \pi\right)} \]
8. Step-by-step derivation

   1. sqrt-pow1 98.1%

      \[\leadsto 0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}} + \color{blue}{\sqrt{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s}\right)}}}}{r \cdot \left(s \cdot \pi\right)} \]

   2. pow1/2 98.1%

      \[\leadsto 0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}} + \color{blue}{{\left({\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s}\right)}\right)}^{0.5}}}{r \cdot \left(s \cdot \pi\right)} \]

   3. pow-to-exp 98.1%

      \[\leadsto 0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}} + \color{blue}{e^{\log \left({\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s}\right)}\right) \cdot 0.5}}}{r \cdot \left(s \cdot \pi\right)} \]

   4. pow-to-exp 98.0%

      \[\leadsto 0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}} + e^{\log \color{blue}{\left(e^{\log \left(e^{-0.6666666666666666}\right) \cdot \frac{r}{s}}\right)} \cdot 0.5}}{r \cdot \left(s \cdot \pi\right)} \]

   5. rem-log-exp 99.2%

      \[\leadsto 0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}} + e^{\color{blue}{\left(\log \left(e^{-0.6666666666666666}\right) \cdot \frac{r}{s}\right)} \cdot 0.5}}{r \cdot \left(s \cdot \pi\right)} \]

   6. clear-num 99.3%

      \[\leadsto 0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}} + e^{\left(\log \left(e^{-0.6666666666666666}\right) \cdot \color{blue}{\frac{1}{\frac{s}{r}}}\right) \cdot 0.5}}{r \cdot \left(s \cdot \pi\right)} \]

   7. un-div-inv 99.4%

      \[\leadsto 0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}} + e^{\color{blue}{\frac{\log \left(e^{-0.6666666666666666}\right)}{\frac{s}{r}}} \cdot 0.5}}{r \cdot \left(s \cdot \pi\right)} \]

   8. rem-log-exp 99.3%

      \[\leadsto 0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}} + e^{\frac{\color{blue}{-0.6666666666666666}}{\frac{s}{r}} \cdot 0.5}}{r \cdot \left(s \cdot \pi\right)} \]

9. Applied egg-rr 99.3%

   \[\leadsto 0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}} + \color{blue}{e^{\frac{-0.6666666666666666}{\frac{s}{r}} \cdot 0.5}}}{r \cdot \left(s \cdot \pi\right)} \]
10. Step-by-step derivation

    1. associate-*l/ 99.3%

       \[\leadsto 0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}} + e^{\color{blue}{\frac{-0.6666666666666666 \cdot 0.5}{\frac{s}{r}}}}}{r \cdot \left(s \cdot \pi\right)} \]

    2. metadata-eval 99.3%

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

    3. associate-/r/ 99.3%

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

11. Simplified 99.3%

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

12. Final simplification 99.3%

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

Alternative 10: 99.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

3. Simplified 99.1%

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

5. Taylor expanded in r around inf 99.3%

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

   1. add-cube-cbrt 99.3%

      \[\leadsto 0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}} + e^{-0.3333333333333333 \cdot \frac{r}{s}}}{\color{blue}{\left(\sqrt[3]{r \cdot \left(s \cdot \pi\right)} \cdot \sqrt[3]{r \cdot \left(s \cdot \pi\right)}\right) \cdot \sqrt[3]{r \cdot \left(s \cdot \pi\right)}}} \]

   2. pow3 99.3%

      \[\leadsto 0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}} + e^{-0.3333333333333333 \cdot \frac{r}{s}}}{\color{blue}{{\left(\sqrt[3]{r \cdot \left(s \cdot \pi\right)}\right)}^{3}}} \]

7. Applied egg-rr 99.3%

   \[\leadsto 0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}} + e^{-0.3333333333333333 \cdot \frac{r}{s}}}{\color{blue}{{\left(\sqrt[3]{r \cdot \left(s \cdot \pi\right)}\right)}^{3}}} \]

8. Taylor expanded in r around 0 99.3%

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

   1. *-commutative 99.3%

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

   2. associate-*l* 99.3%

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

   3. *-commutative 99.3%

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

10. Simplified 99.3%

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

11. Final simplification 99.3%

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

Alternative 11: 99.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

3. Simplified 99.1%

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

5. Taylor expanded in r around inf 99.3%

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

   1. mul-1-neg 99.3%

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

   2. distribute-frac-neg2 99.3%

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

   3. expm1-log1p-u 99.3%

      \[\leadsto 0.125 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{r}{-s}}\right)\right)} + e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r \cdot \left(s \cdot \pi\right)} \]

   4. expm1-undefine 99.3%

      \[\leadsto 0.125 \cdot \frac{\color{blue}{\left(e^{\mathsf{log1p}\left(e^{\frac{r}{-s}}\right)} - 1\right)} + e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r \cdot \left(s \cdot \pi\right)} \]

7. Applied egg-rr 99.3%

   \[\leadsto 0.125 \cdot \frac{\color{blue}{\left(e^{\mathsf{log1p}\left(e^{\frac{r}{-s}}\right)} - 1\right)} + e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r \cdot \left(s \cdot \pi\right)} \]
8. Step-by-step derivation

   1. log1p-undefine 99.3%

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

   2. rem-exp-log 99.3%

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

   3. associate-+r- 99.3%

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

   4. expm1-undefine 99.3%

      \[\leadsto 0.125 \cdot \frac{\left(1 + \color{blue}{\mathsf{expm1}\left(\frac{r}{-s}\right)}\right) + e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r \cdot \left(s \cdot \pi\right)} \]

   5. rem-exp-log 99.3%

      \[\leadsto 0.125 \cdot \frac{\color{blue}{e^{\log \left(1 + \mathsf{expm1}\left(\frac{r}{-s}\right)\right)}} + e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r \cdot \left(s \cdot \pi\right)} \]

   6. log1p-define 99.3%

      \[\leadsto 0.125 \cdot \frac{e^{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{r}{-s}\right)\right)}} + e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r \cdot \left(s \cdot \pi\right)} \]

   7. log1p-expm1 99.3%

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

9. Simplified 99.3%

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

    1. associate-*r/ 99.3%

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

11. Applied egg-rr 99.3%

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

12. Final simplification 99.3%

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

Alternative 12: 99.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

3. Simplified 99.1%

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

5. Taylor expanded in r around inf 99.3%

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

   1. mul-1-neg 99.3%

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

   2. distribute-frac-neg2 99.3%

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

   3. expm1-log1p-u 99.3%

      \[\leadsto 0.125 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{r}{-s}}\right)\right)} + e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r \cdot \left(s \cdot \pi\right)} \]

   4. expm1-undefine 99.3%

      \[\leadsto 0.125 \cdot \frac{\color{blue}{\left(e^{\mathsf{log1p}\left(e^{\frac{r}{-s}}\right)} - 1\right)} + e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r \cdot \left(s \cdot \pi\right)} \]

7. Applied egg-rr 99.3%

   \[\leadsto 0.125 \cdot \frac{\color{blue}{\left(e^{\mathsf{log1p}\left(e^{\frac{r}{-s}}\right)} - 1\right)} + e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r \cdot \left(s \cdot \pi\right)} \]
8. Step-by-step derivation

   1. log1p-undefine 99.3%

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

   2. rem-exp-log 99.3%

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

   3. associate-+r- 99.3%

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

   4. expm1-undefine 99.3%

      \[\leadsto 0.125 \cdot \frac{\left(1 + \color{blue}{\mathsf{expm1}\left(\frac{r}{-s}\right)}\right) + e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r \cdot \left(s \cdot \pi\right)} \]

   5. rem-exp-log 99.3%

      \[\leadsto 0.125 \cdot \frac{\color{blue}{e^{\log \left(1 + \mathsf{expm1}\left(\frac{r}{-s}\right)\right)}} + e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r \cdot \left(s \cdot \pi\right)} \]

   6. log1p-define 99.3%

      \[\leadsto 0.125 \cdot \frac{e^{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{r}{-s}\right)\right)}} + e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r \cdot \left(s \cdot \pi\right)} \]

   7. log1p-expm1 99.3%

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

9. Simplified 99.3%

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

10. Final simplification 99.3%

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

Alternative 13: 44.2% accurate, 1.1× 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 99.4%

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

3. Simplified 99.1%

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

5. Taylor expanded in s around inf 8.1%

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

   1. *-commutative 8.1%

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

   2. associate-*l* 8.1%

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

   3. *-commutative 8.1%

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

7. Simplified 8.1%

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

   1. log1p-expm1-u 41.3%

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

9. Applied egg-rr 41.3%

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

10. Final simplification 41.3%

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

Alternative 14: 11.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.4%

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

3. Simplified 99.1%

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

5. Taylor expanded in s around inf 8.1%

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

   1. log1p-expm1-u 9.7%

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

7. Applied egg-rr 9.7%

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

8. Final simplification 9.7%

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

Alternative 15: 10.1% accurate, 11.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.0625 \cdot \frac{r}{s \cdot \pi} - \frac{0.16666666666666666}{\pi}}{s} + \frac{0.25}{\pi \cdot r}}{s} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (/
  (+
   (/ (- (* 0.0625 (/ r (* s PI))) (/ 0.16666666666666666 PI)) s)
   (/ 0.25 (* PI r)))
  s))

C

float code(float s, float r) {
	return ((((0.0625f * (r / (s * ((float) M_PI)))) - (0.16666666666666666f / ((float) M_PI))) / s) + (0.25f / (((float) M_PI) * r))) / s;
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(Float32(Float32(0.0625) * Float32(r / Float32(s * Float32(pi)))) - Float32(Float32(0.16666666666666666) / Float32(pi))) / s) + Float32(Float32(0.25) / Float32(Float32(pi) * r))) / s)
end

MATLAB

function tmp = code(s, r)
	tmp = ((((single(0.0625) * (r / (s * single(pi)))) - (single(0.16666666666666666) / single(pi))) / s) + (single(0.25) / (single(pi) * r))) / s;
end

Derivation

1. Initial program 99.4%

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

3. Simplified 99.1%

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

5. Taylor expanded in r around 0 8.6%

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

6. Taylor expanded in s around 0 8.6%

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

7. Taylor expanded in s around -inf 8.8%

   \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{0.0625 \cdot \frac{r}{s \cdot \pi} - 0.16666666666666666 \cdot \frac{1}{\pi}}{s} - 0.25 \cdot \frac{1}{r \cdot \pi}}{s}} \]
8. Step-by-step derivation

   1. mul-1-neg 8.8%

      \[\leadsto \color{blue}{-\frac{-1 \cdot \frac{0.0625 \cdot \frac{r}{s \cdot \pi} - 0.16666666666666666 \cdot \frac{1}{\pi}}{s} - 0.25 \cdot \frac{1}{r \cdot \pi}}{s}} \]

   2. mul-1-neg 8.8%

      \[\leadsto -\frac{\color{blue}{\left(-\frac{0.0625 \cdot \frac{r}{s \cdot \pi} - 0.16666666666666666 \cdot \frac{1}{\pi}}{s}\right)} - 0.25 \cdot \frac{1}{r \cdot \pi}}{s} \]

   3. associate-*r/ 8.8%

      \[\leadsto -\frac{\left(-\frac{0.0625 \cdot \frac{r}{s \cdot \pi} - \color{blue}{\frac{0.16666666666666666 \cdot 1}{\pi}}}{s}\right) - 0.25 \cdot \frac{1}{r \cdot \pi}}{s} \]

   4. metadata-eval 8.8%

      \[\leadsto -\frac{\left(-\frac{0.0625 \cdot \frac{r}{s \cdot \pi} - \frac{\color{blue}{0.16666666666666666}}{\pi}}{s}\right) - 0.25 \cdot \frac{1}{r \cdot \pi}}{s} \]

   5. associate-*r/ 8.8%

      \[\leadsto -\frac{\left(-\frac{0.0625 \cdot \frac{r}{s \cdot \pi} - \frac{0.16666666666666666}{\pi}}{s}\right) - \color{blue}{\frac{0.25 \cdot 1}{r \cdot \pi}}}{s} \]

   6. metadata-eval 8.8%

      \[\leadsto -\frac{\left(-\frac{0.0625 \cdot \frac{r}{s \cdot \pi} - \frac{0.16666666666666666}{\pi}}{s}\right) - \frac{\color{blue}{0.25}}{r \cdot \pi}}{s} \]

9. Simplified 8.8%

   \[\leadsto \color{blue}{-\frac{\left(-\frac{0.0625 \cdot \frac{r}{s \cdot \pi} - \frac{0.16666666666666666}{\pi}}{s}\right) - \frac{0.25}{r \cdot \pi}}{s}} \]

10. Final simplification 8.8%

    \[\leadsto \frac{\frac{0.0625 \cdot \frac{r}{s \cdot \pi} - \frac{0.16666666666666666}{\pi}}{s} + \frac{0.25}{\pi \cdot r}}{s} \]
11. Add Preprocessing

Alternative 16: 10.1% accurate, 12.2× speedup

Math

\[\begin{array}{l} \\ \frac{0.25 \cdot \frac{1}{r} - \frac{0.16666666666666666 + \frac{r}{s} \cdot -0.0625}{s}}{s \cdot \pi} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (/
  (- (* 0.25 (/ 1.0 r)) (/ (+ 0.16666666666666666 (* (/ r s) -0.0625)) s))
  (* s PI)))

C

float code(float s, float r) {
	return ((0.25f * (1.0f / r)) - ((0.16666666666666666f + ((r / s) * -0.0625f)) / s)) / (s * ((float) M_PI));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.25) * Float32(Float32(1.0) / r)) - Float32(Float32(Float32(0.16666666666666666) + Float32(Float32(r / s) * Float32(-0.0625))) / s)) / Float32(s * Float32(pi)))
end

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.25) * (single(1.0) / r)) - ((single(0.16666666666666666) + ((r / s) * single(-0.0625))) / s)) / (s * single(pi));
end

Derivation

1. Initial program 99.4%

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

3. Simplified 99.1%

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

5. Taylor expanded in r around 0 8.6%

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

6. Taylor expanded in s around 0 8.6%

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

   1. associate-*l/ 8.6%

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

   2. sub-neg 8.6%

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

   3. metadata-eval 8.6%

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

8. Applied egg-rr 8.6%

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

9. Taylor expanded in s around -inf 8.8%

   \[\leadsto \frac{\color{blue}{-1 \cdot \frac{0.16666666666666666 + -0.0625 \cdot \frac{r}{s}}{s} + 0.25 \cdot \frac{1}{r}}}{s \cdot \pi} \]

10. Final simplification 8.8%

    \[\leadsto \frac{0.25 \cdot \frac{1}{r} - \frac{0.16666666666666666 + \frac{r}{s} \cdot -0.0625}{s}}{s \cdot \pi} \]
11. Add Preprocessing

Alternative 17: 9.1% accurate, 33.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

3. Simplified 99.1%

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

   1. add-sqr-sqrt 99.1%

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

   2. sqrt-unprod 97.9%

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

   3. pow-prod-down 97.9%

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

   4. prod-exp 98.1%

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

   5. metadata-eval 98.1%

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

6. Applied egg-rr 98.1%

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

7. Taylor expanded in s around inf 8.1%

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

   1. associate-/r* 8.1%

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

9. Simplified 8.1%

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

    1. *-un-lft-identity 8.1%

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

    2. associate-/l/ 8.1%

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

    3. *-commutative 8.1%

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

    4. associate-*r* 8.1%

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

11. Applied egg-rr 8.1%

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

12. Final simplification 8.1%

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

Alternative 18: 9.1% accurate, 33.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

3. Simplified 99.1%

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

5. Taylor expanded in s around inf 8.1%

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

   1. *-commutative 8.1%

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

   2. associate-*l* 8.1%

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

   3. *-commutative 8.1%

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

7. Simplified 8.1%

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

8. Final simplification 8.1%

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

Alternative 19: 9.1% accurate, 33.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

3. Simplified 99.1%

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

5. Taylor expanded in s around inf 8.1%

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

6. Final simplification 8.1%

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

Reproduce

herbie shell --seed 2024114 
(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))))