Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.6%

Time: 12.7s

Alternatives: 11

Speedup: N/A×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = (single(0.125) / ((s * single(pi)) * (r * exp((r / s))))) + (single(0.75) * (exp(((r * single(-0.3333333333333333)) / s)) / (r * (s * (single(pi) * single(6.0))))));
end

Derivation

1. Initial program 99.6%

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

   1. times-frac 99.6%

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

   2. *-commutative 99.6%

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

   3. distribute-frac-neg 99.6%

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

   4. associate-/l* 99.6%

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

   5. *-commutative 99.6%

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

   6. *-commutative 99.6%

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

   7. associate-*l* 99.5%

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

3. Simplified 99.5%

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

5. Taylor expanded in s around 0 99.5%

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

   1. *-commutative 99.5%

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

   2. associate-*r* 99.6%

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

7. Simplified 99.6%

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

8. Taylor expanded in r around 0 99.6%

   \[\leadsto \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{-\frac{r}{s}}}{r} + 0.75 \cdot \frac{e^{\color{blue}{-0.3333333333333333 \cdot \frac{r}{s}}}}{r \cdot \left(s \cdot \left(\pi \cdot 6\right)\right)} \]
9. Step-by-step derivation

   1. associate-*r/ 99.6%

      \[\leadsto \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{-\frac{r}{s}}}{r} + 0.75 \cdot \frac{e^{\color{blue}{\frac{-0.3333333333333333 \cdot r}{s}}}}{r \cdot \left(s \cdot \left(\pi \cdot 6\right)\right)} \]

   2. *-commutative 99.6%

      \[\leadsto \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{-\frac{r}{s}}}{r} + 0.75 \cdot \frac{e^{\frac{\color{blue}{r \cdot -0.3333333333333333}}{s}}}{r \cdot \left(s \cdot \left(\pi \cdot 6\right)\right)} \]

10. Simplified 99.6%

    \[\leadsto \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{-\frac{r}{s}}}{r} + 0.75 \cdot \frac{e^{\color{blue}{\frac{r \cdot -0.3333333333333333}{s}}}}{r \cdot \left(s \cdot \left(\pi \cdot 6\right)\right)} \]

11. Taylor expanded in s around 0 99.6%

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

    1. clear-num 99.7%

       \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \color{blue}{\frac{1}{\frac{r}{e^{-\frac{r}{s}}}}} + 0.75 \cdot \frac{e^{\frac{r \cdot -0.3333333333333333}{s}}}{r \cdot \left(s \cdot \left(\pi \cdot 6\right)\right)} \]

    2. frac-times 99.7%

       \[\leadsto \color{blue}{\frac{0.125 \cdot 1}{\left(s \cdot \pi\right) \cdot \frac{r}{e^{-\frac{r}{s}}}}} + 0.75 \cdot \frac{e^{\frac{r \cdot -0.3333333333333333}{s}}}{r \cdot \left(s \cdot \left(\pi \cdot 6\right)\right)} \]

    3. metadata-eval 99.7%

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

    4. div-inv 99.7%

       \[\leadsto \frac{0.125}{\left(s \cdot \pi\right) \cdot \color{blue}{\left(r \cdot \frac{1}{e^{-\frac{r}{s}}}\right)}} + 0.75 \cdot \frac{e^{\frac{r \cdot -0.3333333333333333}{s}}}{r \cdot \left(s \cdot \left(\pi \cdot 6\right)\right)} \]

    5. add-sqr-sqrt -0.0%

       \[\leadsto \frac{0.125}{\left(s \cdot \pi\right) \cdot \left(r \cdot \frac{1}{e^{\color{blue}{\sqrt{-\frac{r}{s}} \cdot \sqrt{-\frac{r}{s}}}}}\right)} + 0.75 \cdot \frac{e^{\frac{r \cdot -0.3333333333333333}{s}}}{r \cdot \left(s \cdot \left(\pi \cdot 6\right)\right)} \]

    6. sqrt-unprod 7.1%

       \[\leadsto \frac{0.125}{\left(s \cdot \pi\right) \cdot \left(r \cdot \frac{1}{e^{\color{blue}{\sqrt{\left(-\frac{r}{s}\right) \cdot \left(-\frac{r}{s}\right)}}}}\right)} + 0.75 \cdot \frac{e^{\frac{r \cdot -0.3333333333333333}{s}}}{r \cdot \left(s \cdot \left(\pi \cdot 6\right)\right)} \]

    7. sqr-neg 7.1%

       \[\leadsto \frac{0.125}{\left(s \cdot \pi\right) \cdot \left(r \cdot \frac{1}{e^{\sqrt{\color{blue}{\frac{r}{s} \cdot \frac{r}{s}}}}}\right)} + 0.75 \cdot \frac{e^{\frac{r \cdot -0.3333333333333333}{s}}}{r \cdot \left(s \cdot \left(\pi \cdot 6\right)\right)} \]

    8. sqrt-unprod 7.1%

       \[\leadsto \frac{0.125}{\left(s \cdot \pi\right) \cdot \left(r \cdot \frac{1}{e^{\color{blue}{\sqrt{\frac{r}{s}} \cdot \sqrt{\frac{r}{s}}}}}\right)} + 0.75 \cdot \frac{e^{\frac{r \cdot -0.3333333333333333}{s}}}{r \cdot \left(s \cdot \left(\pi \cdot 6\right)\right)} \]

    9. add-sqr-sqrt 7.1%

       \[\leadsto \frac{0.125}{\left(s \cdot \pi\right) \cdot \left(r \cdot \frac{1}{e^{\color{blue}{\frac{r}{s}}}}\right)} + 0.75 \cdot \frac{e^{\frac{r \cdot -0.3333333333333333}{s}}}{r \cdot \left(s \cdot \left(\pi \cdot 6\right)\right)} \]

    10. exp-neg 7.1%

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

    11. add-sqr-sqrt -0.0%

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

    12. sqrt-unprod 99.7%

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

    13. sqr-neg 99.7%

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

    14. sqrt-unprod 99.7%

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

    15. add-sqr-sqrt 99.7%

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

13. Applied egg-rr 99.7%

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

Alternative 2: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Simplified 99.4%

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

5. Taylor expanded in r around inf 99.6%

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

   1. associate-*r/ 99.6%

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

7. Applied egg-rr 99.6%

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

8. Final simplification 99.6%

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

Alternative 3: 99.5% accurate, 1.1× speedup

Math

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

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 99.6%

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

3. Simplified 99.4%

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

5. Taylor expanded in r around inf 99.6%

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

   1. mul-1-neg 99.6%

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

   2. distribute-frac-neg2 99.6%

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

   3. add-sqr-sqrt -0.0%

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

   4. sqrt-unprod 7.1%

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

   5. distribute-frac-neg2 7.1%

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

   6. distribute-frac-neg2 7.1%

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

   7. sqr-neg 7.1%

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

   8. sqrt-unprod 7.1%

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

   9. add-sqr-sqrt 7.1%

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

   10. frac-2neg 7.1%

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

   11. add-sqr-sqrt -0.0%

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

   12. sqrt-unprod 99.6%

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

   13. sqr-neg 99.6%

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

   14. sqrt-prod 99.6%

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

   15. add-sqr-sqrt 99.6%

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

7. Applied egg-rr 99.6%

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

8. Final simplification 99.6%

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

Alternative 4: 45.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 4:\\ \;\;\;\;\frac{\frac{0.25}{\pi \cdot r} - \frac{\frac{\frac{r}{\pi} \cdot -0.06944444444444445}{s} + \frac{0.16666666666666666}{\pi}}{s}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-0.25}{s}}{\mathsf{log1p}\left(\mathsf{expm1}\left(\pi \cdot r\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (if (<= r 4.0)
   (/
    (-
     (/ 0.25 (* PI r))
     (/
      (+ (/ (* (/ r PI) -0.06944444444444445) s) (/ 0.16666666666666666 PI))
      s))
    s)
   (/ (/ -0.25 s) (log1p (expm1 (* PI r))))))

C

float code(float s, float r) {
	float tmp;
	if (r <= 4.0f) {
		tmp = ((0.25f / (((float) M_PI) * r)) - (((((r / ((float) M_PI)) * -0.06944444444444445f) / s) + (0.16666666666666666f / ((float) M_PI))) / s)) / s;
	} else {
		tmp = (-0.25f / s) / log1pf(expm1f((((float) M_PI) * r)));
	}
	return tmp;
}

Julia

function code(s, r)
	tmp = Float32(0.0)
	if (r <= Float32(4.0))
		tmp = Float32(Float32(Float32(Float32(0.25) / Float32(Float32(pi) * r)) - Float32(Float32(Float32(Float32(Float32(r / Float32(pi)) * Float32(-0.06944444444444445)) / s) + Float32(Float32(0.16666666666666666) / Float32(pi))) / s)) / s);
	else
		tmp = Float32(Float32(Float32(-0.25) / s) / log1p(expm1(Float32(Float32(pi) * r))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if r < 4

   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. +-commutative 99.4%

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

      2. times-frac 99.4%

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

      3. fma-define 99.4%

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

      4. associate-*l* 99.2%

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

      5. associate-/r* 99.2%

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

      6. metadata-eval 99.2%

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

      7. *-commutative 99.2%

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

      8. neg-mul-1 99.2%

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

      9. times-frac 99.3%

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

      10. metadata-eval 99.3%

          \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\color{blue}{-0.3333333333333333} \cdot \frac{r}{s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

      11. times-frac 99.2%

          \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r}, \color{blue}{\frac{0.25}{\left(2 \cdot \pi\right) \cdot s} \cdot \frac{e^{\frac{-r}{s}}}{r}}\right) \]

   3. Simplified 99.2%

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

   5. Taylor expanded in s around -inf 14.9%

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{-0.0625 \cdot \frac{r}{\pi} + -0.006944444444444444 \cdot \frac{r}{\pi}}{s} - 0.16666666666666666 \cdot \frac{1}{\pi}}{s} - 0.25 \cdot \frac{1}{r \cdot \pi}}{s}} \]
   6. Step-by-step derivation

      1. mul-1-neg 14.9%

         \[\leadsto \color{blue}{-\frac{-1 \cdot \frac{-1 \cdot \frac{-0.0625 \cdot \frac{r}{\pi} + -0.006944444444444444 \cdot \frac{r}{\pi}}{s} - 0.16666666666666666 \cdot \frac{1}{\pi}}{s} - 0.25 \cdot \frac{1}{r \cdot \pi}}{s}} \]

   7. Simplified 14.9%

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

   if 4 < r

   1. Initial program 99.9%

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

   3. Simplified 99.9%

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

      1. add-sqr-sqrt 99.8%

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

      2. sqrt-unprod 99.9%

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

      3. prod-exp 99.9%

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

      4. metadata-eval 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in s around inf 5.0%

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

      1. *-commutative 5.0%

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

      2. associate-/r* 5.0%

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

      3. *-commutative 5.0%

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

      4. associate-/r* 5.0%

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

   9. Simplified 5.0%

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

       1. div-inv 5.0%

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

       2. frac-2neg 5.0%

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

       3. distribute-neg-frac 5.0%

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

       4. metadata-eval 5.0%

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

       5. add-sqr-sqrt -0.0%

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

       6. sqrt-unprod 4.6%

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

       7. sqr-neg 4.6%

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

       8. sqrt-unprod 4.9%

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

       9. add-sqr-sqrt 4.9%

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

   11. Applied egg-rr 4.9%

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

       1. associate-*r/ 4.9%

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

       2. associate-*l/ 4.9%

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

       3. associate-*r/ 4.9%

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

       4. associate-*l/ 4.9%

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

       5. associate-/l/ 4.9%

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

       6. associate-*r/ 4.9%

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

       7. metadata-eval 4.9%

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

   13. Simplified 4.9%

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

       1. log1p-expm1-u 88.0%

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

   15. Applied egg-rr 88.0%

       \[\leadsto \frac{\frac{-0.25}{s}}{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(r \cdot \pi\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 46.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 4:\\ \;\;\;\;\frac{\frac{0.25}{\pi \cdot r} - \frac{\frac{\frac{r}{\pi} \cdot -0.06944444444444445}{s} + \frac{0.16666666666666666}{\pi}}{s}}{s}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-0.25}{s}}{\mathsf{log1p}\left(\mathsf{expm1}\left(\pi \cdot r\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 12.0% 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 99.6%

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

3. Simplified 99.4%

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

5. Taylor expanded in s around inf 8.7%

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

   1. log1p-expm1-u 10.3%

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

7. Applied egg-rr 10.3%

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

8. Final simplification 10.3%

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

Alternative 6: 9.9% accurate, 11.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.25}{\pi \cdot r} - \frac{\frac{\frac{r}{\pi} \cdot -0.06944444444444445}{s} + \frac{0.16666666666666666}{\pi}}{s}}{s} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.25) / (single(pi) * r)) - (((((r / single(pi)) * single(-0.06944444444444445)) / s) + (single(0.16666666666666666) / single(pi))) / s)) / s;
end

Derivation

1. Initial program 99.6%

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

   1. +-commutative 99.6%

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

   2. times-frac 99.7%

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

   3. fma-define 99.6%

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

   4. associate-*l* 99.5%

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

   5. associate-/r* 99.5%

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

   6. metadata-eval 99.5%

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

   7. *-commutative 99.5%

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

   8. neg-mul-1 99.5%

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

   9. times-frac 99.5%

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

   10. metadata-eval 99.5%

       \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\color{blue}{-0.3333333333333333} \cdot \frac{r}{s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   11. times-frac 99.5%

       \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r}, \color{blue}{\frac{0.25}{\left(2 \cdot \pi\right) \cdot s} \cdot \frac{e^{\frac{-r}{s}}}{r}}\right) \]

3. Simplified 99.5%

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

5. Taylor expanded in s around -inf 10.0%

   \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{-0.0625 \cdot \frac{r}{\pi} + -0.006944444444444444 \cdot \frac{r}{\pi}}{s} - 0.16666666666666666 \cdot \frac{1}{\pi}}{s} - 0.25 \cdot \frac{1}{r \cdot \pi}}{s}} \]
6. Step-by-step derivation

   1. mul-1-neg 10.0%

      \[\leadsto \color{blue}{-\frac{-1 \cdot \frac{-1 \cdot \frac{-0.0625 \cdot \frac{r}{\pi} + -0.006944444444444444 \cdot \frac{r}{\pi}}{s} - 0.16666666666666666 \cdot \frac{1}{\pi}}{s} - 0.25 \cdot \frac{1}{r \cdot \pi}}{s}} \]

7. Simplified 10.0%

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

8. Final simplification 10.0%

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

Alternative 7: 9.0% accurate, 15.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Simplified 99.4%

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

5. Taylor expanded in s around inf 8.9%

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

   1. clear-num 8.9%

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

   2. inv-pow 8.9%

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

   3. cancel-sign-sub-inv 8.9%

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

   4. un-div-inv 8.9%

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

   5. *-commutative 8.9%

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

   6. un-div-inv 8.9%

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

   7. metadata-eval 8.9%

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

7. Applied egg-rr 8.9%

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

   1. unpow-1 8.9%

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

   2. +-commutative 8.9%

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

   3. metadata-eval 8.9%

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

   4. distribute-neg-frac 8.9%

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

   5. associate-/l/ 8.9%

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

   6. unsub-neg 8.9%

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

   7. associate-/l/ 8.9%

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

   8. *-commutative 8.9%

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

9. Simplified 8.9%

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

10. Final simplification 8.9%

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

Alternative 8: 9.0% accurate, 17.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Simplified 99.4%

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

5. Taylor expanded in s around inf 8.9%

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

   1. associate-*r/ 8.9%

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

   2. metadata-eval 8.9%

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

   3. associate-*r/ 8.9%

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

   4. metadata-eval 8.9%

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

7. Simplified 8.9%

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

8. Final simplification 8.9%

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

Alternative 9: 9.0% accurate, 33.0× speedup

Math

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

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(Float32(0.25) / 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 99.6%

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

3. Simplified 99.4%

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

5. Taylor expanded in s around inf 8.9%

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

6. Taylor expanded in r around 0 8.7%

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

   1. *-commutative 8.7%

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

   2. associate-*r* 8.7%

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

   3. *-commutative 8.7%

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

   4. associate-/l/ 8.7%

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

   5. *-commutative 8.7%

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

8. Simplified 8.7%

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

9. Final simplification 8.7%

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

Alternative 10: 9.0% accurate, 33.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Simplified 99.4%

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

   1. add-sqr-sqrt 99.2%

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

   2. sqrt-unprod 99.4%

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

   3. prod-exp 99.5%

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

   4. metadata-eval 99.5%

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

6. Applied egg-rr 99.5%

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

7. Taylor expanded in s around inf 8.7%

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

   1. *-commutative 8.7%

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

   2. associate-/r* 8.7%

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

   3. *-commutative 8.7%

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

   4. associate-/r* 8.7%

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

9. Simplified 8.7%

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

10. Taylor expanded in s around 0 8.7%

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

    1. associate-/r* 8.7%

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

12. Simplified 8.7%

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

Alternative 11: 9.0% 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 99.6%

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

3. Simplified 99.4%

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

5. Taylor expanded in s around inf 8.7%

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

6. Final simplification 8.7%

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

Reproduce

herbie shell --seed 2024145 
(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))))