Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.6%

Time: 15.1s

Alternatives: 5

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

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

   1. times-frac 99.5%

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

   2. fma-def 99.6%

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

   3. associate-*l* 99.6%

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

   4. associate-/r* 99.6%

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

   5. metadata-eval 99.6%

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

   6. metadata-eval 99.6%

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

   7. associate-/r* 99.5%

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

   8. associate-*l* 99.5%

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

   9. /-rgt-identity 99.5%

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

   10. fma-def 99.5%

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

3. Simplified 99.5%

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

   1. *-commutative 99.5%

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

   2. add-sqr-sqrt 99.5%

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

   3. pow2 99.5%

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

   4. *-commutative 99.5%

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

6. Applied egg-rr 99.5%

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

   1. pow1 99.5%

      \[\leadsto \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{\frac{-r}{s}}}{r} + \frac{0.75}{\color{blue}{{\left(6 \cdot {\left(\sqrt{\pi \cdot s}\right)}^{2}\right)}^{1}}} \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r} \]

   2. *-commutative 99.5%

      \[\leadsto \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{\frac{-r}{s}}}{r} + \frac{0.75}{{\color{blue}{\left({\left(\sqrt{\pi \cdot s}\right)}^{2} \cdot 6\right)}}^{1}} \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r} \]

   3. unpow2 99.5%

      \[\leadsto \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{\frac{-r}{s}}}{r} + \frac{0.75}{{\left(\color{blue}{\left(\sqrt{\pi \cdot s} \cdot \sqrt{\pi \cdot s}\right)} \cdot 6\right)}^{1}} \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r} \]

   4. add-sqr-sqrt 99.5%

      \[\leadsto \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{\frac{-r}{s}}}{r} + \frac{0.75}{{\left(\color{blue}{\left(\pi \cdot s\right)} \cdot 6\right)}^{1}} \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r} \]

   5. *-commutative 99.5%

      \[\leadsto \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{\frac{-r}{s}}}{r} + \frac{0.75}{{\left(\color{blue}{\left(s \cdot \pi\right)} \cdot 6\right)}^{1}} \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r} \]

   6. associate-*l* 99.6%

      \[\leadsto \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{\frac{-r}{s}}}{r} + \frac{0.75}{{\color{blue}{\left(s \cdot \left(\pi \cdot 6\right)\right)}}^{1}} \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r} \]

8. Applied egg-rr 99.6%

   \[\leadsto \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{\frac{-r}{s}}}{r} + \frac{0.75}{\color{blue}{{\left(s \cdot \left(\pi \cdot 6\right)\right)}^{1}}} \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r} \]
9. Step-by-step derivation

   1. pow1 99.6%

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

   2. clear-num 99.5%

      \[\leadsto \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{\frac{-r}{s}}}{r} + \color{blue}{\frac{1}{\frac{s \cdot \left(\pi \cdot 6\right)}{0.75}}} \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r} \]

   3. associate-/r/ 99.5%

      \[\leadsto \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{\frac{-r}{s}}}{r} + \color{blue}{\left(\frac{1}{s \cdot \left(\pi \cdot 6\right)} \cdot 0.75\right)} \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r} \]

   4. associate-*r* 99.5%

      \[\leadsto \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{\frac{-r}{s}}}{r} + \left(\frac{1}{\color{blue}{\left(s \cdot \pi\right) \cdot 6}} \cdot 0.75\right) \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r} \]

   5. *-commutative 99.5%

      \[\leadsto \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{\frac{-r}{s}}}{r} + \left(\frac{1}{\color{blue}{\left(\pi \cdot s\right)} \cdot 6} \cdot 0.75\right) \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r} \]

   6. associate-*l* 99.5%

      \[\leadsto \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{\frac{-r}{s}}}{r} + \left(\frac{1}{\color{blue}{\pi \cdot \left(s \cdot 6\right)}} \cdot 0.75\right) \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r} \]

10. Applied egg-rr 99.5%

    \[\leadsto \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{\frac{-r}{s}}}{r} + \color{blue}{\left(\frac{1}{\pi \cdot \left(s \cdot 6\right)} \cdot 0.75\right)} \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r} \]
11. Step-by-step derivation

    1. associate-*l/ 99.5%

       \[\leadsto \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{\frac{-r}{s}}}{r} + \color{blue}{\frac{1 \cdot 0.75}{\pi \cdot \left(s \cdot 6\right)}} \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r} \]

    2. metadata-eval 99.5%

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

    3. *-commutative 99.5%

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

    4. associate-*l* 99.6%

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

12. Applied egg-rr 99.6%

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

13. Final simplification 99.6%

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

Alternative 2: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

   1. times-frac 99.5%

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

   2. fma-def 99.6%

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

   3. associate-*l* 99.6%

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

   4. associate-/r* 99.6%

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

   5. metadata-eval 99.6%

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

   6. metadata-eval 99.6%

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

   7. associate-/r* 99.5%

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

   8. associate-*l* 99.5%

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

   9. /-rgt-identity 99.5%

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

   10. fma-def 99.5%

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

3. Simplified 99.5%

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

   1. *-commutative 99.5%

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

   2. add-sqr-sqrt 99.5%

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

   3. pow2 99.5%

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

   4. *-commutative 99.5%

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

6. Applied egg-rr 99.5%

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

   1. pow1 99.5%

      \[\leadsto \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{\frac{-r}{s}}}{r} + \frac{0.75}{\color{blue}{{\left(6 \cdot {\left(\sqrt{\pi \cdot s}\right)}^{2}\right)}^{1}}} \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r} \]

   2. *-commutative 99.5%

      \[\leadsto \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{\frac{-r}{s}}}{r} + \frac{0.75}{{\color{blue}{\left({\left(\sqrt{\pi \cdot s}\right)}^{2} \cdot 6\right)}}^{1}} \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r} \]

   3. unpow2 99.5%

      \[\leadsto \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{\frac{-r}{s}}}{r} + \frac{0.75}{{\left(\color{blue}{\left(\sqrt{\pi \cdot s} \cdot \sqrt{\pi \cdot s}\right)} \cdot 6\right)}^{1}} \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r} \]

   4. add-sqr-sqrt 99.5%

      \[\leadsto \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{\frac{-r}{s}}}{r} + \frac{0.75}{{\left(\color{blue}{\left(\pi \cdot s\right)} \cdot 6\right)}^{1}} \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r} \]

   5. *-commutative 99.5%

      \[\leadsto \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{\frac{-r}{s}}}{r} + \frac{0.75}{{\left(\color{blue}{\left(s \cdot \pi\right)} \cdot 6\right)}^{1}} \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r} \]

   6. associate-*l* 99.6%

      \[\leadsto \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{\frac{-r}{s}}}{r} + \frac{0.75}{{\color{blue}{\left(s \cdot \left(\pi \cdot 6\right)\right)}}^{1}} \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r} \]

8. Applied egg-rr 99.6%

   \[\leadsto \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{\frac{-r}{s}}}{r} + \frac{0.75}{\color{blue}{{\left(s \cdot \left(\pi \cdot 6\right)\right)}^{1}}} \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r} \]
9. Step-by-step derivation

   1. pow1 99.6%

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

   2. clear-num 99.5%

      \[\leadsto \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{\frac{-r}{s}}}{r} + \color{blue}{\frac{1}{\frac{s \cdot \left(\pi \cdot 6\right)}{0.75}}} \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r} \]

   3. associate-*r* 99.5%

      \[\leadsto \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{\frac{-r}{s}}}{r} + \frac{1}{\frac{\color{blue}{\left(s \cdot \pi\right) \cdot 6}}{0.75}} \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r} \]

   4. *-commutative 99.5%

      \[\leadsto \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{\frac{-r}{s}}}{r} + \frac{1}{\frac{\color{blue}{\left(\pi \cdot s\right)} \cdot 6}{0.75}} \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r} \]

   5. associate-/l* 99.5%

      \[\leadsto \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{\frac{-r}{s}}}{r} + \frac{1}{\color{blue}{\frac{\pi \cdot s}{\frac{0.75}{6}}}} \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r} \]

   6. metadata-eval 99.5%

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

   7. clear-num 99.5%

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

   8. associate-/l/ 99.5%

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

   9. un-div-inv 99.5%

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

   10. add-sqr-sqrt 99.5%

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

   11. pow2 99.5%

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

   12. un-div-inv 99.5%

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

10. Applied egg-rr 99.5%

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

    1. unpow2 99.5%

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

    2. add-sqr-sqrt 99.5%

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

    3. frac-2neg 99.5%

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

    4. distribute-neg-frac 99.5%

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

    5. metadata-eval 99.5%

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

12. Applied egg-rr 99.5%

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

13. Final simplification 99.5%

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

Alternative 3: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

   1. times-frac 99.5%

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

   2. fma-def 99.6%

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

   3. associate-*l* 99.6%

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

   4. associate-/r* 99.6%

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

   5. metadata-eval 99.6%

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

   6. metadata-eval 99.6%

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

   7. associate-/r* 99.5%

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

   8. associate-*l* 99.5%

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

   9. /-rgt-identity 99.5%

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

   10. fma-def 99.5%

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

3. Simplified 99.5%

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

   1. *-commutative 99.5%

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

   2. add-sqr-sqrt 99.5%

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

   3. pow2 99.5%

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

   4. *-commutative 99.5%

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

6. Applied egg-rr 99.5%

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

   1. pow1 99.5%

      \[\leadsto \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{\frac{-r}{s}}}{r} + \frac{0.75}{\color{blue}{{\left(6 \cdot {\left(\sqrt{\pi \cdot s}\right)}^{2}\right)}^{1}}} \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r} \]

   2. *-commutative 99.5%

      \[\leadsto \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{\frac{-r}{s}}}{r} + \frac{0.75}{{\color{blue}{\left({\left(\sqrt{\pi \cdot s}\right)}^{2} \cdot 6\right)}}^{1}} \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r} \]

   3. unpow2 99.5%

      \[\leadsto \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{\frac{-r}{s}}}{r} + \frac{0.75}{{\left(\color{blue}{\left(\sqrt{\pi \cdot s} \cdot \sqrt{\pi \cdot s}\right)} \cdot 6\right)}^{1}} \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r} \]

   4. add-sqr-sqrt 99.5%

      \[\leadsto \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{\frac{-r}{s}}}{r} + \frac{0.75}{{\left(\color{blue}{\left(\pi \cdot s\right)} \cdot 6\right)}^{1}} \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r} \]

   5. *-commutative 99.5%

      \[\leadsto \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{\frac{-r}{s}}}{r} + \frac{0.75}{{\left(\color{blue}{\left(s \cdot \pi\right)} \cdot 6\right)}^{1}} \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r} \]

   6. associate-*l* 99.6%

      \[\leadsto \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{\frac{-r}{s}}}{r} + \frac{0.75}{{\color{blue}{\left(s \cdot \left(\pi \cdot 6\right)\right)}}^{1}} \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r} \]

8. Applied egg-rr 99.6%

   \[\leadsto \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{\frac{-r}{s}}}{r} + \frac{0.75}{\color{blue}{{\left(s \cdot \left(\pi \cdot 6\right)\right)}^{1}}} \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r} \]
9. Step-by-step derivation

   1. pow1 99.6%

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

   2. clear-num 99.5%

      \[\leadsto \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{\frac{-r}{s}}}{r} + \color{blue}{\frac{1}{\frac{s \cdot \left(\pi \cdot 6\right)}{0.75}}} \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r} \]

   3. associate-*r* 99.5%

      \[\leadsto \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{\frac{-r}{s}}}{r} + \frac{1}{\frac{\color{blue}{\left(s \cdot \pi\right) \cdot 6}}{0.75}} \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r} \]

   4. *-commutative 99.5%

      \[\leadsto \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{\frac{-r}{s}}}{r} + \frac{1}{\frac{\color{blue}{\left(\pi \cdot s\right)} \cdot 6}{0.75}} \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r} \]

   5. associate-/l* 99.5%

      \[\leadsto \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{\frac{-r}{s}}}{r} + \frac{1}{\color{blue}{\frac{\pi \cdot s}{\frac{0.75}{6}}}} \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r} \]

   6. metadata-eval 99.5%

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

   7. clear-num 99.5%

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

   8. associate-/l/ 99.5%

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

   9. un-div-inv 99.5%

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

   10. add-sqr-sqrt 99.5%

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

   11. pow2 99.5%

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

   12. un-div-inv 99.5%

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

10. Applied egg-rr 99.5%

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

    1. unpow2 99.5%

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

    2. add-sqr-sqrt 99.5%

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

    3. associate-/l/ 99.5%

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

    4. associate-/r* 99.5%

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

12. Applied egg-rr 99.5%

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

13. Final simplification 99.5%

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

Alternative 4: 8.9% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

   1. times-frac 99.5%

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

   2. fma-def 99.6%

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

   3. associate-*l* 99.6%

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

   4. associate-/r* 99.6%

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

   5. metadata-eval 99.6%

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

   6. metadata-eval 99.6%

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

   7. associate-/r* 99.5%

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

   8. associate-*l* 99.5%

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

   9. /-rgt-identity 99.5%

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

   10. fma-def 99.5%

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

3. Simplified 99.5%

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

5. Taylor expanded in r around 0 9.2%

   \[\leadsto \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{\color{blue}{1}}{r} + \frac{0.75}{6 \cdot \left(\pi \cdot s\right)} \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r} \]
6. Step-by-step derivation

   1. associate-*l/ 9.2%

      \[\leadsto \color{blue}{\frac{0.25 \cdot \frac{1}{r}}{s \cdot \left(2 \cdot \pi\right)}} + \frac{0.75}{6 \cdot \left(\pi \cdot s\right)} \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r} \]

   2. *-commutative 9.2%

      \[\leadsto \frac{0.25 \cdot \frac{1}{r}}{\color{blue}{\left(2 \cdot \pi\right) \cdot s}} + \frac{0.75}{6 \cdot \left(\pi \cdot s\right)} \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r} \]

   3. associate-*l* 9.2%

      \[\leadsto \frac{0.25 \cdot \frac{1}{r}}{\color{blue}{2 \cdot \left(\pi \cdot s\right)}} + \frac{0.75}{6 \cdot \left(\pi \cdot s\right)} \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r} \]

   4. add-sqr-sqrt 9.2%

      \[\leadsto \frac{0.25 \cdot \frac{1}{r}}{2 \cdot \color{blue}{\left(\sqrt{\pi \cdot s} \cdot \sqrt{\pi \cdot s}\right)}} + \frac{0.75}{6 \cdot \left(\pi \cdot s\right)} \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r} \]

   5. unpow2 9.2%

      \[\leadsto \frac{0.25 \cdot \frac{1}{r}}{2 \cdot \color{blue}{{\left(\sqrt{\pi \cdot s}\right)}^{2}}} + \frac{0.75}{6 \cdot \left(\pi \cdot s\right)} \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r} \]

   6. associate-/r* 9.2%

      \[\leadsto \color{blue}{\frac{\frac{0.25 \cdot \frac{1}{r}}{2}}{{\left(\sqrt{\pi \cdot s}\right)}^{2}}} + \frac{0.75}{6 \cdot \left(\pi \cdot s\right)} \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r} \]

   7. un-div-inv 9.2%

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

   8. unpow2 9.2%

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

   9. add-sqr-sqrt 9.2%

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

7. Applied egg-rr 9.2%

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

8. Taylor expanded in r around 0 9.2%

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

   1. *-commutative 9.2%

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

   2. associate-*r* 9.2%

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

10. Simplified 9.2%

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

11. Final simplification 9.2%

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

Alternative 5: 8.9% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

   1. times-frac 99.5%

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

   2. fma-def 99.6%

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

   3. associate-*l* 99.6%

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

   4. associate-/r* 99.6%

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

   5. metadata-eval 99.6%

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

   6. metadata-eval 99.6%

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

   7. associate-/r* 99.5%

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

   8. associate-*l* 99.5%

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

   9. /-rgt-identity 99.5%

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

   10. fma-def 99.5%

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

3. Simplified 99.5%

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

5. Taylor expanded in r around 0 9.2%

   \[\leadsto \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{\color{blue}{1}}{r} + \frac{0.75}{6 \cdot \left(\pi \cdot s\right)} \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r} \]

6. Taylor expanded in s around 0 9.2%

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

   1. neg-mul-1 9.2%

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

   2. *-commutative 9.2%

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

   3. times-frac 9.2%

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

   4. metadata-eval 9.2%

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

   5. *-commutative 9.2%

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

8. Applied egg-rr 9.2%

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

9. Final simplification 9.2%

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

Reproduce

herbie shell --seed 2024027 
(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))))