Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.5%

Time: 12.1s

Alternatives: 16

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: 99.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{-r}{s}}}{r} + \frac{0.75}{{\left(\sqrt[3]{s \cdot \left(\pi \cdot 6\right)}\right)}^{3}} \cdot \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (* (/ 0.125 (* s PI)) (/ (exp (/ (- r) s)) r))
  (*
   (/ 0.75 (pow (cbrt (* s (* PI 6.0))) 3.0))
   (/ (exp (* -0.3333333333333333 (/ r s))) r))))

C

float code(float s, float r) {
	return ((0.125f / (s * ((float) M_PI))) * (expf((-r / s)) / r)) + ((0.75f / powf(cbrtf((s * (((float) M_PI) * 6.0f))), 3.0f)) * (expf((-0.3333333333333333f * (r / s))) / r));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.125) / Float32(s * Float32(pi))) * Float32(exp(Float32(Float32(-r) / s)) / r)) + Float32(Float32(Float32(0.75) / (cbrt(Float32(s * Float32(Float32(pi) * Float32(6.0)))) ^ Float32(3.0))) * Float32(exp(Float32(Float32(-0.3333333333333333) * Float32(r / s))) / 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 99.5%

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

   3. associate-*l* 99.5%

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

   4. associate-/r* 99.5%

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

   5. metadata-eval 99.5%

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

   6. metadata-eval 99.5%

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

   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. Step-by-step derivation

   1. associate-*r* 99.6%

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

   2. add-cube-cbrt 99.5%

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

   3. pow3 99.6%

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

   4. *-commutative 99.6%

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

   5. *-commutative 99.6%

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

5. Applied egg-rr 99.6%

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

6. Taylor expanded in s around 0 99.6%

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

7. Taylor expanded in r around 0 99.6%

   \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{-r}{s}}}{r} + \frac{0.75}{{\left(\sqrt[3]{s \cdot \left(\pi \cdot 6\right)}\right)}^{3}} \cdot \frac{e^{\color{blue}{-0.3333333333333333 \cdot \frac{r}{s}}}}{r} \]

8. Final simplification 99.6%

   \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{-r}{s}}}{r} + \frac{0.75}{{\left(\sqrt[3]{s \cdot \left(\pi \cdot 6\right)}\right)}^{3}} \cdot \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r} \]

Alternative 2: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.125) / (s * single(pi))) * (exp((-r / s)) / r)) + ((exp((single(-0.3333333333333333) * (r / s))) / r) * (single(0.75) / ((s * single(pi)) * single(6.0))));
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 99.5%

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

   3. associate-*l* 99.5%

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

   4. associate-/r* 99.5%

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

   5. metadata-eval 99.5%

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

   6. metadata-eval 99.5%

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

   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. Step-by-step derivation

   1. associate-*r* 99.6%

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

   2. add-cube-cbrt 99.5%

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

   3. pow3 99.6%

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

   4. *-commutative 99.6%

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

   5. *-commutative 99.6%

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

5. Applied egg-rr 99.6%

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

6. Taylor expanded in s around 0 99.6%

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

7. Taylor expanded in r around 0 99.6%

   \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{-r}{s}}}{r} + \frac{0.75}{{\left(\sqrt[3]{s \cdot \left(\pi \cdot 6\right)}\right)}^{3}} \cdot \frac{e^{\color{blue}{-0.3333333333333333 \cdot \frac{r}{s}}}}{r} \]
8. Step-by-step derivation

   1. rem-cube-cbrt 99.6%

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

   2. associate-*r* 99.5%

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

9. Applied egg-rr 99.5%

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

10. Final simplification 99.5%

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

Alternative 3: 99.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \left(\frac{0.25}{s} \cdot \frac{0.5}{\pi}\right) \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{e^{\frac{r \cdot -0.3333333333333333}{s}}}{r}\right) \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (*
  (* (/ 0.25 s) (/ 0.5 PI))
  (+ (/ (exp (/ r (- s))) r) (/ (exp (/ (* r -0.3333333333333333) s)) r))))

C

float code(float s, float r) {
	return ((0.25f / s) * (0.5f / ((float) M_PI))) * ((expf((r / -s)) / r) + (expf(((r * -0.3333333333333333f) / s)) / r));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.25) / s) * Float32(Float32(0.5) / Float32(pi))) * Float32(Float32(exp(Float32(r / Float32(-s))) / r) + Float32(exp(Float32(Float32(r * Float32(-0.3333333333333333)) / s)) / r)))
end

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.25) / s) * (single(0.5) / single(pi))) * ((exp((r / -s)) / r) + (exp(((r * single(-0.3333333333333333)) / s)) / r));
end

Derivation

1. Initial program 99.5%

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

3. Simplified 99.3%

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

   1. metadata-eval 99.3%

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

   2. associate-/r* 99.3%

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

   3. *-commutative 99.3%

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

   4. associate-*l* 99.3%

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

   5. associate-/l/ 99.3%

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

   6. div-inv 99.3%

      \[\leadsto \color{blue}{\left(\frac{0.25}{s} \cdot \frac{1}{2 \cdot \pi}\right)} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{{\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}}{r}\right) \]

   7. *-commutative 99.3%

      \[\leadsto \left(\frac{0.25}{s} \cdot \frac{1}{\color{blue}{\pi \cdot 2}}\right) \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{{\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}}{r}\right) \]

5. Applied egg-rr 99.3%

   \[\leadsto \color{blue}{\left(\frac{0.25}{s} \cdot \frac{1}{\pi \cdot 2}\right)} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{{\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}}{r}\right) \]
6. Step-by-step derivation

   1. *-commutative 99.3%

      \[\leadsto \left(\frac{0.25}{s} \cdot \frac{1}{\color{blue}{2 \cdot \pi}}\right) \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{{\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}}{r}\right) \]

   2. associate-/r* 99.3%

      \[\leadsto \left(\frac{0.25}{s} \cdot \color{blue}{\frac{\frac{1}{2}}{\pi}}\right) \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{{\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}}{r}\right) \]

   3. metadata-eval 99.3%

      \[\leadsto \left(\frac{0.25}{s} \cdot \frac{\color{blue}{0.5}}{\pi}\right) \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{{\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}}{r}\right) \]

7. Simplified 99.3%

   \[\leadsto \color{blue}{\left(\frac{0.25}{s} \cdot \frac{0.5}{\pi}\right)} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{{\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}}{r}\right) \]

8. Taylor expanded in r around inf 99.5%

   \[\leadsto \left(\frac{0.25}{s} \cdot \frac{0.5}{\pi}\right) \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\color{blue}{e^{-0.3333333333333333 \cdot \frac{r}{s}}}}{r}\right) \]
9. Step-by-step derivation

   1. associate-*r/ 99.5%

      \[\leadsto \left(\frac{0.25}{s} \cdot \frac{0.5}{\pi}\right) \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{e^{\color{blue}{\frac{-0.3333333333333333 \cdot r}{s}}}}{r}\right) \]

10. Simplified 99.5%

    \[\leadsto \left(\frac{0.25}{s} \cdot \frac{0.5}{\pi}\right) \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\color{blue}{e^{\frac{-0.3333333333333333 \cdot r}{s}}}}{r}\right) \]

11. Final simplification 99.5%

    \[\leadsto \left(\frac{0.25}{s} \cdot \frac{0.5}{\pi}\right) \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{e^{\frac{r \cdot -0.3333333333333333}{s}}}{r}\right) \]

Alternative 4: 99.5% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Simplified 99.3%

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

4. Taylor expanded in r around inf 99.5%

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

5. Final simplification 99.5%

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

Alternative 5: 11.6% accurate, 1.4× 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 99.5%

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

3. Simplified 99.3%

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

4. Taylor expanded in r around 0 8.9%

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

5. Taylor expanded in s around inf 8.5%

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

   1. log1p-expm1-u 13.9%

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

7. Applied egg-rr 13.9%

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

8. Final simplification 13.9%

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

Alternative 6: 9.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \left(\frac{0.25}{s} \cdot \frac{0.5}{\pi}\right) \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{-0.3333333333333333 \cdot \frac{r}{s} + 1}{r}\right) \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (*
  (* (/ 0.25 s) (/ 0.5 PI))
  (+ (/ (exp (/ r (- s))) r) (/ (+ (* -0.3333333333333333 (/ r s)) 1.0) r))))

C

float code(float s, float r) {
	return ((0.25f / s) * (0.5f / ((float) M_PI))) * ((expf((r / -s)) / r) + (((-0.3333333333333333f * (r / s)) + 1.0f) / r));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.25) / s) * Float32(Float32(0.5) / Float32(pi))) * Float32(Float32(exp(Float32(r / Float32(-s))) / r) + Float32(Float32(Float32(Float32(-0.3333333333333333) * Float32(r / s)) + Float32(1.0)) / r)))
end

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.25) / s) * (single(0.5) / single(pi))) * ((exp((r / -s)) / r) + (((single(-0.3333333333333333) * (r / s)) + single(1.0)) / r));
end

Derivation

1. Initial program 99.5%

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

3. Simplified 99.3%

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

   1. metadata-eval 99.3%

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

   2. associate-/r* 99.3%

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

   3. *-commutative 99.3%

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

   4. associate-*l* 99.3%

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

   5. associate-/l/ 99.3%

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

   6. div-inv 99.3%

      \[\leadsto \color{blue}{\left(\frac{0.25}{s} \cdot \frac{1}{2 \cdot \pi}\right)} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{{\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}}{r}\right) \]

   7. *-commutative 99.3%

      \[\leadsto \left(\frac{0.25}{s} \cdot \frac{1}{\color{blue}{\pi \cdot 2}}\right) \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{{\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}}{r}\right) \]

5. Applied egg-rr 99.3%

   \[\leadsto \color{blue}{\left(\frac{0.25}{s} \cdot \frac{1}{\pi \cdot 2}\right)} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{{\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}}{r}\right) \]
6. Step-by-step derivation

   1. *-commutative 99.3%

      \[\leadsto \left(\frac{0.25}{s} \cdot \frac{1}{\color{blue}{2 \cdot \pi}}\right) \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{{\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}}{r}\right) \]

   2. associate-/r* 99.3%

      \[\leadsto \left(\frac{0.25}{s} \cdot \color{blue}{\frac{\frac{1}{2}}{\pi}}\right) \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{{\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}}{r}\right) \]

   3. metadata-eval 99.3%

      \[\leadsto \left(\frac{0.25}{s} \cdot \frac{\color{blue}{0.5}}{\pi}\right) \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{{\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}}{r}\right) \]

7. Simplified 99.3%

   \[\leadsto \color{blue}{\left(\frac{0.25}{s} \cdot \frac{0.5}{\pi}\right)} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{{\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}}{r}\right) \]

8. Taylor expanded in r around 0 9.0%

   \[\leadsto \left(\frac{0.25}{s} \cdot \frac{0.5}{\pi}\right) \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\color{blue}{1 + -0.3333333333333333 \cdot \frac{r}{s}}}{r}\right) \]

9. Final simplification 9.0%

   \[\leadsto \left(\frac{0.25}{s} \cdot \frac{0.5}{\pi}\right) \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{-0.3333333333333333 \cdot \frac{r}{s} + 1}{r}\right) \]

Alternative 7: 9.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Simplified 99.3%

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

4. Taylor expanded in r around 0 9.0%

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

5. Final simplification 9.0%

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

Alternative 8: 9.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Simplified 99.3%

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

4. Taylor expanded in r around 0 9.0%

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

   1. metadata-eval 9.0%

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

   2. times-frac 9.0%

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

   3. neg-mul-1 9.0%

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

   4. frac-2neg 9.0%

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

   5. remove-double-neg 9.0%

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

   6. *-commutative 9.0%

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

   7. distribute-rgt-neg-in 9.0%

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

   8. metadata-eval 9.0%

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

6. Applied egg-rr 9.0%

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

7. Final simplification 9.0%

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

Alternative 9: 9.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Simplified 99.3%

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

4. Taylor expanded in r around 0 9.0%

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

5. Taylor expanded in r around 0 9.0%

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

   1. associate-*r/ 9.0%

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

   2. metadata-eval 9.0%

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

7. Simplified 9.0%

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

8. Final simplification 9.0%

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

Alternative 10: 9.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Simplified 99.3%

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

4. Taylor expanded in r around 0 8.9%

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

5. Taylor expanded in s around 0 8.9%

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

   1. associate-*r/ 8.9%

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

   2. times-frac 8.9%

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

   3. +-commutative 8.9%

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

   4. associate-*r/ 8.9%

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

   5. mul-1-neg 8.9%

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

7. Simplified 8.9%

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

8. Final simplification 8.9%

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

Alternative 11: 9.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Simplified 99.3%

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

4. Taylor expanded in r around 0 8.9%

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

5. Taylor expanded in r around inf 8.9%

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

   1. mul-1-neg 8.9%

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

7. Simplified 8.9%

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

8. Final simplification 8.9%

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

Alternative 12: 9.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Simplified 99.3%

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

4. Taylor expanded in r around 0 8.9%

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

5. Taylor expanded in s around 0 8.9%

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

   1. associate-*r/ 8.9%

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

   2. times-frac 8.9%

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

   3. +-commutative 8.9%

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

   4. associate-*r/ 8.9%

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

   5. mul-1-neg 8.9%

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

7. Simplified 8.9%

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

8. Taylor expanded in r around inf 8.9%

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

   1. associate-*r/ 8.9%

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

   2. neg-mul-1 8.9%

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

10. Simplified 8.9%

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

11. Final simplification 8.9%

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

Alternative 13: 9.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Simplified 99.3%

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

4. Taylor expanded in r around 0 8.9%

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

5. Taylor expanded in s around 0 8.9%

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

   1. associate-*r/ 8.9%

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

   2. times-frac 8.9%

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

   3. +-commutative 8.9%

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

   4. associate-*r/ 8.9%

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

   5. mul-1-neg 8.9%

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

7. Simplified 8.9%

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

8. Taylor expanded in r around inf 8.9%

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

   1. associate-*r/ 8.9%

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

   2. neg-mul-1 8.9%

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

10. Simplified 8.9%

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

11. Final simplification 8.9%

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

Alternative 14: 8.9% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \frac{0.125}{s} \cdot \frac{2}{\pi \cdot r} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Simplified 99.3%

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

4. Taylor expanded in r around 0 8.9%

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

5. Taylor expanded in s around 0 8.9%

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

   1. associate-*r/ 8.9%

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

   2. times-frac 8.9%

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

   3. +-commutative 8.9%

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

   4. associate-*r/ 8.9%

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

   5. mul-1-neg 8.9%

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

7. Simplified 8.9%

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

8. Taylor expanded in r around 0 8.5%

   \[\leadsto \frac{0.125}{s} \cdot \color{blue}{\frac{2}{r \cdot \pi}} \]

9. Final simplification 8.5%

   \[\leadsto \frac{0.125}{s} \cdot \frac{2}{\pi \cdot r} \]

Alternative 15: 8.9% accurate, 4.0× 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 99.5%

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

3. Simplified 99.3%

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

4. Taylor expanded in r around 0 8.9%

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

5. Taylor expanded in s around inf 8.5%

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

6. Final simplification 8.5%

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

Alternative 16: 8.9% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Simplified 99.3%

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

4. Taylor expanded in r around 0 8.9%

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

5. Taylor expanded in s around 0 8.9%

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

   1. associate-*r/ 8.9%

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

   2. times-frac 8.9%

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

   3. +-commutative 8.9%

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

   4. associate-*r/ 8.9%

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

   5. mul-1-neg 8.9%

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

7. Simplified 8.9%

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

8. Taylor expanded in s around inf 8.5%

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

   1. associate-/r* 8.5%

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

   2. associate-/r* 8.5%

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

   3. associate-/r* 8.5%

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

10. Simplified 8.5%

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

11. Final simplification 8.5%

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

Reproduce

herbie shell --seed 2023315 
(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))))