Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.6%

Time: 11.7s

Alternatives: 15

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{\frac{r}{-s}}}{r} + 0.75 \cdot \frac{{e}^{\left(\frac{r}{s} \cdot -0.3333333333333333\right)}}{6 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(s \cdot \left(\pi \cdot r\right)\right)\right)} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (* (/ 0.25 (* s (* 2.0 PI))) (/ (exp (/ r (- s))) r))
  (*
   0.75
   (/
    (pow E (* (/ r s) -0.3333333333333333))
    (* 6.0 (expm1 (log1p (* s (* PI r)))))))))

C

float code(float s, float r) {
	return ((0.25f / (s * (2.0f * ((float) M_PI)))) * (expf((r / -s)) / r)) + (0.75f * (powf(((float) M_E), ((r / s) * -0.3333333333333333f)) / (6.0f * expm1f(log1pf((s * (((float) M_PI) * r)))))));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.25) / Float32(s * Float32(Float32(2.0) * Float32(pi)))) * Float32(exp(Float32(r / Float32(-s))) / r)) + Float32(Float32(0.75) * Float32((Float32(exp(1)) ^ Float32(Float32(r / s) * Float32(-0.3333333333333333))) / Float32(Float32(6.0) * expm1(log1p(Float32(s * Float32(Float32(pi) * r))))))))
end

Derivation

1. Initial program 99.6%

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

   1. times-frac 99.6%

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

   2. *-commutative 99.6%

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

   3. distribute-frac-neg 99.6%

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

   4. associate-/l* 99.5%

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

   5. *-commutative 99.5%

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

   6. *-commutative 99.5%

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

   7. associate-*l* 99.5%

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

3. Simplified 99.5%

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

5. Taylor expanded in r around 0 99.5%

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

   1. *-un-lft-identity 99.5%

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

   2. exp-prod 99.5%

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

   3. neg-mul-1 99.5%

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

   4. *-commutative 99.5%

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

   5. times-frac 99.6%

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

   6. metadata-eval 99.6%

      \[\leadsto \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{-\frac{r}{s}}}{r} + 0.75 \cdot \frac{{\left(e^{1}\right)}^{\left(\color{blue}{-0.3333333333333333} \cdot \frac{r}{s}\right)}}{6 \cdot \left(r \cdot \left(s \cdot \pi\right)\right)} \]

   7. add-log-exp 98.0%

      \[\leadsto \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{-\frac{r}{s}}}{r} + 0.75 \cdot \frac{{\left(e^{1}\right)}^{\left(-0.3333333333333333 \cdot \color{blue}{\log \left(e^{\frac{r}{s}}\right)}\right)}}{6 \cdot \left(r \cdot \left(s \cdot \pi\right)\right)} \]

   8. log-pow 98.0%

      \[\leadsto \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{-\frac{r}{s}}}{r} + 0.75 \cdot \frac{{\left(e^{1}\right)}^{\color{blue}{\log \left({\left(e^{\frac{r}{s}}\right)}^{-0.3333333333333333}\right)}}}{6 \cdot \left(r \cdot \left(s \cdot \pi\right)\right)} \]

   9. pow-exp 99.6%

      \[\leadsto \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{-\frac{r}{s}}}{r} + 0.75 \cdot \frac{{\left(e^{1}\right)}^{\log \color{blue}{\left(e^{\frac{r}{s} \cdot -0.3333333333333333}\right)}}}{6 \cdot \left(r \cdot \left(s \cdot \pi\right)\right)} \]

   10. rem-log-exp 99.6%

       \[\leadsto \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{-\frac{r}{s}}}{r} + 0.75 \cdot \frac{{\left(e^{1}\right)}^{\color{blue}{\left(\frac{r}{s} \cdot -0.3333333333333333\right)}}}{6 \cdot \left(r \cdot \left(s \cdot \pi\right)\right)} \]

7. Applied egg-rr 99.6%

   \[\leadsto \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{-\frac{r}{s}}}{r} + 0.75 \cdot \frac{\color{blue}{{\left(e^{1}\right)}^{\left(\frac{r}{s} \cdot -0.3333333333333333\right)}}}{6 \cdot \left(r \cdot \left(s \cdot \pi\right)\right)} \]
8. Step-by-step derivation

   1. expm1-log1p-u 99.6%

      \[\leadsto \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{-\frac{r}{s}}}{r} + 0.75 \cdot \frac{{\left(e^{1}\right)}^{\left(\frac{r}{s} \cdot -0.3333333333333333\right)}}{6 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(r \cdot \left(s \cdot \pi\right)\right)\right)}} \]

   2. expm1-undefine 23.0%

      \[\leadsto \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{-\frac{r}{s}}}{r} + 0.75 \cdot \frac{{\left(e^{1}\right)}^{\left(\frac{r}{s} \cdot -0.3333333333333333\right)}}{6 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(r \cdot \left(s \cdot \pi\right)\right)} - 1\right)}} \]

   3. *-commutative 23.0%

      \[\leadsto \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{-\frac{r}{s}}}{r} + 0.75 \cdot \frac{{\left(e^{1}\right)}^{\left(\frac{r}{s} \cdot -0.3333333333333333\right)}}{6 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\left(s \cdot \pi\right) \cdot r}\right)} - 1\right)} \]

   4. associate-*l* 23.0%

      \[\leadsto \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{-\frac{r}{s}}}{r} + 0.75 \cdot \frac{{\left(e^{1}\right)}^{\left(\frac{r}{s} \cdot -0.3333333333333333\right)}}{6 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{s \cdot \left(\pi \cdot r\right)}\right)} - 1\right)} \]

9. Applied egg-rr 23.0%

   \[\leadsto \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{-\frac{r}{s}}}{r} + 0.75 \cdot \frac{{\left(e^{1}\right)}^{\left(\frac{r}{s} \cdot -0.3333333333333333\right)}}{6 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(s \cdot \left(\pi \cdot r\right)\right)} - 1\right)}} \]
10. Step-by-step derivation

    1. expm1-define 99.7%

       \[\leadsto \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{-\frac{r}{s}}}{r} + 0.75 \cdot \frac{{\left(e^{1}\right)}^{\left(\frac{r}{s} \cdot -0.3333333333333333\right)}}{6 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(s \cdot \left(\pi \cdot r\right)\right)\right)}} \]

11. Simplified 99.7%

    \[\leadsto \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{-\frac{r}{s}}}{r} + 0.75 \cdot \frac{{\left(e^{1}\right)}^{\left(\frac{r}{s} \cdot -0.3333333333333333\right)}}{6 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(s \cdot \left(\pi \cdot r\right)\right)\right)}} \]

12. Final simplification 99.7%

    \[\leadsto \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{\frac{r}{-s}}}{r} + 0.75 \cdot \frac{{e}^{\left(\frac{r}{s} \cdot -0.3333333333333333\right)}}{6 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(s \cdot \left(\pi \cdot r\right)\right)\right)} \]
13. Add Preprocessing

Alternative 2: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.125}{s \cdot \pi}\\ \mathsf{fma}\left(t\_0, \frac{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r \cdot 0.5}{s}\right)}}{r}, t\_0 \cdot \frac{e^{\frac{r}{-s}}}{r}\right) \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (let* ((t_0 (/ 0.125 (* s PI))))
   (fma
    t_0
    (/ (pow (exp -0.6666666666666666) (/ (* r 0.5) s)) r)
    (* t_0 (/ (exp (/ r (- s))) r)))))

C

float code(float s, float r) {
	float t_0 = 0.125f / (s * ((float) M_PI));
	return fmaf(t_0, (powf(expf(-0.6666666666666666f), ((r * 0.5f) / s)) / r), (t_0 * (expf((r / -s)) / r)));
}

Julia

function code(s, r)
	t_0 = Float32(Float32(0.125) / Float32(s * Float32(pi)))
	return fma(t_0, Float32((exp(Float32(-0.6666666666666666)) ^ Float32(Float32(r * Float32(0.5)) / s)) / r), Float32(t_0 * Float32(exp(Float32(r / Float32(-s))) / r)))
end

Derivation

1. Initial program 99.6%

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

   1. +-commutative 99.6%

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

   2. times-frac 99.6%

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

   3. fma-define 99.6%

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

   4. associate-*l* 99.6%

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

   5. associate-/r* 99.6%

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

   6. metadata-eval 99.6%

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

   7. *-commutative 99.6%

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

   8. neg-mul-1 99.6%

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

   9. times-frac 99.5%

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

   10. metadata-eval 99.5%

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

   11. times-frac 99.5%

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

3. Simplified 99.5%

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

   1. pow-exp 99.4%

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

   2. sqr-pow 99.4%

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

   3. pow-prod-down 99.4%

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

   4. prod-exp 99.6%

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

   5. metadata-eval 99.6%

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

   6. div-inv 99.6%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{{\left(e^{-0.6666666666666666}\right)}^{\color{blue}{\left(\frac{r}{s} \cdot \frac{1}{2}\right)}}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{r}{-s}}}{r}\right) \]

   7. metadata-eval 99.6%

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

6. Applied egg-rr 99.6%

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

   1. associate-*l/ 99.6%

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

8. Simplified 99.6%

   \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{\color{blue}{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r \cdot 0.5}{s}\right)}}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{r}{-s}}}{r}\right) \]
9. Add Preprocessing

Alternative 3: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{\frac{r}{-s}}}{r} + 0.75 \cdot \frac{{e}^{\left(\frac{r}{s} \cdot -0.3333333333333333\right)}}{\left(s \cdot r\right) \cdot \left(\pi \cdot 6\right)} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (* (/ 0.25 (* s (* 2.0 PI))) (/ (exp (/ r (- s))) r))
  (* 0.75 (/ (pow E (* (/ r s) -0.3333333333333333)) (* (* s r) (* PI 6.0))))))

C

float code(float s, float r) {
	return ((0.25f / (s * (2.0f * ((float) M_PI)))) * (expf((r / -s)) / r)) + (0.75f * (powf(((float) M_E), ((r / s) * -0.3333333333333333f)) / ((s * r) * (((float) M_PI) * 6.0f))));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.25) / Float32(s * Float32(Float32(2.0) * Float32(pi)))) * Float32(exp(Float32(r / Float32(-s))) / r)) + Float32(Float32(0.75) * Float32((Float32(exp(1)) ^ Float32(Float32(r / s) * Float32(-0.3333333333333333))) / Float32(Float32(s * r) * Float32(Float32(pi) * Float32(6.0))))))
end

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.25) / (s * (single(2.0) * single(pi)))) * (exp((r / -s)) / r)) + (single(0.75) * ((single(2.71828182845904523536) ^ ((r / s) * single(-0.3333333333333333))) / ((s * r) * (single(pi) * single(6.0)))));
end

Derivation

1. Initial program 99.6%

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

   1. times-frac 99.6%

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

   2. *-commutative 99.6%

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

   3. distribute-frac-neg 99.6%

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

   4. associate-/l* 99.5%

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

   5. *-commutative 99.5%

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

   6. *-commutative 99.5%

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

   7. associate-*l* 99.5%

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

3. Simplified 99.5%

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

5. Taylor expanded in r around 0 99.5%

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

   1. *-un-lft-identity 99.5%

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

   2. exp-prod 99.5%

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

   3. neg-mul-1 99.5%

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

   4. *-commutative 99.5%

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

   5. times-frac 99.6%

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

   6. metadata-eval 99.6%

      \[\leadsto \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{-\frac{r}{s}}}{r} + 0.75 \cdot \frac{{\left(e^{1}\right)}^{\left(\color{blue}{-0.3333333333333333} \cdot \frac{r}{s}\right)}}{6 \cdot \left(r \cdot \left(s \cdot \pi\right)\right)} \]

   7. add-log-exp 98.0%

      \[\leadsto \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{-\frac{r}{s}}}{r} + 0.75 \cdot \frac{{\left(e^{1}\right)}^{\left(-0.3333333333333333 \cdot \color{blue}{\log \left(e^{\frac{r}{s}}\right)}\right)}}{6 \cdot \left(r \cdot \left(s \cdot \pi\right)\right)} \]

   8. log-pow 98.0%

      \[\leadsto \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{-\frac{r}{s}}}{r} + 0.75 \cdot \frac{{\left(e^{1}\right)}^{\color{blue}{\log \left({\left(e^{\frac{r}{s}}\right)}^{-0.3333333333333333}\right)}}}{6 \cdot \left(r \cdot \left(s \cdot \pi\right)\right)} \]

   9. pow-exp 99.6%

      \[\leadsto \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{-\frac{r}{s}}}{r} + 0.75 \cdot \frac{{\left(e^{1}\right)}^{\log \color{blue}{\left(e^{\frac{r}{s} \cdot -0.3333333333333333}\right)}}}{6 \cdot \left(r \cdot \left(s \cdot \pi\right)\right)} \]

   10. rem-log-exp 99.6%

       \[\leadsto \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{-\frac{r}{s}}}{r} + 0.75 \cdot \frac{{\left(e^{1}\right)}^{\color{blue}{\left(\frac{r}{s} \cdot -0.3333333333333333\right)}}}{6 \cdot \left(r \cdot \left(s \cdot \pi\right)\right)} \]

7. Applied egg-rr 99.6%

   \[\leadsto \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{-\frac{r}{s}}}{r} + 0.75 \cdot \frac{\color{blue}{{\left(e^{1}\right)}^{\left(\frac{r}{s} \cdot -0.3333333333333333\right)}}}{6 \cdot \left(r \cdot \left(s \cdot \pi\right)\right)} \]
8. Step-by-step derivation

   1. expm1-log1p-u 99.6%

      \[\leadsto \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{-\frac{r}{s}}}{r} + 0.75 \cdot \frac{{\left(e^{1}\right)}^{\left(\frac{r}{s} \cdot -0.3333333333333333\right)}}{6 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(r \cdot \left(s \cdot \pi\right)\right)\right)}} \]

   2. expm1-undefine 23.0%

      \[\leadsto \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{-\frac{r}{s}}}{r} + 0.75 \cdot \frac{{\left(e^{1}\right)}^{\left(\frac{r}{s} \cdot -0.3333333333333333\right)}}{6 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(r \cdot \left(s \cdot \pi\right)\right)} - 1\right)}} \]

   3. *-commutative 23.0%

      \[\leadsto \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{-\frac{r}{s}}}{r} + 0.75 \cdot \frac{{\left(e^{1}\right)}^{\left(\frac{r}{s} \cdot -0.3333333333333333\right)}}{6 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\left(s \cdot \pi\right) \cdot r}\right)} - 1\right)} \]

   4. associate-*l* 23.0%

      \[\leadsto \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{-\frac{r}{s}}}{r} + 0.75 \cdot \frac{{\left(e^{1}\right)}^{\left(\frac{r}{s} \cdot -0.3333333333333333\right)}}{6 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{s \cdot \left(\pi \cdot r\right)}\right)} - 1\right)} \]

9. Applied egg-rr 23.0%

   \[\leadsto \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{-\frac{r}{s}}}{r} + 0.75 \cdot \frac{{\left(e^{1}\right)}^{\left(\frac{r}{s} \cdot -0.3333333333333333\right)}}{6 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(s \cdot \left(\pi \cdot r\right)\right)} - 1\right)}} \]
10. Step-by-step derivation

    1. expm1-define 99.7%

       \[\leadsto \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{-\frac{r}{s}}}{r} + 0.75 \cdot \frac{{\left(e^{1}\right)}^{\left(\frac{r}{s} \cdot -0.3333333333333333\right)}}{6 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(s \cdot \left(\pi \cdot r\right)\right)\right)}} \]

11. Simplified 99.7%

    \[\leadsto \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{-\frac{r}{s}}}{r} + 0.75 \cdot \frac{{\left(e^{1}\right)}^{\left(\frac{r}{s} \cdot -0.3333333333333333\right)}}{6 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(s \cdot \left(\pi \cdot r\right)\right)\right)}} \]

12. Taylor expanded in s around 0 99.6%

    \[\leadsto \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{-\frac{r}{s}}}{r} + 0.75 \cdot \frac{{\left(e^{1}\right)}^{\left(\frac{r}{s} \cdot -0.3333333333333333\right)}}{\color{blue}{6 \cdot \left(r \cdot \left(s \cdot \pi\right)\right)}} \]
13. Step-by-step derivation

    1. *-commutative 99.6%

       \[\leadsto \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{-\frac{r}{s}}}{r} + 0.75 \cdot \frac{{\left(e^{1}\right)}^{\left(\frac{r}{s} \cdot -0.3333333333333333\right)}}{\color{blue}{\left(r \cdot \left(s \cdot \pi\right)\right) \cdot 6}} \]

    2. associate-*r* 99.6%

       \[\leadsto \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{-\frac{r}{s}}}{r} + 0.75 \cdot \frac{{\left(e^{1}\right)}^{\left(\frac{r}{s} \cdot -0.3333333333333333\right)}}{\color{blue}{\left(\left(r \cdot s\right) \cdot \pi\right)} \cdot 6} \]

    3. associate-*l* 99.6%

       \[\leadsto \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{-\frac{r}{s}}}{r} + 0.75 \cdot \frac{{\left(e^{1}\right)}^{\left(\frac{r}{s} \cdot -0.3333333333333333\right)}}{\color{blue}{\left(r \cdot s\right) \cdot \left(\pi \cdot 6\right)}} \]

14. Simplified 99.6%

    \[\leadsto \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{-\frac{r}{s}}}{r} + 0.75 \cdot \frac{{\left(e^{1}\right)}^{\left(\frac{r}{s} \cdot -0.3333333333333333\right)}}{\color{blue}{\left(r \cdot s\right) \cdot \left(\pi \cdot 6\right)}} \]

15. Final simplification 99.6%

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

Alternative 4: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{\frac{r}{-s}}}{r} + 0.75 \cdot \frac{{e}^{\left(\frac{r}{s} \cdot -0.3333333333333333\right)}}{6 \cdot \left(r \cdot \left(s \cdot \pi\right)\right)} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (* (/ 0.25 (* s (* 2.0 PI))) (/ (exp (/ r (- s))) r))
  (* 0.75 (/ (pow E (* (/ r s) -0.3333333333333333)) (* 6.0 (* r (* s PI)))))))

C

float code(float s, float r) {
	return ((0.25f / (s * (2.0f * ((float) M_PI)))) * (expf((r / -s)) / r)) + (0.75f * (powf(((float) M_E), ((r / s) * -0.3333333333333333f)) / (6.0f * (r * (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(r / Float32(-s))) / r)) + Float32(Float32(0.75) * Float32((Float32(exp(1)) ^ Float32(Float32(r / s) * Float32(-0.3333333333333333))) / Float32(Float32(6.0) * Float32(r * Float32(s * Float32(pi)))))))
end

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.25) / (s * (single(2.0) * single(pi)))) * (exp((r / -s)) / r)) + (single(0.75) * ((single(2.71828182845904523536) ^ ((r / s) * single(-0.3333333333333333))) / (single(6.0) * (r * (s * single(pi))))));
end

Derivation

1. Initial program 99.6%

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

   1. times-frac 99.6%

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

   2. *-commutative 99.6%

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

   3. distribute-frac-neg 99.6%

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

   4. associate-/l* 99.5%

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

   5. *-commutative 99.5%

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

   6. *-commutative 99.5%

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

   7. associate-*l* 99.5%

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

3. Simplified 99.5%

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

5. Taylor expanded in r around 0 99.5%

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

   1. *-un-lft-identity 99.5%

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

   2. exp-prod 99.5%

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

   3. neg-mul-1 99.5%

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

   4. *-commutative 99.5%

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

   5. times-frac 99.6%

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

   6. metadata-eval 99.6%

      \[\leadsto \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{-\frac{r}{s}}}{r} + 0.75 \cdot \frac{{\left(e^{1}\right)}^{\left(\color{blue}{-0.3333333333333333} \cdot \frac{r}{s}\right)}}{6 \cdot \left(r \cdot \left(s \cdot \pi\right)\right)} \]

   7. add-log-exp 98.0%

      \[\leadsto \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{-\frac{r}{s}}}{r} + 0.75 \cdot \frac{{\left(e^{1}\right)}^{\left(-0.3333333333333333 \cdot \color{blue}{\log \left(e^{\frac{r}{s}}\right)}\right)}}{6 \cdot \left(r \cdot \left(s \cdot \pi\right)\right)} \]

   8. log-pow 98.0%

      \[\leadsto \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{-\frac{r}{s}}}{r} + 0.75 \cdot \frac{{\left(e^{1}\right)}^{\color{blue}{\log \left({\left(e^{\frac{r}{s}}\right)}^{-0.3333333333333333}\right)}}}{6 \cdot \left(r \cdot \left(s \cdot \pi\right)\right)} \]

   9. pow-exp 99.6%

      \[\leadsto \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{-\frac{r}{s}}}{r} + 0.75 \cdot \frac{{\left(e^{1}\right)}^{\log \color{blue}{\left(e^{\frac{r}{s} \cdot -0.3333333333333333}\right)}}}{6 \cdot \left(r \cdot \left(s \cdot \pi\right)\right)} \]

   10. rem-log-exp 99.6%

       \[\leadsto \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{-\frac{r}{s}}}{r} + 0.75 \cdot \frac{{\left(e^{1}\right)}^{\color{blue}{\left(\frac{r}{s} \cdot -0.3333333333333333\right)}}}{6 \cdot \left(r \cdot \left(s \cdot \pi\right)\right)} \]

7. Applied egg-rr 99.6%

   \[\leadsto \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{-\frac{r}{s}}}{r} + 0.75 \cdot \frac{\color{blue}{{\left(e^{1}\right)}^{\left(\frac{r}{s} \cdot -0.3333333333333333\right)}}}{6 \cdot \left(r \cdot \left(s \cdot \pi\right)\right)} \]

8. Final simplification 99.6%

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

Alternative 5: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in r around 0 99.5%

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

   1. *-commutative 99.5%

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

   2. metadata-eval 99.5%

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

   3. times-frac 99.6%

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

   4. *-rgt-identity 99.6%

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

5. Simplified 99.6%

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

6. Final simplification 99.6%

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

Alternative 6: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Simplified 99.4%

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

5. Taylor expanded in r around inf 99.5%

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

   1. associate-*r/ 99.5%

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

   2. associate-*l/ 99.5%

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

   3. associate-/r/ 99.5%

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

7. Simplified 99.5%

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

   1. associate-/r* 99.5%

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

   2. div-inv 99.6%

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

9. Applied egg-rr 99.6%

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

Alternative 7: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Simplified 99.4%

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

5. Taylor expanded in r around inf 99.5%

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

   1. associate-*r/ 99.5%

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

   2. associate-*l/ 99.5%

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

   3. associate-/r/ 99.5%

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

7. Simplified 99.5%

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

   1. *-un-lft-identity 99.5%

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

   2. exp-prod 99.6%

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

   3. associate-/r/ 99.6%

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

   4. *-commutative 99.6%

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

   5. associate-/l* 99.6%

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

9. Applied egg-rr 99.6%

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

10. Final simplification 99.6%

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

Alternative 8: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Simplified 99.4%

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

5. Taylor expanded in r around inf 99.5%

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

   1. associate-*r/ 99.5%

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

   2. associate-*l/ 99.5%

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

   3. associate-/r/ 99.5%

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

7. Simplified 99.5%

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

Alternative 9: 43.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.6%

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

3. Simplified 99.4%

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

5. Taylor expanded in r around 0 7.0%

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

6. Taylor expanded in s around 0 7.0%

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

   1. mul-1-neg 7.0%

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

8. Simplified 7.0%

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

9. Taylor expanded in r around 0 6.8%

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

    1. *-commutative 6.8%

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

    2. associate-*r* 6.8%

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

11. Simplified 6.8%

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

    1. log1p-expm1-u 44.8%

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

13. Applied egg-rr 44.8%

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

Alternative 10: 11.8% accurate, 1.1× speedup

Math

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

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

Derivation

1. Initial program 99.6%

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

3. Simplified 99.4%

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

5. Taylor expanded in r around 0 7.0%

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

6. Taylor expanded in s around inf 6.8%

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

   1. log1p-expm1-u 9.9%

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

   2. *-commutative 9.9%

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

   3. associate-*l* 9.9%

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

8. Applied egg-rr 9.9%

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

Alternative 11: 9.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Simplified 99.4%

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

5. Taylor expanded in r around 0 7.0%

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

6. Taylor expanded in r around -inf 7.0%

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

   1. associate-*r/ 7.0%

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

   2. sub-neg 7.0%

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

   3. metadata-eval 7.0%

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

   4. distribute-lft-in 7.0%

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

   5. mul-1-neg 7.0%

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

   6. rec-exp 7.0%

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

   7. associate-*r/ 7.0%

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

   8. metadata-eval 7.0%

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

   9. metadata-eval 7.0%

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

8. Simplified 7.0%

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

9. Final simplification 7.0%

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

Alternative 12: 9.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Simplified 99.4%

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

5. Taylor expanded in r around 0 7.0%

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

6. Taylor expanded in r around inf 7.0%

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

   1. mul-1-neg 7.0%

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

   2. associate-*r* 7.0%

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

8. Simplified 7.0%

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

9. Final simplification 7.0%

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

Alternative 13: 9.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Simplified 99.4%

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

5. Taylor expanded in r around 0 7.0%

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

6. Taylor expanded in s around 0 7.0%

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

   1. mul-1-neg 7.0%

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

8. Simplified 7.0%

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

9. Taylor expanded in r around inf 7.0%

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

10. Final simplification 7.0%

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

Alternative 14: 9.1% accurate, 33.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Simplified 99.4%

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

5. Taylor expanded in r around 0 7.0%

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

6. Taylor expanded in s around inf 6.8%

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

   1. associate-*r* 6.8%

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

8. Simplified 6.8%

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

9. Final simplification 6.8%

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

Alternative 15: 9.1% accurate, 33.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Simplified 99.4%

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

5. Taylor expanded in r around 0 7.0%

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

6. Taylor expanded in s around inf 6.8%

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

Reproduce

herbie shell --seed 2024098 
(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))))