Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.6%

Time: 14.0s

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, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

   1. lift-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r}} + \frac{\frac{3}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   2. lift-*.f32 N/A

      \[\leadsto \frac{\color{blue}{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{3}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   3. lift-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\color{blue}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r}} + \frac{\frac{3}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   4. times-frac N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{4}}{\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{r}} + \frac{\frac{3}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   5. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{\frac{\frac{1}{4}}{\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{r}} + \frac{\frac{3}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   6. lower-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\frac{\frac{1}{4}}{\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{r}} + \frac{\frac{3}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

4. Applied rewrites 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Final simplification 99.8%

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

Alternative 3: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

4. Applied rewrites 99.7%

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

Alternative 4: 72.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := r \cdot \left(r \cdot r\right)\\ \mathbf{if}\;s \leq 3.2000000625327404 \cdot 10^{-24}:\\ \;\;\;\;\frac{1}{s \cdot \mathsf{fma}\left(r \cdot r, \frac{\pi}{s} \cdot 2.6666666666666665, r \cdot \left(\pi \cdot 4\right) - \frac{t\_0 \cdot \left(\pi \cdot -0.6666666666666666\right)}{s \cdot s}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(-s\right) \cdot \mathsf{fma}\left(r, \pi \cdot -4, \frac{\mathsf{fma}\left(r, r \cdot \left(\pi \cdot 2.6666666666666665\right), \frac{\mathsf{fma}\left(\pi, t\_0 \cdot -0.6666666666666666, \mathsf{fma}\left(r \cdot -0.6666666666666666, t\_0 \cdot \left(\frac{\pi}{s} \cdot 0.6666666666666666\right), \frac{\left(\pi \cdot {r}^{4}\right) \cdot 0.3950617283950617}{s}\right)\right)}{-s}\right)}{-s}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (let* ((t_0 (* r (* r r))))
   (if (<= s 3.2000000625327404e-24)
     (/
      1.0
      (*
       s
       (fma
        (* r r)
        (* (/ PI s) 2.6666666666666665)
        (- (* r (* PI 4.0)) (/ (* t_0 (* PI -0.6666666666666666)) (* s s))))))
     (/
      1.0
      (*
       (- s)
       (fma
        r
        (* PI -4.0)
        (/
         (fma
          r
          (* r (* PI 2.6666666666666665))
          (/
           (fma
            PI
            (* t_0 -0.6666666666666666)
            (fma
             (* r -0.6666666666666666)
             (* t_0 (* (/ PI s) 0.6666666666666666))
             (/ (* (* PI (pow r 4.0)) 0.3950617283950617) s)))
           (- s)))
         (- s))))))))

C

float code(float s, float r) {
	float t_0 = r * (r * r);
	float tmp;
	if (s <= 3.2000000625327404e-24f) {
		tmp = 1.0f / (s * fmaf((r * r), ((((float) M_PI) / s) * 2.6666666666666665f), ((r * (((float) M_PI) * 4.0f)) - ((t_0 * (((float) M_PI) * -0.6666666666666666f)) / (s * s)))));
	} else {
		tmp = 1.0f / (-s * fmaf(r, (((float) M_PI) * -4.0f), (fmaf(r, (r * (((float) M_PI) * 2.6666666666666665f)), (fmaf(((float) M_PI), (t_0 * -0.6666666666666666f), fmaf((r * -0.6666666666666666f), (t_0 * ((((float) M_PI) / s) * 0.6666666666666666f)), (((((float) M_PI) * powf(r, 4.0f)) * 0.3950617283950617f) / s))) / -s)) / -s)));
	}
	return tmp;
}

Julia

function code(s, r)
	t_0 = Float32(r * Float32(r * r))
	tmp = Float32(0.0)
	if (s <= Float32(3.2000000625327404e-24))
		tmp = Float32(Float32(1.0) / Float32(s * fma(Float32(r * r), Float32(Float32(Float32(pi) / s) * Float32(2.6666666666666665)), Float32(Float32(r * Float32(Float32(pi) * Float32(4.0))) - Float32(Float32(t_0 * Float32(Float32(pi) * Float32(-0.6666666666666666))) / Float32(s * s))))));
	else
		tmp = Float32(Float32(1.0) / Float32(Float32(-s) * fma(r, Float32(Float32(pi) * Float32(-4.0)), Float32(fma(r, Float32(r * Float32(Float32(pi) * Float32(2.6666666666666665))), Float32(fma(Float32(pi), Float32(t_0 * Float32(-0.6666666666666666)), fma(Float32(r * Float32(-0.6666666666666666)), Float32(t_0 * Float32(Float32(Float32(pi) / s) * Float32(0.6666666666666666))), Float32(Float32(Float32(Float32(pi) * (r ^ Float32(4.0))) * Float32(0.3950617283950617)) / s))) / Float32(-s))) / Float32(-s)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if s < 3.20000006e-24

   1. Initial program 100.0%

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

   3. Taylor expanded in s around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{\frac{-1}{48} \cdot \frac{{r}^{2}}{\mathsf{PI}\left(\right)} + \frac{-1}{1296} \cdot \frac{{r}^{2}}{\mathsf{PI}\left(\right)}}{s} + \left(\frac{-1}{16} \cdot \frac{r}{\mathsf{PI}\left(\right)} + \frac{-1}{144} \cdot \frac{r}{\mathsf{PI}\left(\right)}\right)}{s} - \frac{1}{6} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{s} - \frac{1}{4} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s}} \]

   5. Applied rewrites 3.1%

      \[\leadsto \color{blue}{\frac{\frac{0.25}{r \cdot \pi} + \left(\frac{\mathsf{fma}\left(-0.021604938271604937, \frac{r \cdot r}{s \cdot \pi}, \frac{r \cdot 0.06944444444444445}{\pi}\right)}{s \cdot s} + \frac{-0.16666666666666666}{s \cdot \pi}\right)}{s}} \]
   6. Step-by-step derivation

   7. Applied rewrites 3.1%

      \[\leadsto \frac{1}{\color{blue}{\frac{s}{\frac{\mathsf{fma}\left(r, \frac{0.06944444444444445}{\pi}, \frac{-0.021604938271604937 \cdot \left(r \cdot r\right)}{s \cdot \pi}\right)}{s \cdot s} + \left(\frac{-0.16666666666666666}{s \cdot \pi} + \frac{0.25}{\pi \cdot r}\right)}}} \]

   8. Taylor expanded in s around inf

      \[\leadsto \frac{1}{s \cdot \color{blue}{\left(\left(-1 \cdot \frac{\frac{-16}{9} \cdot \left({r}^{3} \cdot \mathsf{PI}\left(\right)\right) + \frac{10}{9} \cdot \left({r}^{3} \cdot \mathsf{PI}\left(\right)\right)}{{s}^{2}} + 4 \cdot \left(r \cdot \mathsf{PI}\left(\right)\right)\right) - \frac{-8}{3} \cdot \frac{{r}^{2} \cdot \mathsf{PI}\left(\right)}{s}\right)}} \]

   10. Applied rewrites 100.0%

       \[\leadsto \frac{1}{s \cdot \color{blue}{\mathsf{fma}\left(r \cdot r, \frac{\pi}{s} \cdot 2.6666666666666665, r \cdot \left(\pi \cdot 4\right) - \frac{\left(r \cdot \left(r \cdot r\right)\right) \cdot \left(\pi \cdot -0.6666666666666666\right)}{s \cdot s}\right)}} \]

   if 3.20000006e-24 < s

   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 s around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{\frac{-1}{48} \cdot \frac{{r}^{2}}{\mathsf{PI}\left(\right)} + \frac{-1}{1296} \cdot \frac{{r}^{2}}{\mathsf{PI}\left(\right)}}{s} + \left(\frac{-1}{16} \cdot \frac{r}{\mathsf{PI}\left(\right)} + \frac{-1}{144} \cdot \frac{r}{\mathsf{PI}\left(\right)}\right)}{s} - \frac{1}{6} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{s} - \frac{1}{4} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s}} \]

   5. Applied rewrites 13.7%

      \[\leadsto \color{blue}{\frac{\frac{0.25}{r \cdot \pi} + \left(\frac{\mathsf{fma}\left(-0.021604938271604937, \frac{r \cdot r}{s \cdot \pi}, \frac{r \cdot 0.06944444444444445}{\pi}\right)}{s \cdot s} + \frac{-0.16666666666666666}{s \cdot \pi}\right)}{s}} \]
   6. Step-by-step derivation

   7. Applied rewrites 13.8%

      \[\leadsto \frac{1}{\color{blue}{\frac{s}{\frac{\mathsf{fma}\left(r, \frac{0.06944444444444445}{\pi}, \frac{-0.021604938271604937 \cdot \left(r \cdot r\right)}{s \cdot \pi}\right)}{s \cdot s} + \left(\frac{-0.16666666666666666}{s \cdot \pi} + \frac{0.25}{\pi \cdot r}\right)}}} \]

   8. Taylor expanded in s around -inf

      \[\leadsto \frac{1}{-1 \cdot \color{blue}{\left(s \cdot \left(-4 \cdot \left(r \cdot \mathsf{PI}\left(\right)\right) + -1 \cdot \frac{-1 \cdot \frac{\left(\frac{-2}{3} \cdot \frac{r \cdot \left(\frac{-10}{9} \cdot \left({r}^{3} \cdot \mathsf{PI}\left(\right)\right) + \frac{16}{9} \cdot \left({r}^{3} \cdot \mathsf{PI}\left(\right)\right)\right)}{s} + \left(\frac{-28}{81} \cdot \frac{{r}^{4} \cdot \mathsf{PI}\left(\right)}{s} + \frac{20}{27} \cdot \frac{{r}^{4} \cdot \mathsf{PI}\left(\right)}{s}\right)\right) - \left(\frac{-10}{9} \cdot \left({r}^{3} \cdot \mathsf{PI}\left(\right)\right) + \frac{16}{9} \cdot \left({r}^{3} \cdot \mathsf{PI}\left(\right)\right)\right)}{s} - \frac{-8}{3} \cdot \left({r}^{2} \cdot \mathsf{PI}\left(\right)\right)}{s}\right)\right)}} \]

   10. Applied rewrites 55.8%

       \[\leadsto \frac{1}{-s \cdot \mathsf{fma}\left(r, \pi \cdot -4, \frac{\mathsf{fma}\left(r, r \cdot \left(\pi \cdot 2.6666666666666665\right), \frac{\mathsf{fma}\left(\pi, \left(r \cdot \left(r \cdot r\right)\right) \cdot -0.6666666666666666, \mathsf{fma}\left(-0.6666666666666666 \cdot r, \left(r \cdot \left(r \cdot r\right)\right) \cdot \left(\frac{\pi}{s} \cdot 0.6666666666666666\right), \frac{\left(\pi \cdot {r}^{4}\right) \cdot 0.3950617283950617}{s}\right)\right)}{-s}\right)}{-s}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \leq 3.2000000625327404 \cdot 10^{-24}:\\ \;\;\;\;\frac{1}{s \cdot \mathsf{fma}\left(r \cdot r, \frac{\pi}{s} \cdot 2.6666666666666665, r \cdot \left(\pi \cdot 4\right) - \frac{\left(r \cdot \left(r \cdot r\right)\right) \cdot \left(\pi \cdot -0.6666666666666666\right)}{s \cdot s}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(-s\right) \cdot \mathsf{fma}\left(r, \pi \cdot -4, \frac{\mathsf{fma}\left(r, r \cdot \left(\pi \cdot 2.6666666666666665\right), \frac{\mathsf{fma}\left(\pi, \left(r \cdot \left(r \cdot r\right)\right) \cdot -0.6666666666666666, \mathsf{fma}\left(r \cdot -0.6666666666666666, \left(r \cdot \left(r \cdot r\right)\right) \cdot \left(\frac{\pi}{s} \cdot 0.6666666666666666\right), \frac{\left(\pi \cdot {r}^{4}\right) \cdot 0.3950617283950617}{s}\right)\right)}{-s}\right)}{-s}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

   1. lift-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r}} + \frac{\frac{3}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   2. lift-*.f32 N/A

      \[\leadsto \frac{\color{blue}{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\frac{3}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   3. lift-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\color{blue}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r}} + \frac{\frac{3}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   4. times-frac N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{4}}{\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s} \cdot \frac{e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{r}} + \frac{\frac{3}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   5. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{\frac{\frac{1}{4}}{\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{r}} + \frac{\frac{3}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

   6. lower-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\frac{\frac{1}{4}}{\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{r}} + \frac{\frac{3}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{3 \cdot s}}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

4. Applied rewrites 99.8%

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

6. Applied rewrites 99.7%

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

   1. lift-*.f32 N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \cdot \left(\frac{1}{r} \cdot \left(e^{\frac{r}{s \cdot -3}} + e^{\frac{r}{\mathsf{neg}\left(s\right)}}\right)\right)} \]

   2. lift-*.f32 N/A

      \[\leadsto \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \cdot \color{blue}{\left(\frac{1}{r} \cdot \left(e^{\frac{r}{s \cdot -3}} + e^{\frac{r}{\mathsf{neg}\left(s\right)}}\right)\right)} \]

   3. lift-/.f32 N/A

      \[\leadsto \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \cdot \left(\color{blue}{\frac{1}{r}} \cdot \left(e^{\frac{r}{s \cdot -3}} + e^{\frac{r}{\mathsf{neg}\left(s\right)}}\right)\right) \]

   4. associate-*l/ N/A

      \[\leadsto \frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \cdot \color{blue}{\frac{1 \cdot \left(e^{\frac{r}{s \cdot -3}} + e^{\frac{r}{\mathsf{neg}\left(s\right)}}\right)}{r}} \]

   5. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{\frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \cdot \left(1 \cdot \left(e^{\frac{r}{s \cdot -3}} + e^{\frac{r}{\mathsf{neg}\left(s\right)}}\right)\right)}{r}} \]

   6. lower-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \cdot \left(1 \cdot \left(e^{\frac{r}{s \cdot -3}} + e^{\frac{r}{\mathsf{neg}\left(s\right)}}\right)\right)}{r}} \]

8. Applied rewrites 99.7%

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

   1. lift-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \cdot \left(e^{\frac{r}{\mathsf{neg}\left(s\right)}} + e^{\frac{r}{s} \cdot \frac{-1}{3}}\right)}{r}} \]

   2. lift-*.f32 N/A

      \[\leadsto \frac{\color{blue}{\frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)} \cdot \left(e^{\frac{r}{\mathsf{neg}\left(s\right)}} + e^{\frac{r}{s} \cdot \frac{-1}{3}}\right)}}{r} \]

   3. lift-/.f32 N/A

      \[\leadsto \frac{\color{blue}{\frac{\frac{1}{8}}{s \cdot \mathsf{PI}\left(\right)}} \cdot \left(e^{\frac{r}{\mathsf{neg}\left(s\right)}} + e^{\frac{r}{s} \cdot \frac{-1}{3}}\right)}{r} \]

   4. associate-*l/ N/A

      \[\leadsto \frac{\color{blue}{\frac{\frac{1}{8} \cdot \left(e^{\frac{r}{\mathsf{neg}\left(s\right)}} + e^{\frac{r}{s} \cdot \frac{-1}{3}}\right)}{s \cdot \mathsf{PI}\left(\right)}}}{r} \]

   5. lift-*.f32 N/A

      \[\leadsto \frac{\frac{\frac{1}{8} \cdot \left(e^{\frac{r}{\mathsf{neg}\left(s\right)}} + e^{\color{blue}{\frac{r}{s} \cdot \frac{-1}{3}}}\right)}{s \cdot \mathsf{PI}\left(\right)}}{r} \]

   6. lift-/.f32 N/A

      \[\leadsto \frac{\frac{\frac{1}{8} \cdot \left(e^{\frac{r}{\mathsf{neg}\left(s\right)}} + e^{\color{blue}{\frac{r}{s}} \cdot \frac{-1}{3}}\right)}{s \cdot \mathsf{PI}\left(\right)}}{r} \]

   7. associate-*l/ N/A

      \[\leadsto \frac{\frac{\frac{1}{8} \cdot \left(e^{\frac{r}{\mathsf{neg}\left(s\right)}} + e^{\color{blue}{\frac{r \cdot \frac{-1}{3}}{s}}}\right)}{s \cdot \mathsf{PI}\left(\right)}}{r} \]

   8. lift-*.f32 N/A

      \[\leadsto \frac{\frac{\frac{1}{8} \cdot \left(e^{\frac{r}{\mathsf{neg}\left(s\right)}} + e^{\frac{\color{blue}{r \cdot \frac{-1}{3}}}{s}}\right)}{s \cdot \mathsf{PI}\left(\right)}}{r} \]

   9. lift-/.f32 N/A

      \[\leadsto \frac{\frac{\frac{1}{8} \cdot \left(e^{\frac{r}{\mathsf{neg}\left(s\right)}} + e^{\color{blue}{\frac{r \cdot \frac{-1}{3}}{s}}}\right)}{s \cdot \mathsf{PI}\left(\right)}}{r} \]

   10. lift-*.f32 N/A

       \[\leadsto \frac{\frac{\color{blue}{\frac{1}{8} \cdot \left(e^{\frac{r}{\mathsf{neg}\left(s\right)}} + e^{\frac{r \cdot \frac{-1}{3}}{s}}\right)}}{s \cdot \mathsf{PI}\left(\right)}}{r} \]

10. Applied rewrites 99.7%

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

Alternative 6: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in r around inf

   \[\leadsto \color{blue}{\frac{1}{8} \cdot \frac{e^{-1 \cdot \frac{r}{s}} + e^{\frac{-1}{3} \cdot \frac{r}{s}}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}} \]
6. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{8} \cdot \left(e^{-1 \cdot \frac{r}{s}} + e^{\frac{-1}{3} \cdot \frac{r}{s}}\right)}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}} \]

   2. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{\left(\frac{-1}{8} \cdot -1\right)} \cdot \left(e^{-1 \cdot \frac{r}{s}} + e^{\frac{-1}{3} \cdot \frac{r}{s}}\right)}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} \]

   3. associate-*r* N/A

      \[\leadsto \frac{\color{blue}{\frac{-1}{8} \cdot \left(-1 \cdot \left(e^{-1 \cdot \frac{r}{s}} + e^{\frac{-1}{3} \cdot \frac{r}{s}}\right)\right)}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} \]

   4. distribute-lft-out N/A

      \[\leadsto \frac{\frac{-1}{8} \cdot \color{blue}{\left(-1 \cdot e^{-1 \cdot \frac{r}{s}} + -1 \cdot e^{\frac{-1}{3} \cdot \frac{r}{s}}\right)}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} \]

   5. lower-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(-1 \cdot e^{-1 \cdot \frac{r}{s}} + -1 \cdot e^{\frac{-1}{3} \cdot \frac{r}{s}}\right)}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}} \]

7. Applied rewrites 99.6%

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

9. Applied rewrites 99.7%

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

10. Final simplification 99.7%

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

Alternative 7: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in r around inf

   \[\leadsto \color{blue}{\frac{1}{8} \cdot \frac{e^{-1 \cdot \frac{r}{s}} + e^{\frac{-1}{3} \cdot \frac{r}{s}}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}} \]
6. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{8} \cdot \left(e^{-1 \cdot \frac{r}{s}} + e^{\frac{-1}{3} \cdot \frac{r}{s}}\right)}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}} \]

   2. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{\left(\frac{-1}{8} \cdot -1\right)} \cdot \left(e^{-1 \cdot \frac{r}{s}} + e^{\frac{-1}{3} \cdot \frac{r}{s}}\right)}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} \]

   3. associate-*r* N/A

      \[\leadsto \frac{\color{blue}{\frac{-1}{8} \cdot \left(-1 \cdot \left(e^{-1 \cdot \frac{r}{s}} + e^{\frac{-1}{3} \cdot \frac{r}{s}}\right)\right)}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} \]

   4. distribute-lft-out N/A

      \[\leadsto \frac{\frac{-1}{8} \cdot \color{blue}{\left(-1 \cdot e^{-1 \cdot \frac{r}{s}} + -1 \cdot e^{\frac{-1}{3} \cdot \frac{r}{s}}\right)}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} \]

   5. lower-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\frac{-1}{8} \cdot \left(-1 \cdot e^{-1 \cdot \frac{r}{s}} + -1 \cdot e^{\frac{-1}{3} \cdot \frac{r}{s}}\right)}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}} \]

7. Applied rewrites 99.6%

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

8. Final simplification 99.6%

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

Alternative 8: 59.8% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \frac{1}{s \cdot \mathsf{fma}\left(r \cdot r, \frac{\pi}{s} \cdot 2.6666666666666665, r \cdot \left(\pi \cdot 4\right) - \frac{\left(r \cdot \left(r \cdot r\right)\right) \cdot \left(\pi \cdot -0.6666666666666666\right)}{s \cdot s}\right)} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (/
  1.0
  (*
   s
   (fma
    (* r r)
    (* (/ PI s) 2.6666666666666665)
    (-
     (* r (* PI 4.0))
     (/ (* (* r (* r r)) (* PI -0.6666666666666666)) (* s s)))))))

C

float code(float s, float r) {
	return 1.0f / (s * fmaf((r * r), ((((float) M_PI) / s) * 2.6666666666666665f), ((r * (((float) M_PI) * 4.0f)) - (((r * (r * r)) * (((float) M_PI) * -0.6666666666666666f)) / (s * s)))));
}

Julia

function code(s, r)
	return Float32(Float32(1.0) / Float32(s * fma(Float32(r * r), Float32(Float32(Float32(pi) / s) * Float32(2.6666666666666665)), Float32(Float32(r * Float32(Float32(pi) * Float32(4.0))) - Float32(Float32(Float32(r * Float32(r * r)) * Float32(Float32(pi) * Float32(-0.6666666666666666))) / Float32(s * s))))))
end

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in s around -inf

   \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{\frac{-1}{48} \cdot \frac{{r}^{2}}{\mathsf{PI}\left(\right)} + \frac{-1}{1296} \cdot \frac{{r}^{2}}{\mathsf{PI}\left(\right)}}{s} + \left(\frac{-1}{16} \cdot \frac{r}{\mathsf{PI}\left(\right)} + \frac{-1}{144} \cdot \frac{r}{\mathsf{PI}\left(\right)}\right)}{s} - \frac{1}{6} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{s} - \frac{1}{4} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s}} \]

5. Applied rewrites 9.7%

   \[\leadsto \color{blue}{\frac{\frac{0.25}{r \cdot \pi} + \left(\frac{\mathsf{fma}\left(-0.021604938271604937, \frac{r \cdot r}{s \cdot \pi}, \frac{r \cdot 0.06944444444444445}{\pi}\right)}{s \cdot s} + \frac{-0.16666666666666666}{s \cdot \pi}\right)}{s}} \]
6. Step-by-step derivation

7. Applied rewrites 9.7%

   \[\leadsto \frac{1}{\color{blue}{\frac{s}{\frac{\mathsf{fma}\left(r, \frac{0.06944444444444445}{\pi}, \frac{-0.021604938271604937 \cdot \left(r \cdot r\right)}{s \cdot \pi}\right)}{s \cdot s} + \left(\frac{-0.16666666666666666}{s \cdot \pi} + \frac{0.25}{\pi \cdot r}\right)}}} \]

8. Taylor expanded in s around inf

   \[\leadsto \frac{1}{s \cdot \color{blue}{\left(\left(-1 \cdot \frac{\frac{-16}{9} \cdot \left({r}^{3} \cdot \mathsf{PI}\left(\right)\right) + \frac{10}{9} \cdot \left({r}^{3} \cdot \mathsf{PI}\left(\right)\right)}{{s}^{2}} + 4 \cdot \left(r \cdot \mathsf{PI}\left(\right)\right)\right) - \frac{-8}{3} \cdot \frac{{r}^{2} \cdot \mathsf{PI}\left(\right)}{s}\right)}} \]

10. Applied rewrites 60.6%

    \[\leadsto \frac{1}{s \cdot \color{blue}{\mathsf{fma}\left(r \cdot r, \frac{\pi}{s} \cdot 2.6666666666666665, r \cdot \left(\pi \cdot 4\right) - \frac{\left(r \cdot \left(r \cdot r\right)\right) \cdot \left(\pi \cdot -0.6666666666666666\right)}{s \cdot s}\right)}} \]
11. Add Preprocessing

Alternative 9: 57.9% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \frac{1}{s \cdot \left(\frac{\mathsf{fma}\left(r, r \cdot \left(\pi \cdot 2.6666666666666665\right), \left(r \cdot \left(r \cdot r\right)\right) \cdot \left(\frac{\pi}{s} \cdot 0.6666666666666666\right)\right)}{s} - \left(\pi \cdot r\right) \cdot -4\right)} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (/
  1.0
  (*
   s
   (-
    (/
     (fma
      r
      (* r (* PI 2.6666666666666665))
      (* (* r (* r r)) (* (/ PI s) 0.6666666666666666)))
     s)
    (* (* PI r) -4.0)))))

C

float code(float s, float r) {
	return 1.0f / (s * ((fmaf(r, (r * (((float) M_PI) * 2.6666666666666665f)), ((r * (r * r)) * ((((float) M_PI) / s) * 0.6666666666666666f))) / s) - ((((float) M_PI) * r) * -4.0f)));
}

Julia

function code(s, r)
	return Float32(Float32(1.0) / Float32(s * Float32(Float32(fma(r, Float32(r * Float32(Float32(pi) * Float32(2.6666666666666665))), Float32(Float32(r * Float32(r * r)) * Float32(Float32(Float32(pi) / s) * Float32(0.6666666666666666)))) / s) - Float32(Float32(Float32(pi) * r) * Float32(-4.0)))))
end

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in s around -inf

   \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{\frac{-1}{48} \cdot \frac{{r}^{2}}{\mathsf{PI}\left(\right)} + \frac{-1}{1296} \cdot \frac{{r}^{2}}{\mathsf{PI}\left(\right)}}{s} + \left(\frac{-1}{16} \cdot \frac{r}{\mathsf{PI}\left(\right)} + \frac{-1}{144} \cdot \frac{r}{\mathsf{PI}\left(\right)}\right)}{s} - \frac{1}{6} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{s} - \frac{1}{4} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s}} \]

5. Applied rewrites 9.7%

   \[\leadsto \color{blue}{\frac{\frac{0.25}{r \cdot \pi} + \left(\frac{\mathsf{fma}\left(-0.021604938271604937, \frac{r \cdot r}{s \cdot \pi}, \frac{r \cdot 0.06944444444444445}{\pi}\right)}{s \cdot s} + \frac{-0.16666666666666666}{s \cdot \pi}\right)}{s}} \]
6. Step-by-step derivation

7. Applied rewrites 9.7%

   \[\leadsto \frac{1}{\color{blue}{\frac{s}{\frac{\mathsf{fma}\left(r, \frac{0.06944444444444445}{\pi}, \frac{-0.021604938271604937 \cdot \left(r \cdot r\right)}{s \cdot \pi}\right)}{s \cdot s} + \left(\frac{-0.16666666666666666}{s \cdot \pi} + \frac{0.25}{\pi \cdot r}\right)}}} \]

8. Taylor expanded in s around -inf

   \[\leadsto \frac{1}{-1 \cdot \color{blue}{\left(s \cdot \left(-4 \cdot \left(r \cdot \mathsf{PI}\left(\right)\right) + -1 \cdot \frac{\left(\frac{-10}{9} \cdot \frac{{r}^{3} \cdot \mathsf{PI}\left(\right)}{s} + \frac{16}{9} \cdot \frac{{r}^{3} \cdot \mathsf{PI}\left(\right)}{s}\right) - \frac{-8}{3} \cdot \left({r}^{2} \cdot \mathsf{PI}\left(\right)\right)}{s}\right)\right)}} \]

10. Applied rewrites 58.4%

    \[\leadsto \frac{1}{\left(\left(r \cdot \pi\right) \cdot -4 - \frac{\mathsf{fma}\left(r, r \cdot \left(\pi \cdot 2.6666666666666665\right), \left(r \cdot \left(r \cdot r\right)\right) \cdot \left(\frac{\pi}{s} \cdot 0.6666666666666666\right)\right)}{s}\right) \cdot \color{blue}{\left(-s\right)}} \]

11. Final simplification 58.4%

    \[\leadsto \frac{1}{s \cdot \left(\frac{\mathsf{fma}\left(r, r \cdot \left(\pi \cdot 2.6666666666666665\right), \left(r \cdot \left(r \cdot r\right)\right) \cdot \left(\frac{\pi}{s} \cdot 0.6666666666666666\right)\right)}{s} - \left(\pi \cdot r\right) \cdot -4\right)} \]
12. Add Preprocessing

Alternative 10: 27.3% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{r \cdot \mathsf{fma}\left(r, \mathsf{fma}\left(r, \frac{\pi}{s} \cdot 0.6666666666666666, \pi \cdot 2.6666666666666665\right), s \cdot \left(\pi \cdot 4\right)\right)} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (/
  1.0
  (*
   r
   (fma
    r
    (fma r (* (/ PI s) 0.6666666666666666) (* PI 2.6666666666666665))
    (* s (* PI 4.0))))))

C

float code(float s, float r) {
	return 1.0f / (r * fmaf(r, fmaf(r, ((((float) M_PI) / s) * 0.6666666666666666f), (((float) M_PI) * 2.6666666666666665f)), (s * (((float) M_PI) * 4.0f))));
}

Julia

function code(s, r)
	return Float32(Float32(1.0) / Float32(r * fma(r, fma(r, Float32(Float32(Float32(pi) / s) * Float32(0.6666666666666666)), Float32(Float32(pi) * Float32(2.6666666666666665))), Float32(s * Float32(Float32(pi) * Float32(4.0))))))
end

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in s around -inf

   \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{\frac{-1}{48} \cdot \frac{{r}^{2}}{\mathsf{PI}\left(\right)} + \frac{-1}{1296} \cdot \frac{{r}^{2}}{\mathsf{PI}\left(\right)}}{s} + \left(\frac{-1}{16} \cdot \frac{r}{\mathsf{PI}\left(\right)} + \frac{-1}{144} \cdot \frac{r}{\mathsf{PI}\left(\right)}\right)}{s} - \frac{1}{6} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{s} - \frac{1}{4} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s}} \]

5. Applied rewrites 9.7%

   \[\leadsto \color{blue}{\frac{\frac{0.25}{r \cdot \pi} + \left(\frac{\mathsf{fma}\left(-0.021604938271604937, \frac{r \cdot r}{s \cdot \pi}, \frac{r \cdot 0.06944444444444445}{\pi}\right)}{s \cdot s} + \frac{-0.16666666666666666}{s \cdot \pi}\right)}{s}} \]
6. Step-by-step derivation

7. Applied rewrites 9.7%

   \[\leadsto \frac{1}{\color{blue}{\frac{s}{\frac{\mathsf{fma}\left(r, \frac{0.06944444444444445}{\pi}, \frac{-0.021604938271604937 \cdot \left(r \cdot r\right)}{s \cdot \pi}\right)}{s \cdot s} + \left(\frac{-0.16666666666666666}{s \cdot \pi} + \frac{0.25}{\pi \cdot r}\right)}}} \]

8. Taylor expanded in r around 0

   \[\leadsto \frac{1}{r \cdot \color{blue}{\left(4 \cdot \left(s \cdot \mathsf{PI}\left(\right)\right) + r \cdot \left(-1 \cdot \left(r \cdot \left(\frac{-16}{9} \cdot \frac{\mathsf{PI}\left(\right)}{s} + \frac{10}{9} \cdot \frac{\mathsf{PI}\left(\right)}{s}\right)\right) - \frac{-8}{3} \cdot \mathsf{PI}\left(\right)\right)\right)}} \]
9. Step-by-step derivation

10. Applied rewrites 26.0%

    \[\leadsto \frac{1}{r \cdot \color{blue}{\mathsf{fma}\left(r, \mathsf{fma}\left(r, \frac{\pi}{s} \cdot 0.6666666666666666, \pi \cdot 2.6666666666666665\right), s \cdot \left(\pi \cdot 4\right)\right)}} \]
11. Add Preprocessing

Alternative 11: 20.9% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \frac{1}{s \cdot \mathsf{fma}\left(r, \pi \cdot 4, \left(r \cdot r\right) \cdot \left(\frac{\pi}{s} \cdot 2.6666666666666665\right)\right)} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (/ 1.0 (* s (fma r (* PI 4.0) (* (* r r) (* (/ PI s) 2.6666666666666665))))))

C

float code(float s, float r) {
	return 1.0f / (s * fmaf(r, (((float) M_PI) * 4.0f), ((r * r) * ((((float) M_PI) / s) * 2.6666666666666665f))));
}

Julia

function code(s, r)
	return Float32(Float32(1.0) / Float32(s * fma(r, Float32(Float32(pi) * Float32(4.0)), Float32(Float32(r * r) * Float32(Float32(Float32(pi) / s) * Float32(2.6666666666666665))))))
end

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in s around -inf

   \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{\frac{-1}{48} \cdot \frac{{r}^{2}}{\mathsf{PI}\left(\right)} + \frac{-1}{1296} \cdot \frac{{r}^{2}}{\mathsf{PI}\left(\right)}}{s} + \left(\frac{-1}{16} \cdot \frac{r}{\mathsf{PI}\left(\right)} + \frac{-1}{144} \cdot \frac{r}{\mathsf{PI}\left(\right)}\right)}{s} - \frac{1}{6} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{s} - \frac{1}{4} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s}} \]

5. Applied rewrites 9.7%

   \[\leadsto \color{blue}{\frac{\frac{0.25}{r \cdot \pi} + \left(\frac{\mathsf{fma}\left(-0.021604938271604937, \frac{r \cdot r}{s \cdot \pi}, \frac{r \cdot 0.06944444444444445}{\pi}\right)}{s \cdot s} + \frac{-0.16666666666666666}{s \cdot \pi}\right)}{s}} \]
6. Step-by-step derivation

7. Applied rewrites 9.7%

   \[\leadsto \frac{1}{\color{blue}{\frac{s}{\frac{\mathsf{fma}\left(r, \frac{0.06944444444444445}{\pi}, \frac{-0.021604938271604937 \cdot \left(r \cdot r\right)}{s \cdot \pi}\right)}{s \cdot s} + \left(\frac{-0.16666666666666666}{s \cdot \pi} + \frac{0.25}{\pi \cdot r}\right)}}} \]

8. Taylor expanded in s around inf

   \[\leadsto \frac{1}{s \cdot \color{blue}{\left(\frac{8}{3} \cdot \frac{{r}^{2} \cdot \mathsf{PI}\left(\right)}{s} + 4 \cdot \left(r \cdot \mathsf{PI}\left(\right)\right)\right)}} \]
9. Step-by-step derivation

10. Applied rewrites 19.3%

    \[\leadsto \frac{1}{s \cdot \color{blue}{\mathsf{fma}\left(r, \pi \cdot 4, \left(r \cdot r\right) \cdot \left(\frac{\pi}{s} \cdot 2.6666666666666665\right)\right)}} \]
11. Add Preprocessing

Alternative 12: 12.6% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{r \cdot \left(\pi \cdot \mathsf{fma}\left(r, 2.6666666666666665, s \cdot 4\right)\right)} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (/ 1.0 (* r (* PI (fma r 2.6666666666666665 (* s 4.0))))))

C

float code(float s, float r) {
	return 1.0f / (r * (((float) M_PI) * fmaf(r, 2.6666666666666665f, (s * 4.0f))));
}

Julia

function code(s, r)
	return Float32(Float32(1.0) / Float32(r * Float32(Float32(pi) * fma(r, Float32(2.6666666666666665), Float32(s * Float32(4.0))))))
end

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in s around -inf

   \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{\frac{-1}{48} \cdot \frac{{r}^{2}}{\mathsf{PI}\left(\right)} + \frac{-1}{1296} \cdot \frac{{r}^{2}}{\mathsf{PI}\left(\right)}}{s} + \left(\frac{-1}{16} \cdot \frac{r}{\mathsf{PI}\left(\right)} + \frac{-1}{144} \cdot \frac{r}{\mathsf{PI}\left(\right)}\right)}{s} - \frac{1}{6} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{s} - \frac{1}{4} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s}} \]

5. Applied rewrites 9.7%

   \[\leadsto \color{blue}{\frac{\frac{0.25}{r \cdot \pi} + \left(\frac{\mathsf{fma}\left(-0.021604938271604937, \frac{r \cdot r}{s \cdot \pi}, \frac{r \cdot 0.06944444444444445}{\pi}\right)}{s \cdot s} + \frac{-0.16666666666666666}{s \cdot \pi}\right)}{s}} \]
6. Step-by-step derivation

7. Applied rewrites 9.7%

   \[\leadsto \frac{1}{\color{blue}{\frac{s}{\frac{\mathsf{fma}\left(r, \frac{0.06944444444444445}{\pi}, \frac{-0.021604938271604937 \cdot \left(r \cdot r\right)}{s \cdot \pi}\right)}{s \cdot s} + \left(\frac{-0.16666666666666666}{s \cdot \pi} + \frac{0.25}{\pi \cdot r}\right)}}} \]

8. Taylor expanded in r around 0

   \[\leadsto \frac{1}{r \cdot \color{blue}{\left(\frac{8}{3} \cdot \left(r \cdot \mathsf{PI}\left(\right)\right) + 4 \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)\right)}} \]
9. Step-by-step derivation

10. Applied rewrites 12.0%

    \[\leadsto \frac{1}{r \cdot \color{blue}{\left(\pi \cdot \mathsf{fma}\left(r, 2.6666666666666665, s \cdot 4\right)\right)}} \]
11. Add Preprocessing

Alternative 13: 9.2% accurate, 10.6× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in r around 0

   \[\leadsto \color{blue}{\frac{\frac{1}{4}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}} \]
4. Step-by-step derivation

   1. lower-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{4}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}} \]

   2. lower-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4}}{\color{blue}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}} \]

   3. lower-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4}}{r \cdot \color{blue}{\left(s \cdot \mathsf{PI}\left(\right)\right)}} \]

   4. lower-PI.f32 8.7

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

5. Applied rewrites 8.7%

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

7. Applied rewrites 8.7%

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

Alternative 14: 9.2% accurate, 13.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in r around 0

   \[\leadsto \color{blue}{\frac{\frac{1}{4}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}} \]
4. Step-by-step derivation

   1. lower-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{4}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}} \]

   2. lower-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4}}{\color{blue}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}} \]

   3. lower-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4}}{r \cdot \color{blue}{\left(s \cdot \mathsf{PI}\left(\right)\right)}} \]

   4. lower-PI.f32 8.7

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

5. Applied rewrites 8.7%

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

7. Applied rewrites 8.7%

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

8. Final simplification 8.7%

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

Alternative 15: 9.2% accurate, 13.5× 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 99.8%

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

3. Taylor expanded in r around 0

   \[\leadsto \color{blue}{\frac{\frac{1}{4}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}} \]
4. Step-by-step derivation

   1. lower-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{4}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}} \]

   2. lower-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4}}{\color{blue}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}} \]

   3. lower-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4}}{r \cdot \color{blue}{\left(s \cdot \mathsf{PI}\left(\right)\right)}} \]

   4. lower-PI.f32 8.7

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

5. Applied rewrites 8.7%

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

6. Final simplification 8.7%

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

Reproduce

herbie shell --seed 2024233 
(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))))