Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.5%

Time: 13.2s

Alternatives: 10

Speedup: N/A×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.5% accurate, 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(-Float32(r / 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 99.6%

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

4. Applied rewrites 99.6%

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

5. Final simplification 99.6%

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

Alternative 2: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = (single(0.125) / (s * single(pi))) * ((exp(-(r / s)) + exp(((r / s) * single(-0.3333333333333333)))) / 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. Add Preprocessing

4. Applied rewrites 99.6%

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

5. Taylor expanded in r around inf

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

   1. lower-/.f32 N/A

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

   2. lower-+.f32 N/A

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

   3. lower-exp.f32 N/A

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

   4. mul-1-neg N/A

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

   5. distribute-neg-frac2 N/A

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

   6. lower-/.f32 N/A

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

   7. lower-neg.f32 N/A

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

   8. lower-exp.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f32 N/A

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

   11. lower-/.f32 99.6

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

7. Applied rewrites 99.6%

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

8. Final simplification 99.6%

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

Alternative 3: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

4. Applied rewrites 99.6%

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

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%

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

8. Final simplification 99.6%

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

Alternative 4: 73.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := r \cdot \left(r \cdot r\right)\\ t_1 := t\_0 \cdot 0.5555555555555556\\ \mathbf{if}\;s \leq 9.999999682655225 \cdot 10^{-20}:\\ \;\;\;\;\frac{\frac{1}{\pi}}{r \cdot \mathsf{fma}\left(r, \mathsf{fma}\left(r, \mathsf{fma}\left(\frac{r}{s \cdot s}, 0.04938271604938271, \frac{0.6666666666666666}{s}\right), 2.6666666666666665\right), s \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\pi}}{s \cdot \mathsf{fma}\left(r, 4, \frac{\mathsf{fma}\left(2, r \cdot \left(r \cdot \left(r \cdot \left(r \cdot -0.1728395061728395\right)\right)\right), \mathsf{fma}\left(r, 0.6666666666666666 \cdot \mathsf{fma}\left(t\_0, -1.7777777777777777, 2 \cdot t\_1\right), t\_0 \cdot \left(\left(r \cdot 0.5555555555555556\right) \cdot 1.3333333333333333\right)\right)\right)}{-s \cdot \left(s \cdot s\right)} - \mathsf{fma}\left(t\_1, \frac{2}{s \cdot s}, \mathsf{fma}\left(r \cdot r, \frac{-2.6666666666666665}{s}, \frac{r \cdot \left(\left(r \cdot r\right) \cdot -1.7777777777777777\right)}{s \cdot s}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (let* ((t_0 (* r (* r r))) (t_1 (* t_0 0.5555555555555556)))
   (if (<= s 9.999999682655225e-20)
     (/
      (/ 1.0 PI)
      (*
       r
       (fma
        r
        (fma
         r
         (fma (/ r (* s s)) 0.04938271604938271 (/ 0.6666666666666666 s))
         2.6666666666666665)
        (* s 4.0))))
     (/
      (/ 1.0 PI)
      (*
       s
       (fma
        r
        4.0
        (-
         (/
          (fma
           2.0
           (* r (* r (* r (* r -0.1728395061728395))))
           (fma
            r
            (* 0.6666666666666666 (fma t_0 -1.7777777777777777 (* 2.0 t_1)))
            (* t_0 (* (* r 0.5555555555555556) 1.3333333333333333))))
          (- (* s (* s s))))
         (fma
          t_1
          (/ 2.0 (* s s))
          (fma
           (* r r)
           (/ -2.6666666666666665 s)
           (/ (* r (* (* r r) -1.7777777777777777)) (* s s)))))))))))

C

float code(float s, float r) {
	float t_0 = r * (r * r);
	float t_1 = t_0 * 0.5555555555555556f;
	float tmp;
	if (s <= 9.999999682655225e-20f) {
		tmp = (1.0f / ((float) M_PI)) / (r * fmaf(r, fmaf(r, fmaf((r / (s * s)), 0.04938271604938271f, (0.6666666666666666f / s)), 2.6666666666666665f), (s * 4.0f)));
	} else {
		tmp = (1.0f / ((float) M_PI)) / (s * fmaf(r, 4.0f, ((fmaf(2.0f, (r * (r * (r * (r * -0.1728395061728395f)))), fmaf(r, (0.6666666666666666f * fmaf(t_0, -1.7777777777777777f, (2.0f * t_1))), (t_0 * ((r * 0.5555555555555556f) * 1.3333333333333333f)))) / -(s * (s * s))) - fmaf(t_1, (2.0f / (s * s)), fmaf((r * r), (-2.6666666666666665f / s), ((r * ((r * r) * -1.7777777777777777f)) / (s * s)))))));
	}
	return tmp;
}

Julia

function code(s, r)
	t_0 = Float32(r * Float32(r * r))
	t_1 = Float32(t_0 * Float32(0.5555555555555556))
	tmp = Float32(0.0)
	if (s <= Float32(9.999999682655225e-20))
		tmp = Float32(Float32(Float32(1.0) / Float32(pi)) / Float32(r * fma(r, fma(r, fma(Float32(r / Float32(s * s)), Float32(0.04938271604938271), Float32(Float32(0.6666666666666666) / s)), Float32(2.6666666666666665)), Float32(s * Float32(4.0)))));
	else
		tmp = Float32(Float32(Float32(1.0) / Float32(pi)) / Float32(s * fma(r, Float32(4.0), Float32(Float32(fma(Float32(2.0), Float32(r * Float32(r * Float32(r * Float32(r * Float32(-0.1728395061728395))))), fma(r, Float32(Float32(0.6666666666666666) * fma(t_0, Float32(-1.7777777777777777), Float32(Float32(2.0) * t_1))), Float32(t_0 * Float32(Float32(r * Float32(0.5555555555555556)) * Float32(1.3333333333333333))))) / Float32(-Float32(s * Float32(s * s)))) - fma(t_1, Float32(Float32(2.0) / Float32(s * s)), fma(Float32(r * r), Float32(Float32(-2.6666666666666665) / s), Float32(Float32(r * Float32(Float32(r * r) * Float32(-1.7777777777777777))) / Float32(s * s))))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if s < 9.99999968e-20

   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

   4. Applied rewrites 100.0%

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

      1. *-commutative N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft-neg-in N/A

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

      4. distribute-frac-neg2 N/A

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

      5. neg-mul-1 N/A

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

      6. exp-prod N/A

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

      7. lower-pow.f32 N/A

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

      8. lower-exp.f32 N/A

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

      9. div-inv N/A

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

      10. lower-*.f32 N/A

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

      11. associate-/r* N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f32 100.0

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

   6. Applied rewrites 100.0%

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

   8. Applied rewrites 3.9%

      \[\leadsto \color{blue}{\frac{\frac{1}{\pi}}{\frac{s}{0.125 \cdot \frac{2}{r}}}} \]

   9. Taylor expanded in r around 0

      \[\leadsto \frac{\frac{1}{\mathsf{PI}\left(\right)}}{\color{blue}{r \cdot \left(4 \cdot s + r \cdot \left(\frac{8}{3} + r \cdot \left(\frac{4}{81} \cdot \frac{r}{{s}^{2}} + \frac{2}{3} \cdot \frac{1}{s}\right)\right)\right)}} \]
   10. Step-by-step derivation

       1. lower-*.f32 N/A

          \[\leadsto \frac{\frac{1}{\mathsf{PI}\left(\right)}}{\color{blue}{r \cdot \left(4 \cdot s + r \cdot \left(\frac{8}{3} + r \cdot \left(\frac{4}{81} \cdot \frac{r}{{s}^{2}} + \frac{2}{3} \cdot \frac{1}{s}\right)\right)\right)}} \]

       2. +-commutative N/A

          \[\leadsto \frac{\frac{1}{\mathsf{PI}\left(\right)}}{r \cdot \color{blue}{\left(r \cdot \left(\frac{8}{3} + r \cdot \left(\frac{4}{81} \cdot \frac{r}{{s}^{2}} + \frac{2}{3} \cdot \frac{1}{s}\right)\right) + 4 \cdot s\right)}} \]

       3. lower-fma.f32 N/A

          \[\leadsto \frac{\frac{1}{\mathsf{PI}\left(\right)}}{r \cdot \color{blue}{\mathsf{fma}\left(r, \frac{8}{3} + r \cdot \left(\frac{4}{81} \cdot \frac{r}{{s}^{2}} + \frac{2}{3} \cdot \frac{1}{s}\right), 4 \cdot s\right)}} \]

       4. +-commutative N/A

          \[\leadsto \frac{\frac{1}{\mathsf{PI}\left(\right)}}{r \cdot \mathsf{fma}\left(r, \color{blue}{r \cdot \left(\frac{4}{81} \cdot \frac{r}{{s}^{2}} + \frac{2}{3} \cdot \frac{1}{s}\right) + \frac{8}{3}}, 4 \cdot s\right)} \]

       5. lower-fma.f32 N/A

          \[\leadsto \frac{\frac{1}{\mathsf{PI}\left(\right)}}{r \cdot \mathsf{fma}\left(r, \color{blue}{\mathsf{fma}\left(r, \frac{4}{81} \cdot \frac{r}{{s}^{2}} + \frac{2}{3} \cdot \frac{1}{s}, \frac{8}{3}\right)}, 4 \cdot s\right)} \]

       6. *-commutative N/A

          \[\leadsto \frac{\frac{1}{\mathsf{PI}\left(\right)}}{r \cdot \mathsf{fma}\left(r, \mathsf{fma}\left(r, \color{blue}{\frac{r}{{s}^{2}} \cdot \frac{4}{81}} + \frac{2}{3} \cdot \frac{1}{s}, \frac{8}{3}\right), 4 \cdot s\right)} \]

       7. lower-fma.f32 N/A

          \[\leadsto \frac{\frac{1}{\mathsf{PI}\left(\right)}}{r \cdot \mathsf{fma}\left(r, \mathsf{fma}\left(r, \color{blue}{\mathsf{fma}\left(\frac{r}{{s}^{2}}, \frac{4}{81}, \frac{2}{3} \cdot \frac{1}{s}\right)}, \frac{8}{3}\right), 4 \cdot s\right)} \]

       8. lower-/.f32 N/A

          \[\leadsto \frac{\frac{1}{\mathsf{PI}\left(\right)}}{r \cdot \mathsf{fma}\left(r, \mathsf{fma}\left(r, \mathsf{fma}\left(\color{blue}{\frac{r}{{s}^{2}}}, \frac{4}{81}, \frac{2}{3} \cdot \frac{1}{s}\right), \frac{8}{3}\right), 4 \cdot s\right)} \]

       9. unpow2 N/A

          \[\leadsto \frac{\frac{1}{\mathsf{PI}\left(\right)}}{r \cdot \mathsf{fma}\left(r, \mathsf{fma}\left(r, \mathsf{fma}\left(\frac{r}{\color{blue}{s \cdot s}}, \frac{4}{81}, \frac{2}{3} \cdot \frac{1}{s}\right), \frac{8}{3}\right), 4 \cdot s\right)} \]

       10. lower-*.f32 N/A

           \[\leadsto \frac{\frac{1}{\mathsf{PI}\left(\right)}}{r \cdot \mathsf{fma}\left(r, \mathsf{fma}\left(r, \mathsf{fma}\left(\frac{r}{\color{blue}{s \cdot s}}, \frac{4}{81}, \frac{2}{3} \cdot \frac{1}{s}\right), \frac{8}{3}\right), 4 \cdot s\right)} \]

       11. associate-*r/ N/A

           \[\leadsto \frac{\frac{1}{\mathsf{PI}\left(\right)}}{r \cdot \mathsf{fma}\left(r, \mathsf{fma}\left(r, \mathsf{fma}\left(\frac{r}{s \cdot s}, \frac{4}{81}, \color{blue}{\frac{\frac{2}{3} \cdot 1}{s}}\right), \frac{8}{3}\right), 4 \cdot s\right)} \]

       12. metadata-eval N/A

           \[\leadsto \frac{\frac{1}{\mathsf{PI}\left(\right)}}{r \cdot \mathsf{fma}\left(r, \mathsf{fma}\left(r, \mathsf{fma}\left(\frac{r}{s \cdot s}, \frac{4}{81}, \frac{\color{blue}{\frac{2}{3}}}{s}\right), \frac{8}{3}\right), 4 \cdot s\right)} \]

       13. lower-/.f32 N/A

           \[\leadsto \frac{\frac{1}{\mathsf{PI}\left(\right)}}{r \cdot \mathsf{fma}\left(r, \mathsf{fma}\left(r, \mathsf{fma}\left(\frac{r}{s \cdot s}, \frac{4}{81}, \color{blue}{\frac{\frac{2}{3}}{s}}\right), \frac{8}{3}\right), 4 \cdot s\right)} \]

       14. *-commutative N/A

           \[\leadsto \frac{\frac{1}{\mathsf{PI}\left(\right)}}{r \cdot \mathsf{fma}\left(r, \mathsf{fma}\left(r, \mathsf{fma}\left(\frac{r}{s \cdot s}, \frac{4}{81}, \frac{\frac{2}{3}}{s}\right), \frac{8}{3}\right), \color{blue}{s \cdot 4}\right)} \]

       15. lower-*.f32 97.8

           \[\leadsto \frac{\frac{1}{\pi}}{r \cdot \mathsf{fma}\left(r, \mathsf{fma}\left(r, \mathsf{fma}\left(\frac{r}{s \cdot s}, 0.04938271604938271, \frac{0.6666666666666666}{s}\right), 2.6666666666666665\right), \color{blue}{s \cdot 4}\right)} \]

   11. Applied rewrites 97.8%

       \[\leadsto \frac{\frac{1}{\pi}}{\color{blue}{r \cdot \mathsf{fma}\left(r, \mathsf{fma}\left(r, \mathsf{fma}\left(\frac{r}{s \cdot s}, 0.04938271604938271, \frac{0.6666666666666666}{s}\right), 2.6666666666666665\right), s \cdot 4\right)}} \]

   if 9.99999968e-20 < s

   1. Initial program 99.3%

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

   4. Applied rewrites 99.3%

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

      1. *-commutative N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft-neg-in N/A

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

      4. distribute-frac-neg2 N/A

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

      5. neg-mul-1 N/A

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

      6. exp-prod N/A

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

      7. lower-pow.f32 N/A

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

      8. lower-exp.f32 N/A

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

      9. div-inv N/A

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

      10. lower-*.f32 N/A

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

      11. associate-/r* N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f32 99.3

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

   6. Applied rewrites 99.3%

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

   8. Applied rewrites 12.7%

      \[\leadsto \color{blue}{\frac{\frac{1}{\pi}}{\frac{s}{0.125 \cdot \frac{2}{r}}}} \]

   9. Taylor expanded in s around inf

      \[\leadsto \frac{\frac{1}{\mathsf{PI}\left(\right)}}{\color{blue}{s \cdot \left(\left(-1 \cdot \frac{\frac{2}{3} \cdot \left(r \cdot \left(\frac{-16}{9} \cdot {r}^{3} + 2 \cdot \left({r}^{2} \cdot \left(\frac{1}{18} \cdot r + \frac{1}{2} \cdot r\right)\right)\right)\right) + \left(\frac{4}{3} \cdot \left({r}^{3} \cdot \left(\frac{1}{18} \cdot r + \frac{1}{2} \cdot r\right)\right) + 2 \cdot \left({r}^{2} \cdot \left(\frac{-1}{6} \cdot {r}^{2} + \frac{-1}{162} \cdot {r}^{2}\right)\right)\right)}{{s}^{3}} + 4 \cdot r\right) - \left(\frac{-8}{3} \cdot \frac{{r}^{2}}{s} + \left(\frac{-16}{9} \cdot \frac{{r}^{3}}{{s}^{2}} + 2 \cdot \frac{{r}^{2} \cdot \left(\frac{1}{18} \cdot r + \frac{1}{2} \cdot r\right)}{{s}^{2}}\right)\right)\right)}} \]

   11. Applied rewrites 54.8%

       \[\leadsto \frac{\frac{1}{\pi}}{\color{blue}{s \cdot \mathsf{fma}\left(r, 4, \frac{\mathsf{fma}\left(2, r \cdot \left(r \cdot \left(r \cdot \left(r \cdot -0.1728395061728395\right)\right)\right), \mathsf{fma}\left(r, \mathsf{fma}\left(r \cdot \left(r \cdot r\right), -1.7777777777777777, 2 \cdot \left(\left(r \cdot \left(r \cdot r\right)\right) \cdot 0.5555555555555556\right)\right) \cdot 0.6666666666666666, \left(r \cdot \left(r \cdot r\right)\right) \cdot \left(\left(r \cdot 0.5555555555555556\right) \cdot 1.3333333333333333\right)\right)\right)}{-s \cdot \left(s \cdot s\right)} - \mathsf{fma}\left(\left(r \cdot \left(r \cdot r\right)\right) \cdot 0.5555555555555556, \frac{2}{s \cdot s}, \mathsf{fma}\left(r \cdot r, \frac{-2.6666666666666665}{s}, \frac{r \cdot \left(\left(r \cdot r\right) \cdot -1.7777777777777777\right)}{s \cdot s}\right)\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \leq 9.999999682655225 \cdot 10^{-20}:\\ \;\;\;\;\frac{\frac{1}{\pi}}{r \cdot \mathsf{fma}\left(r, \mathsf{fma}\left(r, \mathsf{fma}\left(\frac{r}{s \cdot s}, 0.04938271604938271, \frac{0.6666666666666666}{s}\right), 2.6666666666666665\right), s \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\pi}}{s \cdot \mathsf{fma}\left(r, 4, \frac{\mathsf{fma}\left(2, r \cdot \left(r \cdot \left(r \cdot \left(r \cdot -0.1728395061728395\right)\right)\right), \mathsf{fma}\left(r, 0.6666666666666666 \cdot \mathsf{fma}\left(r \cdot \left(r \cdot r\right), -1.7777777777777777, 2 \cdot \left(\left(r \cdot \left(r \cdot r\right)\right) \cdot 0.5555555555555556\right)\right), \left(r \cdot \left(r \cdot r\right)\right) \cdot \left(\left(r \cdot 0.5555555555555556\right) \cdot 1.3333333333333333\right)\right)\right)}{-s \cdot \left(s \cdot s\right)} - \mathsf{fma}\left(\left(r \cdot \left(r \cdot r\right)\right) \cdot 0.5555555555555556, \frac{2}{s \cdot s}, \mathsf{fma}\left(r \cdot r, \frac{-2.6666666666666665}{s}, \frac{r \cdot \left(\left(r \cdot r\right) \cdot -1.7777777777777777\right)}{s \cdot s}\right)\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 71.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(r \cdot \left(r \cdot r\right)\right) \cdot 0.5555555555555556\\ t_1 := \mathsf{fma}\left(r, \left(r \cdot r\right) \cdot 1.7777777777777777, t\_0 \cdot -2\right)\\ \mathbf{if}\;s \leq 1.7999999428779406 \cdot 10^{-20}:\\ \;\;\;\;\frac{\frac{1}{\pi}}{r \cdot \mathsf{fma}\left(r, \mathsf{fma}\left(r, \mathsf{fma}\left(\frac{r}{s \cdot s}, 0.04938271604938271, \frac{0.6666666666666666}{s}\right), 2.6666666666666665\right), s \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\pi}}{s \cdot \left(\frac{\mathsf{fma}\left(r, r \cdot 2.6666666666666665, \frac{t\_1 - \mathsf{fma}\left(-0.6666666666666666, \frac{r}{s} \cdot t\_1, \mathsf{fma}\left(1.3333333333333333, \frac{r \cdot t\_0}{s}, \left(\left(r \cdot r\right) \cdot -2\right) \cdot \frac{\left(r \cdot r\right) \cdot 0.1728395061728395}{s}\right)\right)}{s}\right)}{s} - r \cdot -4\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (let* ((t_0 (* (* r (* r r)) 0.5555555555555556))
        (t_1 (fma r (* (* r r) 1.7777777777777777) (* t_0 -2.0))))
   (if (<= s 1.7999999428779406e-20)
     (/
      (/ 1.0 PI)
      (*
       r
       (fma
        r
        (fma
         r
         (fma (/ r (* s s)) 0.04938271604938271 (/ 0.6666666666666666 s))
         2.6666666666666665)
        (* s 4.0))))
     (/
      (/ 1.0 PI)
      (*
       s
       (-
        (/
         (fma
          r
          (* r 2.6666666666666665)
          (/
           (-
            t_1
            (fma
             -0.6666666666666666
             (* (/ r s) t_1)
             (fma
              1.3333333333333333
              (/ (* r t_0) s)
              (* (* (* r r) -2.0) (/ (* (* r r) 0.1728395061728395) s)))))
           s))
         s)
        (* r -4.0)))))))

C

float code(float s, float r) {
	float t_0 = (r * (r * r)) * 0.5555555555555556f;
	float t_1 = fmaf(r, ((r * r) * 1.7777777777777777f), (t_0 * -2.0f));
	float tmp;
	if (s <= 1.7999999428779406e-20f) {
		tmp = (1.0f / ((float) M_PI)) / (r * fmaf(r, fmaf(r, fmaf((r / (s * s)), 0.04938271604938271f, (0.6666666666666666f / s)), 2.6666666666666665f), (s * 4.0f)));
	} else {
		tmp = (1.0f / ((float) M_PI)) / (s * ((fmaf(r, (r * 2.6666666666666665f), ((t_1 - fmaf(-0.6666666666666666f, ((r / s) * t_1), fmaf(1.3333333333333333f, ((r * t_0) / s), (((r * r) * -2.0f) * (((r * r) * 0.1728395061728395f) / s))))) / s)) / s) - (r * -4.0f)));
	}
	return tmp;
}

Julia

function code(s, r)
	t_0 = Float32(Float32(r * Float32(r * r)) * Float32(0.5555555555555556))
	t_1 = fma(r, Float32(Float32(r * r) * Float32(1.7777777777777777)), Float32(t_0 * Float32(-2.0)))
	tmp = Float32(0.0)
	if (s <= Float32(1.7999999428779406e-20))
		tmp = Float32(Float32(Float32(1.0) / Float32(pi)) / Float32(r * fma(r, fma(r, fma(Float32(r / Float32(s * s)), Float32(0.04938271604938271), Float32(Float32(0.6666666666666666) / s)), Float32(2.6666666666666665)), Float32(s * Float32(4.0)))));
	else
		tmp = Float32(Float32(Float32(1.0) / Float32(pi)) / Float32(s * Float32(Float32(fma(r, Float32(r * Float32(2.6666666666666665)), Float32(Float32(t_1 - fma(Float32(-0.6666666666666666), Float32(Float32(r / s) * t_1), fma(Float32(1.3333333333333333), Float32(Float32(r * t_0) / s), Float32(Float32(Float32(r * r) * Float32(-2.0)) * Float32(Float32(Float32(r * r) * Float32(0.1728395061728395)) / s))))) / s)) / s) - Float32(r * Float32(-4.0)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if s < 1.79999994e-20

   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

   4. Applied rewrites 100.0%

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

      1. *-commutative N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft-neg-in N/A

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

      4. distribute-frac-neg2 N/A

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

      5. neg-mul-1 N/A

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

      6. exp-prod N/A

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

      7. lower-pow.f32 N/A

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

      8. lower-exp.f32 N/A

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

      9. div-inv N/A

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

      10. lower-*.f32 N/A

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

      11. associate-/r* N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f32 100.0

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

   6. Applied rewrites 100.0%

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

   8. Applied rewrites 3.9%

      \[\leadsto \color{blue}{\frac{\frac{1}{\pi}}{\frac{s}{0.125 \cdot \frac{2}{r}}}} \]

   9. Taylor expanded in r around 0

      \[\leadsto \frac{\frac{1}{\mathsf{PI}\left(\right)}}{\color{blue}{r \cdot \left(4 \cdot s + r \cdot \left(\frac{8}{3} + r \cdot \left(\frac{4}{81} \cdot \frac{r}{{s}^{2}} + \frac{2}{3} \cdot \frac{1}{s}\right)\right)\right)}} \]
   10. Step-by-step derivation

       1. lower-*.f32 N/A

          \[\leadsto \frac{\frac{1}{\mathsf{PI}\left(\right)}}{\color{blue}{r \cdot \left(4 \cdot s + r \cdot \left(\frac{8}{3} + r \cdot \left(\frac{4}{81} \cdot \frac{r}{{s}^{2}} + \frac{2}{3} \cdot \frac{1}{s}\right)\right)\right)}} \]

       2. +-commutative N/A

          \[\leadsto \frac{\frac{1}{\mathsf{PI}\left(\right)}}{r \cdot \color{blue}{\left(r \cdot \left(\frac{8}{3} + r \cdot \left(\frac{4}{81} \cdot \frac{r}{{s}^{2}} + \frac{2}{3} \cdot \frac{1}{s}\right)\right) + 4 \cdot s\right)}} \]

       3. lower-fma.f32 N/A

          \[\leadsto \frac{\frac{1}{\mathsf{PI}\left(\right)}}{r \cdot \color{blue}{\mathsf{fma}\left(r, \frac{8}{3} + r \cdot \left(\frac{4}{81} \cdot \frac{r}{{s}^{2}} + \frac{2}{3} \cdot \frac{1}{s}\right), 4 \cdot s\right)}} \]

       4. +-commutative N/A

          \[\leadsto \frac{\frac{1}{\mathsf{PI}\left(\right)}}{r \cdot \mathsf{fma}\left(r, \color{blue}{r \cdot \left(\frac{4}{81} \cdot \frac{r}{{s}^{2}} + \frac{2}{3} \cdot \frac{1}{s}\right) + \frac{8}{3}}, 4 \cdot s\right)} \]

       5. lower-fma.f32 N/A

          \[\leadsto \frac{\frac{1}{\mathsf{PI}\left(\right)}}{r \cdot \mathsf{fma}\left(r, \color{blue}{\mathsf{fma}\left(r, \frac{4}{81} \cdot \frac{r}{{s}^{2}} + \frac{2}{3} \cdot \frac{1}{s}, \frac{8}{3}\right)}, 4 \cdot s\right)} \]

       6. *-commutative N/A

          \[\leadsto \frac{\frac{1}{\mathsf{PI}\left(\right)}}{r \cdot \mathsf{fma}\left(r, \mathsf{fma}\left(r, \color{blue}{\frac{r}{{s}^{2}} \cdot \frac{4}{81}} + \frac{2}{3} \cdot \frac{1}{s}, \frac{8}{3}\right), 4 \cdot s\right)} \]

       7. lower-fma.f32 N/A

          \[\leadsto \frac{\frac{1}{\mathsf{PI}\left(\right)}}{r \cdot \mathsf{fma}\left(r, \mathsf{fma}\left(r, \color{blue}{\mathsf{fma}\left(\frac{r}{{s}^{2}}, \frac{4}{81}, \frac{2}{3} \cdot \frac{1}{s}\right)}, \frac{8}{3}\right), 4 \cdot s\right)} \]

       8. lower-/.f32 N/A

          \[\leadsto \frac{\frac{1}{\mathsf{PI}\left(\right)}}{r \cdot \mathsf{fma}\left(r, \mathsf{fma}\left(r, \mathsf{fma}\left(\color{blue}{\frac{r}{{s}^{2}}}, \frac{4}{81}, \frac{2}{3} \cdot \frac{1}{s}\right), \frac{8}{3}\right), 4 \cdot s\right)} \]

       9. unpow2 N/A

          \[\leadsto \frac{\frac{1}{\mathsf{PI}\left(\right)}}{r \cdot \mathsf{fma}\left(r, \mathsf{fma}\left(r, \mathsf{fma}\left(\frac{r}{\color{blue}{s \cdot s}}, \frac{4}{81}, \frac{2}{3} \cdot \frac{1}{s}\right), \frac{8}{3}\right), 4 \cdot s\right)} \]

       10. lower-*.f32 N/A

           \[\leadsto \frac{\frac{1}{\mathsf{PI}\left(\right)}}{r \cdot \mathsf{fma}\left(r, \mathsf{fma}\left(r, \mathsf{fma}\left(\frac{r}{\color{blue}{s \cdot s}}, \frac{4}{81}, \frac{2}{3} \cdot \frac{1}{s}\right), \frac{8}{3}\right), 4 \cdot s\right)} \]

       11. associate-*r/ N/A

           \[\leadsto \frac{\frac{1}{\mathsf{PI}\left(\right)}}{r \cdot \mathsf{fma}\left(r, \mathsf{fma}\left(r, \mathsf{fma}\left(\frac{r}{s \cdot s}, \frac{4}{81}, \color{blue}{\frac{\frac{2}{3} \cdot 1}{s}}\right), \frac{8}{3}\right), 4 \cdot s\right)} \]

       12. metadata-eval N/A

           \[\leadsto \frac{\frac{1}{\mathsf{PI}\left(\right)}}{r \cdot \mathsf{fma}\left(r, \mathsf{fma}\left(r, \mathsf{fma}\left(\frac{r}{s \cdot s}, \frac{4}{81}, \frac{\color{blue}{\frac{2}{3}}}{s}\right), \frac{8}{3}\right), 4 \cdot s\right)} \]

       13. lower-/.f32 N/A

           \[\leadsto \frac{\frac{1}{\mathsf{PI}\left(\right)}}{r \cdot \mathsf{fma}\left(r, \mathsf{fma}\left(r, \mathsf{fma}\left(\frac{r}{s \cdot s}, \frac{4}{81}, \color{blue}{\frac{\frac{2}{3}}{s}}\right), \frac{8}{3}\right), 4 \cdot s\right)} \]

       14. *-commutative N/A

           \[\leadsto \frac{\frac{1}{\mathsf{PI}\left(\right)}}{r \cdot \mathsf{fma}\left(r, \mathsf{fma}\left(r, \mathsf{fma}\left(\frac{r}{s \cdot s}, \frac{4}{81}, \frac{\frac{2}{3}}{s}\right), \frac{8}{3}\right), \color{blue}{s \cdot 4}\right)} \]

       15. lower-*.f32 99.2

           \[\leadsto \frac{\frac{1}{\pi}}{r \cdot \mathsf{fma}\left(r, \mathsf{fma}\left(r, \mathsf{fma}\left(\frac{r}{s \cdot s}, 0.04938271604938271, \frac{0.6666666666666666}{s}\right), 2.6666666666666665\right), \color{blue}{s \cdot 4}\right)} \]

   11. Applied rewrites 99.2%

       \[\leadsto \frac{\frac{1}{\pi}}{\color{blue}{r \cdot \mathsf{fma}\left(r, \mathsf{fma}\left(r, \mathsf{fma}\left(\frac{r}{s \cdot s}, 0.04938271604938271, \frac{0.6666666666666666}{s}\right), 2.6666666666666665\right), s \cdot 4\right)}} \]

   if 1.79999994e-20 < s

   1. Initial program 99.3%

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

   4. Applied rewrites 99.3%

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

      1. *-commutative N/A

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

      2. metadata-eval N/A

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

      3. distribute-lft-neg-in N/A

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

      4. distribute-frac-neg2 N/A

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

      5. neg-mul-1 N/A

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

      6. exp-prod N/A

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

      7. lower-pow.f32 N/A

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

      8. lower-exp.f32 N/A

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

      9. div-inv N/A

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

      10. lower-*.f32 N/A

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

      11. associate-/r* N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f32 99.3

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

   6. Applied rewrites 99.3%

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

   8. Applied rewrites 12.4%

      \[\leadsto \color{blue}{\frac{\frac{1}{\pi}}{\frac{s}{0.125 \cdot \frac{2}{r}}}} \]

   9. Taylor expanded in s around -inf

      \[\leadsto \frac{\frac{1}{\mathsf{PI}\left(\right)}}{\color{blue}{-1 \cdot \left(s \cdot \left(-4 \cdot r + -1 \cdot \frac{-1 \cdot \frac{\left(-2 \cdot \frac{{r}^{2} \cdot \left(\frac{1}{162} \cdot {r}^{2} + \frac{1}{6} \cdot {r}^{2}\right)}{s} + \left(\frac{-2}{3} \cdot \frac{r \cdot \left(-2 \cdot \left({r}^{2} \cdot \left(\frac{1}{18} \cdot r + \frac{1}{2} \cdot r\right)\right) + \frac{16}{9} \cdot {r}^{3}\right)}{s} + \frac{4}{3} \cdot \frac{{r}^{3} \cdot \left(\frac{1}{18} \cdot r + \frac{1}{2} \cdot r\right)}{s}\right)\right) - \left(-2 \cdot \left({r}^{2} \cdot \left(\frac{1}{18} \cdot r + \frac{1}{2} \cdot r\right)\right) + \frac{16}{9} \cdot {r}^{3}\right)}{s} - \frac{-8}{3} \cdot {r}^{2}}{s}\right)\right)}} \]

   11. Applied rewrites 53.1%

       \[\leadsto \frac{\frac{1}{\pi}}{\color{blue}{-s \cdot \left(r \cdot -4 - \frac{\mathsf{fma}\left(r, r \cdot 2.6666666666666665, \frac{\mathsf{fma}\left(-0.6666666666666666, \mathsf{fma}\left(r, \left(r \cdot r\right) \cdot 1.7777777777777777, \left(\left(r \cdot \left(r \cdot r\right)\right) \cdot 0.5555555555555556\right) \cdot -2\right) \cdot \frac{r}{s}, \mathsf{fma}\left(1.3333333333333333, \frac{r \cdot \left(\left(r \cdot \left(r \cdot r\right)\right) \cdot 0.5555555555555556\right)}{s}, \left(-2 \cdot \left(r \cdot r\right)\right) \cdot \frac{\left(r \cdot r\right) \cdot 0.1728395061728395}{s}\right)\right) - \mathsf{fma}\left(r, \left(r \cdot r\right) \cdot 1.7777777777777777, \left(\left(r \cdot \left(r \cdot r\right)\right) \cdot 0.5555555555555556\right) \cdot -2\right)}{-s}\right)}{s}\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \leq 1.7999999428779406 \cdot 10^{-20}:\\ \;\;\;\;\frac{\frac{1}{\pi}}{r \cdot \mathsf{fma}\left(r, \mathsf{fma}\left(r, \mathsf{fma}\left(\frac{r}{s \cdot s}, 0.04938271604938271, \frac{0.6666666666666666}{s}\right), 2.6666666666666665\right), s \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\pi}}{s \cdot \left(\frac{\mathsf{fma}\left(r, r \cdot 2.6666666666666665, \frac{\mathsf{fma}\left(r, \left(r \cdot r\right) \cdot 1.7777777777777777, \left(\left(r \cdot \left(r \cdot r\right)\right) \cdot 0.5555555555555556\right) \cdot -2\right) - \mathsf{fma}\left(-0.6666666666666666, \frac{r}{s} \cdot \mathsf{fma}\left(r, \left(r \cdot r\right) \cdot 1.7777777777777777, \left(\left(r \cdot \left(r \cdot r\right)\right) \cdot 0.5555555555555556\right) \cdot -2\right), \mathsf{fma}\left(1.3333333333333333, \frac{r \cdot \left(\left(r \cdot \left(r \cdot r\right)\right) \cdot 0.5555555555555556\right)}{s}, \left(\left(r \cdot r\right) \cdot -2\right) \cdot \frac{\left(r \cdot r\right) \cdot 0.1728395061728395}{s}\right)\right)}{s}\right)}{s} - r \cdot -4\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 64.1% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{\pi}}{r \cdot \mathsf{fma}\left(r, \mathsf{fma}\left(r, \mathsf{fma}\left(\frac{r}{s \cdot s}, 0.04938271604938271, \frac{0.6666666666666666}{s}\right), 2.6666666666666665\right), s \cdot 4\right)} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (/
  (/ 1.0 PI)
  (*
   r
   (fma
    r
    (fma
     r
     (fma (/ r (* s s)) 0.04938271604938271 (/ 0.6666666666666666 s))
     2.6666666666666665)
    (* s 4.0)))))

C

float code(float s, float r) {
	return (1.0f / ((float) M_PI)) / (r * fmaf(r, fmaf(r, fmaf((r / (s * s)), 0.04938271604938271f, (0.6666666666666666f / s)), 2.6666666666666665f), (s * 4.0f)));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(1.0) / Float32(pi)) / Float32(r * fma(r, fma(r, fma(Float32(r / Float32(s * s)), Float32(0.04938271604938271), Float32(Float32(0.6666666666666666) / s)), Float32(2.6666666666666665)), Float32(s * Float32(4.0)))))
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

4. Applied rewrites 99.6%

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

   1. *-commutative N/A

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

   2. metadata-eval N/A

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

   3. distribute-lft-neg-in N/A

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

   4. distribute-frac-neg2 N/A

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

   5. neg-mul-1 N/A

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

   6. exp-prod N/A

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

   7. lower-pow.f32 N/A

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

   8. lower-exp.f32 N/A

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

   9. div-inv N/A

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

   10. lower-*.f32 N/A

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

   11. associate-/r* N/A

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

   12. metadata-eval N/A

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

   13. lower-/.f32 99.6

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

6. Applied rewrites 99.6%

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

8. Applied rewrites 8.6%

   \[\leadsto \color{blue}{\frac{\frac{1}{\pi}}{\frac{s}{0.125 \cdot \frac{2}{r}}}} \]

9. Taylor expanded in r around 0

   \[\leadsto \frac{\frac{1}{\mathsf{PI}\left(\right)}}{\color{blue}{r \cdot \left(4 \cdot s + r \cdot \left(\frac{8}{3} + r \cdot \left(\frac{4}{81} \cdot \frac{r}{{s}^{2}} + \frac{2}{3} \cdot \frac{1}{s}\right)\right)\right)}} \]
10. Step-by-step derivation

    1. lower-*.f32 N/A

       \[\leadsto \frac{\frac{1}{\mathsf{PI}\left(\right)}}{\color{blue}{r \cdot \left(4 \cdot s + r \cdot \left(\frac{8}{3} + r \cdot \left(\frac{4}{81} \cdot \frac{r}{{s}^{2}} + \frac{2}{3} \cdot \frac{1}{s}\right)\right)\right)}} \]

    2. +-commutative N/A

       \[\leadsto \frac{\frac{1}{\mathsf{PI}\left(\right)}}{r \cdot \color{blue}{\left(r \cdot \left(\frac{8}{3} + r \cdot \left(\frac{4}{81} \cdot \frac{r}{{s}^{2}} + \frac{2}{3} \cdot \frac{1}{s}\right)\right) + 4 \cdot s\right)}} \]

    3. lower-fma.f32 N/A

       \[\leadsto \frac{\frac{1}{\mathsf{PI}\left(\right)}}{r \cdot \color{blue}{\mathsf{fma}\left(r, \frac{8}{3} + r \cdot \left(\frac{4}{81} \cdot \frac{r}{{s}^{2}} + \frac{2}{3} \cdot \frac{1}{s}\right), 4 \cdot s\right)}} \]

    4. +-commutative N/A

       \[\leadsto \frac{\frac{1}{\mathsf{PI}\left(\right)}}{r \cdot \mathsf{fma}\left(r, \color{blue}{r \cdot \left(\frac{4}{81} \cdot \frac{r}{{s}^{2}} + \frac{2}{3} \cdot \frac{1}{s}\right) + \frac{8}{3}}, 4 \cdot s\right)} \]

    5. lower-fma.f32 N/A

       \[\leadsto \frac{\frac{1}{\mathsf{PI}\left(\right)}}{r \cdot \mathsf{fma}\left(r, \color{blue}{\mathsf{fma}\left(r, \frac{4}{81} \cdot \frac{r}{{s}^{2}} + \frac{2}{3} \cdot \frac{1}{s}, \frac{8}{3}\right)}, 4 \cdot s\right)} \]

    6. *-commutative N/A

       \[\leadsto \frac{\frac{1}{\mathsf{PI}\left(\right)}}{r \cdot \mathsf{fma}\left(r, \mathsf{fma}\left(r, \color{blue}{\frac{r}{{s}^{2}} \cdot \frac{4}{81}} + \frac{2}{3} \cdot \frac{1}{s}, \frac{8}{3}\right), 4 \cdot s\right)} \]

    7. lower-fma.f32 N/A

       \[\leadsto \frac{\frac{1}{\mathsf{PI}\left(\right)}}{r \cdot \mathsf{fma}\left(r, \mathsf{fma}\left(r, \color{blue}{\mathsf{fma}\left(\frac{r}{{s}^{2}}, \frac{4}{81}, \frac{2}{3} \cdot \frac{1}{s}\right)}, \frac{8}{3}\right), 4 \cdot s\right)} \]

    8. lower-/.f32 N/A

       \[\leadsto \frac{\frac{1}{\mathsf{PI}\left(\right)}}{r \cdot \mathsf{fma}\left(r, \mathsf{fma}\left(r, \mathsf{fma}\left(\color{blue}{\frac{r}{{s}^{2}}}, \frac{4}{81}, \frac{2}{3} \cdot \frac{1}{s}\right), \frac{8}{3}\right), 4 \cdot s\right)} \]

    9. unpow2 N/A

       \[\leadsto \frac{\frac{1}{\mathsf{PI}\left(\right)}}{r \cdot \mathsf{fma}\left(r, \mathsf{fma}\left(r, \mathsf{fma}\left(\frac{r}{\color{blue}{s \cdot s}}, \frac{4}{81}, \frac{2}{3} \cdot \frac{1}{s}\right), \frac{8}{3}\right), 4 \cdot s\right)} \]

    10. lower-*.f32 N/A

        \[\leadsto \frac{\frac{1}{\mathsf{PI}\left(\right)}}{r \cdot \mathsf{fma}\left(r, \mathsf{fma}\left(r, \mathsf{fma}\left(\frac{r}{\color{blue}{s \cdot s}}, \frac{4}{81}, \frac{2}{3} \cdot \frac{1}{s}\right), \frac{8}{3}\right), 4 \cdot s\right)} \]

    11. associate-*r/ N/A

        \[\leadsto \frac{\frac{1}{\mathsf{PI}\left(\right)}}{r \cdot \mathsf{fma}\left(r, \mathsf{fma}\left(r, \mathsf{fma}\left(\frac{r}{s \cdot s}, \frac{4}{81}, \color{blue}{\frac{\frac{2}{3} \cdot 1}{s}}\right), \frac{8}{3}\right), 4 \cdot s\right)} \]

    12. metadata-eval N/A

        \[\leadsto \frac{\frac{1}{\mathsf{PI}\left(\right)}}{r \cdot \mathsf{fma}\left(r, \mathsf{fma}\left(r, \mathsf{fma}\left(\frac{r}{s \cdot s}, \frac{4}{81}, \frac{\color{blue}{\frac{2}{3}}}{s}\right), \frac{8}{3}\right), 4 \cdot s\right)} \]

    13. lower-/.f32 N/A

        \[\leadsto \frac{\frac{1}{\mathsf{PI}\left(\right)}}{r \cdot \mathsf{fma}\left(r, \mathsf{fma}\left(r, \mathsf{fma}\left(\frac{r}{s \cdot s}, \frac{4}{81}, \color{blue}{\frac{\frac{2}{3}}{s}}\right), \frac{8}{3}\right), 4 \cdot s\right)} \]

    14. *-commutative N/A

        \[\leadsto \frac{\frac{1}{\mathsf{PI}\left(\right)}}{r \cdot \mathsf{fma}\left(r, \mathsf{fma}\left(r, \mathsf{fma}\left(\frac{r}{s \cdot s}, \frac{4}{81}, \frac{\frac{2}{3}}{s}\right), \frac{8}{3}\right), \color{blue}{s \cdot 4}\right)} \]

    15. lower-*.f32 66.9

        \[\leadsto \frac{\frac{1}{\pi}}{r \cdot \mathsf{fma}\left(r, \mathsf{fma}\left(r, \mathsf{fma}\left(\frac{r}{s \cdot s}, 0.04938271604938271, \frac{0.6666666666666666}{s}\right), 2.6666666666666665\right), \color{blue}{s \cdot 4}\right)} \]

11. Applied rewrites 66.9%

    \[\leadsto \frac{\frac{1}{\pi}}{\color{blue}{r \cdot \mathsf{fma}\left(r, \mathsf{fma}\left(r, \mathsf{fma}\left(\frac{r}{s \cdot s}, 0.04938271604938271, \frac{0.6666666666666666}{s}\right), 2.6666666666666665\right), s \cdot 4\right)}} \]
12. Add Preprocessing

Alternative 7: 26.9% accurate, 5.3× speedup

Math

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

FPCore

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

C

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

Julia

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

4. Applied rewrites 99.6%

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

   1. *-commutative N/A

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

   2. metadata-eval N/A

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

   3. distribute-lft-neg-in N/A

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

   4. distribute-frac-neg2 N/A

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

   5. neg-mul-1 N/A

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

   6. exp-prod N/A

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

   7. lower-pow.f32 N/A

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

   8. lower-exp.f32 N/A

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

   9. div-inv N/A

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

   10. lower-*.f32 N/A

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

   11. associate-/r* N/A

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

   12. metadata-eval N/A

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

   13. lower-/.f32 99.6

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

6. Applied rewrites 99.6%

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

8. Applied rewrites 8.6%

   \[\leadsto \color{blue}{\frac{\frac{1}{\pi}}{\frac{s}{0.125 \cdot \frac{2}{r}}}} \]

9. Taylor expanded in r around 0

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

    1. lower-*.f32 N/A

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

    2. +-commutative N/A

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

    3. lower-fma.f32 N/A

       \[\leadsto \frac{\frac{1}{\mathsf{PI}\left(\right)}}{r \cdot \color{blue}{\mathsf{fma}\left(r, \frac{8}{3} + \frac{2}{3} \cdot \frac{r}{s}, 4 \cdot s\right)}} \]

    4. +-commutative N/A

       \[\leadsto \frac{\frac{1}{\mathsf{PI}\left(\right)}}{r \cdot \mathsf{fma}\left(r, \color{blue}{\frac{2}{3} \cdot \frac{r}{s} + \frac{8}{3}}, 4 \cdot s\right)} \]

    5. *-commutative N/A

       \[\leadsto \frac{\frac{1}{\mathsf{PI}\left(\right)}}{r \cdot \mathsf{fma}\left(r, \color{blue}{\frac{r}{s} \cdot \frac{2}{3}} + \frac{8}{3}, 4 \cdot s\right)} \]

    6. associate-*l/ N/A

       \[\leadsto \frac{\frac{1}{\mathsf{PI}\left(\right)}}{r \cdot \mathsf{fma}\left(r, \color{blue}{\frac{r \cdot \frac{2}{3}}{s}} + \frac{8}{3}, 4 \cdot s\right)} \]

    7. associate-*r/ N/A

       \[\leadsto \frac{\frac{1}{\mathsf{PI}\left(\right)}}{r \cdot \mathsf{fma}\left(r, \color{blue}{r \cdot \frac{\frac{2}{3}}{s}} + \frac{8}{3}, 4 \cdot s\right)} \]

    8. metadata-eval N/A

       \[\leadsto \frac{\frac{1}{\mathsf{PI}\left(\right)}}{r \cdot \mathsf{fma}\left(r, r \cdot \frac{\color{blue}{\frac{2}{3} \cdot 1}}{s} + \frac{8}{3}, 4 \cdot s\right)} \]

    9. associate-*r/ N/A

       \[\leadsto \frac{\frac{1}{\mathsf{PI}\left(\right)}}{r \cdot \mathsf{fma}\left(r, r \cdot \color{blue}{\left(\frac{2}{3} \cdot \frac{1}{s}\right)} + \frac{8}{3}, 4 \cdot s\right)} \]

    10. lower-fma.f32 N/A

        \[\leadsto \frac{\frac{1}{\mathsf{PI}\left(\right)}}{r \cdot \mathsf{fma}\left(r, \color{blue}{\mathsf{fma}\left(r, \frac{2}{3} \cdot \frac{1}{s}, \frac{8}{3}\right)}, 4 \cdot s\right)} \]

    11. associate-*r/ N/A

        \[\leadsto \frac{\frac{1}{\mathsf{PI}\left(\right)}}{r \cdot \mathsf{fma}\left(r, \mathsf{fma}\left(r, \color{blue}{\frac{\frac{2}{3} \cdot 1}{s}}, \frac{8}{3}\right), 4 \cdot s\right)} \]

    12. metadata-eval N/A

        \[\leadsto \frac{\frac{1}{\mathsf{PI}\left(\right)}}{r \cdot \mathsf{fma}\left(r, \mathsf{fma}\left(r, \frac{\color{blue}{\frac{2}{3}}}{s}, \frac{8}{3}\right), 4 \cdot s\right)} \]

    13. lower-/.f32 N/A

        \[\leadsto \frac{\frac{1}{\mathsf{PI}\left(\right)}}{r \cdot \mathsf{fma}\left(r, \mathsf{fma}\left(r, \color{blue}{\frac{\frac{2}{3}}{s}}, \frac{8}{3}\right), 4 \cdot s\right)} \]

    14. *-commutative N/A

        \[\leadsto \frac{\frac{1}{\mathsf{PI}\left(\right)}}{r \cdot \mathsf{fma}\left(r, \mathsf{fma}\left(r, \frac{\frac{2}{3}}{s}, \frac{8}{3}\right), \color{blue}{s \cdot 4}\right)} \]

    15. lower-*.f32 24.4

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

11. Applied rewrites 24.4%

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

Alternative 8: 19.7% accurate, 5.4× speedup

Math

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

FPCore

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

C

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

Julia

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

4. Applied rewrites 99.6%

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

   1. *-commutative N/A

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

   2. metadata-eval N/A

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

   3. distribute-lft-neg-in N/A

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

   4. distribute-frac-neg2 N/A

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

   5. neg-mul-1 N/A

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

   6. exp-prod N/A

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

   7. lower-pow.f32 N/A

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

   8. lower-exp.f32 N/A

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

   9. div-inv N/A

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

   10. lower-*.f32 N/A

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

   11. associate-/r* N/A

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

   12. metadata-eval N/A

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

   13. lower-/.f32 99.6

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

6. Applied rewrites 99.6%

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

8. Applied rewrites 8.6%

   \[\leadsto \color{blue}{\frac{\frac{1}{\pi}}{\frac{s}{0.125 \cdot \frac{2}{r}}}} \]

9. Taylor expanded in s around inf

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

    1. lower-*.f32 N/A

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

    2. *-commutative N/A

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

    3. lower-fma.f32 N/A

       \[\leadsto \frac{\frac{1}{\mathsf{PI}\left(\right)}}{s \cdot \color{blue}{\mathsf{fma}\left(\frac{{r}^{2}}{s}, \frac{8}{3}, 4 \cdot r\right)}} \]

    4. lower-/.f32 N/A

       \[\leadsto \frac{\frac{1}{\mathsf{PI}\left(\right)}}{s \cdot \mathsf{fma}\left(\color{blue}{\frac{{r}^{2}}{s}}, \frac{8}{3}, 4 \cdot r\right)} \]

    5. unpow2 N/A

       \[\leadsto \frac{\frac{1}{\mathsf{PI}\left(\right)}}{s \cdot \mathsf{fma}\left(\frac{\color{blue}{r \cdot r}}{s}, \frac{8}{3}, 4 \cdot r\right)} \]

    6. lower-*.f32 N/A

       \[\leadsto \frac{\frac{1}{\mathsf{PI}\left(\right)}}{s \cdot \mathsf{fma}\left(\frac{\color{blue}{r \cdot r}}{s}, \frac{8}{3}, 4 \cdot r\right)} \]

    7. *-commutative N/A

       \[\leadsto \frac{\frac{1}{\mathsf{PI}\left(\right)}}{s \cdot \mathsf{fma}\left(\frac{r \cdot r}{s}, \frac{8}{3}, \color{blue}{r \cdot 4}\right)} \]

    8. lower-*.f32 18.6

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

11. Applied rewrites 18.6%

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

Alternative 9: 12.7% accurate, 7.6× speedup

Math

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

FPCore

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

C

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

Julia

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

4. Applied rewrites 99.6%

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

   1. *-commutative N/A

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

   2. metadata-eval N/A

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

   3. distribute-lft-neg-in N/A

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

   4. distribute-frac-neg2 N/A

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

   5. neg-mul-1 N/A

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

   6. exp-prod N/A

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

   7. lower-pow.f32 N/A

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

   8. lower-exp.f32 N/A

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

   9. div-inv N/A

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

   10. lower-*.f32 N/A

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

   11. associate-/r* N/A

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

   12. metadata-eval N/A

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

   13. lower-/.f32 99.6

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

6. Applied rewrites 99.6%

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

8. Applied rewrites 8.6%

   \[\leadsto \color{blue}{\frac{\frac{1}{\pi}}{\frac{s}{0.125 \cdot \frac{2}{r}}}} \]

9. Taylor expanded in r around 0

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

    1. lower-*.f32 N/A

       \[\leadsto \frac{\frac{1}{\mathsf{PI}\left(\right)}}{\color{blue}{r \cdot \left(\frac{8}{3} \cdot r + 4 \cdot s\right)}} \]

    2. *-commutative N/A

       \[\leadsto \frac{\frac{1}{\mathsf{PI}\left(\right)}}{r \cdot \left(\color{blue}{r \cdot \frac{8}{3}} + 4 \cdot s\right)} \]

    3. lower-fma.f32 N/A

       \[\leadsto \frac{\frac{1}{\mathsf{PI}\left(\right)}}{r \cdot \color{blue}{\mathsf{fma}\left(r, \frac{8}{3}, 4 \cdot s\right)}} \]

    4. *-commutative N/A

       \[\leadsto \frac{\frac{1}{\mathsf{PI}\left(\right)}}{r \cdot \mathsf{fma}\left(r, \frac{8}{3}, \color{blue}{s \cdot 4}\right)} \]

    5. lower-*.f32 11.7

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

11. Applied rewrites 11.7%

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

Alternative 10: 9.3% 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 99.6%

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

3. Taylor expanded in r around 0

   \[\leadsto \color{blue}{\frac{\frac{1}{4}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)}} \]
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 8.6

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

5. Applied rewrites 8.6%

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

6. Final simplification 8.6%

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

Reproduce

herbie shell --seed 2024212 
(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))))