Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.5%

Time: 14.2s

Alternatives: 13

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

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
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 99.7%

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

   1. lift-PI.f32 N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f32 N/A

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

   4. lower-*.f32 99.8

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

9. Applied rewrites 99.8%

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

10. Final simplification 99.8%

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

Alternative 2: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
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 99.7%

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

8. Final simplification 99.7%

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

Alternative 3: 97.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
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 99.7%

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

   1. lift-neg.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. lift-exp.f32 N/A

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

   4. lift-/.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-exp.f32 N/A

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

   7. lift-+.f32 N/A

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

   8. lift-PI.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. *-commutative N/A

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

   11. lift-*.f32 N/A

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

   12. associate-/l* N/A

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

   13. lower-*.f32 N/A

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

9. Applied rewrites 99.0%

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

10. Final simplification 99.0%

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

Alternative 4: 10.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \frac{0.25 \cdot e^{\frac{r}{-s}}}{r \cdot \left(s \cdot \left(\pi \cdot 2\right)\right)} + \frac{\mathsf{fma}\left(r, \frac{\mathsf{fma}\left(r, \frac{0.006944444444444444}{s \cdot \pi}, \frac{-0.041666666666666664}{\pi}\right)}{s \cdot s}, \frac{0.125}{s \cdot \pi}\right)}{r} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/ (* 0.25 (exp (/ r (- s)))) (* r (* s (* PI 2.0))))
  (/
   (fma
    r
    (/
     (fma r (/ 0.006944444444444444 (* s PI)) (/ -0.041666666666666664 PI))
     (* s s))
    (/ 0.125 (* s PI)))
   r)))

C

float code(float s, float r) {
	return ((0.25f * expf((r / -s))) / (r * (s * (((float) M_PI) * 2.0f)))) + (fmaf(r, (fmaf(r, (0.006944444444444444f / (s * ((float) M_PI))), (-0.041666666666666664f / ((float) M_PI))) / (s * s)), (0.125f / (s * ((float) M_PI)))) / r);
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.25) * exp(Float32(r / Float32(-s)))) / Float32(r * Float32(s * Float32(Float32(pi) * Float32(2.0))))) + Float32(fma(r, Float32(fma(r, Float32(Float32(0.006944444444444444) / Float32(s * Float32(pi))), Float32(Float32(-0.041666666666666664) / Float32(pi))) / Float32(s * s)), Float32(Float32(0.125) / Float32(s * Float32(pi)))) / r))
end

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in r around 0

   \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \color{blue}{\frac{r \cdot \left(\frac{1}{144} \cdot \frac{r}{{s}^{3} \cdot \mathsf{PI}\left(\right)} - \frac{1}{24} \cdot \frac{1}{{s}^{2} \cdot \mathsf{PI}\left(\right)}\right) + \frac{1}{8} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}}{r}} \]
4. Step-by-step derivation

   1. lower-/.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \color{blue}{\frac{r \cdot \left(\frac{1}{144} \cdot \frac{r}{{s}^{3} \cdot \mathsf{PI}\left(\right)} - \frac{1}{24} \cdot \frac{1}{{s}^{2} \cdot \mathsf{PI}\left(\right)}\right) + \frac{1}{8} \cdot \frac{1}{s \cdot \mathsf{PI}\left(\right)}}{r}} \]

5. Applied rewrites 8.9%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \color{blue}{\frac{\mathsf{fma}\left(r, \frac{\mathsf{fma}\left(r, \frac{0.006944444444444444}{s \cdot \pi}, \frac{-0.041666666666666664}{\pi}\right)}{s \cdot s}, \frac{0.125}{s \cdot \pi}\right)}{r}} \]

6. Final simplification 8.9%

   \[\leadsto \frac{0.25 \cdot e^{\frac{r}{-s}}}{r \cdot \left(s \cdot \left(\pi \cdot 2\right)\right)} + \frac{\mathsf{fma}\left(r, \frac{\mathsf{fma}\left(r, \frac{0.006944444444444444}{s \cdot \pi}, \frac{-0.041666666666666664}{\pi}\right)}{s \cdot s}, \frac{0.125}{s \cdot \pi}\right)}{r} \]
7. Add Preprocessing

Alternative 5: 10.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \frac{0.25 \cdot e^{\frac{r}{-s}}}{r \cdot \left(s \cdot \left(\pi \cdot 2\right)\right)} + \frac{\mathsf{fma}\left(\frac{r}{\pi \cdot \left(s \cdot s\right)}, 0.006944444444444444, \frac{-0.041666666666666664}{s \cdot \pi}\right) + \frac{0.125}{r \cdot \pi}}{s} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/ (* 0.25 (exp (/ r (- s)))) (* r (* s (* PI 2.0))))
  (/
   (+
    (fma
     (/ r (* PI (* s s)))
     0.006944444444444444
     (/ -0.041666666666666664 (* s PI)))
    (/ 0.125 (* r PI)))
   s)))

C

float code(float s, float r) {
	return ((0.25f * expf((r / -s))) / (r * (s * (((float) M_PI) * 2.0f)))) + ((fmaf((r / (((float) M_PI) * (s * s))), 0.006944444444444444f, (-0.041666666666666664f / (s * ((float) M_PI)))) + (0.125f / (r * ((float) M_PI)))) / s);
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.25) * exp(Float32(r / Float32(-s)))) / Float32(r * Float32(s * Float32(Float32(pi) * Float32(2.0))))) + Float32(Float32(fma(Float32(r / Float32(Float32(pi) * Float32(s * s))), Float32(0.006944444444444444), Float32(Float32(-0.041666666666666664) / Float32(s * Float32(pi)))) + Float32(Float32(0.125) / Float32(r * Float32(pi)))) / s))
end

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in r around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower-/.f32 99.8

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

5. Applied rewrites 99.8%

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

6. Taylor expanded in s around -inf

   \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \color{blue}{-1 \cdot \frac{-1 \cdot \frac{\frac{1}{144} \cdot \frac{r}{s \cdot \mathsf{PI}\left(\right)} - \frac{1}{24} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{s} - \frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s}} \]
7. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \color{blue}{\frac{-1 \cdot \left(-1 \cdot \frac{\frac{1}{144} \cdot \frac{r}{s \cdot \mathsf{PI}\left(\right)} - \frac{1}{24} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{s} - \frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}\right)}{s}} \]

   2. mul-1-neg N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\color{blue}{\mathsf{neg}\left(\left(-1 \cdot \frac{\frac{1}{144} \cdot \frac{r}{s \cdot \mathsf{PI}\left(\right)} - \frac{1}{24} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{s} - \frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}\right)\right)}}{s} \]

   3. sub-neg N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\mathsf{neg}\left(\color{blue}{\left(-1 \cdot \frac{\frac{1}{144} \cdot \frac{r}{s \cdot \mathsf{PI}\left(\right)} - \frac{1}{24} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{s} + \left(\mathsf{neg}\left(\frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}\right)\right)\right)}\right)}{s} \]

   4. mul-1-neg N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\mathsf{neg}\left(\left(\color{blue}{\left(\mathsf{neg}\left(\frac{\frac{1}{144} \cdot \frac{r}{s \cdot \mathsf{PI}\left(\right)} - \frac{1}{24} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{s}\right)\right)} + \left(\mathsf{neg}\left(\frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}\right)\right)\right)\right)}{s} \]

   5. distribute-neg-out N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\left(\frac{\frac{1}{144} \cdot \frac{r}{s \cdot \mathsf{PI}\left(\right)} - \frac{1}{24} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{s} + \frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}\right)\right)\right)}\right)}{s} \]

   6. remove-double-neg N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \frac{\color{blue}{\frac{\frac{1}{144} \cdot \frac{r}{s \cdot \mathsf{PI}\left(\right)} - \frac{1}{24} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{s} + \frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}}{s} \]

   7. lower-/.f32 N/A

      \[\leadsto \frac{\frac{1}{4} \cdot e^{\frac{\mathsf{neg}\left(r\right)}{s}}}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} + \color{blue}{\frac{\frac{\frac{1}{144} \cdot \frac{r}{s \cdot \mathsf{PI}\left(\right)} - \frac{1}{24} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{s} + \frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}}{s}} \]

8. Applied rewrites 8.8%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \color{blue}{\frac{\mathsf{fma}\left(\frac{r}{\pi \cdot \left(s \cdot s\right)}, 0.006944444444444444, \frac{-0.041666666666666664}{s \cdot \pi}\right) + \frac{0.125}{r \cdot \pi}}{s}} \]

9. Final simplification 8.8%

   \[\leadsto \frac{0.25 \cdot e^{\frac{r}{-s}}}{r \cdot \left(s \cdot \left(\pi \cdot 2\right)\right)} + \frac{\mathsf{fma}\left(\frac{r}{\pi \cdot \left(s \cdot s\right)}, 0.006944444444444444, \frac{-0.041666666666666664}{s \cdot \pi}\right) + \frac{0.125}{r \cdot \pi}}{s} \]
10. Add Preprocessing

Alternative 6: 10.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \frac{0.125 \cdot e^{\frac{r}{-s}}}{r \cdot \left(s \cdot \pi\right)} + \frac{\frac{0.125}{r \cdot \pi} + \mathsf{fma}\left(\frac{r}{s \cdot \left(s \cdot \pi\right)}, 0.006944444444444444, \frac{-0.041666666666666664}{s \cdot \pi}\right)}{s} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/ (* 0.125 (exp (/ r (- s)))) (* r (* s PI)))
  (/
   (+
    (/ 0.125 (* r PI))
    (fma
     (/ r (* s (* s PI)))
     0.006944444444444444
     (/ -0.041666666666666664 (* s PI))))
   s)))

C

float code(float s, float r) {
	return ((0.125f * expf((r / -s))) / (r * (s * ((float) M_PI)))) + (((0.125f / (r * ((float) M_PI))) + fmaf((r / (s * (s * ((float) M_PI)))), 0.006944444444444444f, (-0.041666666666666664f / (s * ((float) M_PI))))) / s);
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.125) * exp(Float32(r / Float32(-s)))) / Float32(r * Float32(s * Float32(pi)))) + Float32(Float32(Float32(Float32(0.125) / Float32(r * Float32(pi))) + fma(Float32(r / Float32(s * Float32(s * Float32(pi)))), Float32(0.006944444444444444), Float32(Float32(-0.041666666666666664) / Float32(s * Float32(pi))))) / s))
end

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in r around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower-/.f32 99.8

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

5. Applied rewrites 99.8%

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

   1. distribute-frac-neg N/A

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

   2. distribute-frac-neg2 N/A

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

   3. lift-neg.f32 N/A

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

   4. lift-/.f32 N/A

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

   5. lift-exp.f32 N/A

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

   6. *-commutative N/A

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

   7. lift-PI.f32 N/A

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

   8. lift-*.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. *-commutative N/A

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

   11. times-frac N/A

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

   12. lift-*.f32 N/A

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

   13. lift-*.f32 N/A

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

   14. associate-*l* N/A

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

   15. *-commutative N/A

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

   16. lift-*.f32 N/A

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

   17. associate-/r* N/A

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

   18. metadata-eval N/A

       \[\leadsto \frac{e^{\frac{r}{\mathsf{neg}\left(s\right)}}}{r} \cdot \frac{\color{blue}{\frac{1}{8}}}{s \cdot \mathsf{PI}\left(\right)} + \frac{\frac{3}{4} \cdot e^{\frac{r}{s} \cdot \frac{-1}{3}}}{\left(\left(6 \cdot \mathsf{PI}\left(\right)\right) \cdot s\right) \cdot r} \]

7. Applied rewrites 99.8%

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

8. Taylor expanded in s around inf

   \[\leadsto \frac{e^{\mathsf{neg}\left(\frac{r}{s}\right)} \cdot \frac{1}{8}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} + \color{blue}{\frac{\left(\frac{1}{144} \cdot \frac{r}{{s}^{2} \cdot \mathsf{PI}\left(\right)} + \frac{1}{8} \cdot \frac{1}{r \cdot \mathsf{PI}\left(\right)}\right) - \frac{\frac{1}{24}}{s \cdot \mathsf{PI}\left(\right)}}{s}} \]

10. Applied rewrites 8.8%

    \[\leadsto \frac{e^{-\frac{r}{s}} \cdot 0.125}{r \cdot \left(s \cdot \pi\right)} + \color{blue}{\frac{\frac{0.125}{r \cdot \pi} + \mathsf{fma}\left(\frac{r}{s \cdot \left(s \cdot \pi\right)}, 0.006944444444444444, \frac{-0.041666666666666664}{s \cdot \pi}\right)}{s}} \]

11. Final simplification 8.8%

    \[\leadsto \frac{0.125 \cdot e^{\frac{r}{-s}}}{r \cdot \left(s \cdot \pi\right)} + \frac{\frac{0.125}{r \cdot \pi} + \mathsf{fma}\left(\frac{r}{s \cdot \left(s \cdot \pi\right)}, 0.006944444444444444, \frac{-0.041666666666666664}{s \cdot \pi}\right)}{s} \]
12. Add Preprocessing

Alternative 7: 10.0% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \frac{0.125 \cdot \left(2 - \frac{\mathsf{fma}\left(r, 1.3333333333333333, \frac{\left(r \cdot r\right) \cdot -0.5555555555555556}{s}\right)}{s}\right)}{r \cdot \left(s \cdot \pi\right)} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (/
  (*
   0.125
   (-
    2.0
    (/ (fma r 1.3333333333333333 (/ (* (* r r) -0.5555555555555556) s)) s)))
  (* r (* s PI))))

C

float code(float s, float r) {
	return (0.125f * (2.0f - (fmaf(r, 1.3333333333333333f, (((r * r) * -0.5555555555555556f) / s)) / s))) / (r * (s * ((float) M_PI)));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(0.125) * Float32(Float32(2.0) - Float32(fma(r, Float32(1.3333333333333333), Float32(Float32(Float32(r * r) * Float32(-0.5555555555555556)) / s)) / s))) / Float32(r * Float32(s * Float32(pi))))
end

Derivation

1. Initial program 99.7%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
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 99.7%

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

8. Taylor expanded in s around -inf

   \[\leadsto \frac{\frac{1}{8} \cdot \color{blue}{\left(2 + -1 \cdot \frac{r + \left(-1 \cdot \frac{\frac{1}{18} \cdot {r}^{2} + \frac{1}{2} \cdot {r}^{2}}{s} + \frac{1}{3} \cdot r\right)}{s}\right)}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} \]
9. Step-by-step derivation

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f32 N/A

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

   4. lower-/.f32 N/A

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

10. Applied rewrites 8.4%

    \[\leadsto \frac{0.125 \cdot \color{blue}{\left(2 - \frac{\mathsf{fma}\left(r, 1.3333333333333333, \frac{\left(r \cdot r\right) \cdot -0.5555555555555556}{s}\right)}{s}\right)}}{r \cdot \left(s \cdot \pi\right)} \]
11. Add Preprocessing

Alternative 8: 10.0% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(r, \mathsf{fma}\left(r, \frac{0.06944444444444445}{s \cdot s}, \frac{-0.16666666666666666}{s}\right), 0.25\right)}{r \cdot \left(s \cdot \pi\right)} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (/
  (fma
   r
   (fma r (/ 0.06944444444444445 (* s s)) (/ -0.16666666666666666 s))
   0.25)
  (* r (* s PI))))

C

float code(float s, float r) {
	return fmaf(r, fmaf(r, (0.06944444444444445f / (s * s)), (-0.16666666666666666f / s)), 0.25f) / (r * (s * ((float) M_PI)));
}

Julia

function code(s, r)
	return Float32(fma(r, fma(r, Float32(Float32(0.06944444444444445) / Float32(s * s)), Float32(Float32(-0.16666666666666666) / s)), Float32(0.25)) / Float32(r * Float32(s * Float32(pi))))
end

Derivation

1. Initial program 99.7%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
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 99.7%

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

8. Taylor expanded in r around 0

   \[\leadsto \frac{\color{blue}{\frac{1}{4} + r \cdot \left(\frac{5}{72} \cdot \frac{r}{{s}^{2}} - \frac{1}{6} \cdot \frac{1}{s}\right)}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} \]
9. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\color{blue}{r \cdot \left(\frac{5}{72} \cdot \frac{r}{{s}^{2}} - \frac{1}{6} \cdot \frac{1}{s}\right) + \frac{1}{4}}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} \]

   2. lower-fma.f32 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(r, \frac{5}{72} \cdot \frac{r}{{s}^{2}} - \frac{1}{6} \cdot \frac{1}{s}, \frac{1}{4}\right)}}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} \]

   3. sub-neg N/A

      \[\leadsto \frac{\mathsf{fma}\left(r, \color{blue}{\frac{5}{72} \cdot \frac{r}{{s}^{2}} + \left(\mathsf{neg}\left(\frac{1}{6} \cdot \frac{1}{s}\right)\right)}, \frac{1}{4}\right)}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} \]

   4. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(r, \color{blue}{\frac{r}{{s}^{2}} \cdot \frac{5}{72}} + \left(\mathsf{neg}\left(\frac{1}{6} \cdot \frac{1}{s}\right)\right), \frac{1}{4}\right)}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} \]

   5. associate-*l/ N/A

      \[\leadsto \frac{\mathsf{fma}\left(r, \color{blue}{\frac{r \cdot \frac{5}{72}}{{s}^{2}}} + \left(\mathsf{neg}\left(\frac{1}{6} \cdot \frac{1}{s}\right)\right), \frac{1}{4}\right)}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} \]

   6. associate-*r/ N/A

      \[\leadsto \frac{\mathsf{fma}\left(r, \color{blue}{r \cdot \frac{\frac{5}{72}}{{s}^{2}}} + \left(\mathsf{neg}\left(\frac{1}{6} \cdot \frac{1}{s}\right)\right), \frac{1}{4}\right)}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} \]

   7. metadata-eval N/A

      \[\leadsto \frac{\mathsf{fma}\left(r, r \cdot \frac{\color{blue}{\frac{5}{72} \cdot 1}}{{s}^{2}} + \left(\mathsf{neg}\left(\frac{1}{6} \cdot \frac{1}{s}\right)\right), \frac{1}{4}\right)}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} \]

   8. associate-*r/ N/A

      \[\leadsto \frac{\mathsf{fma}\left(r, r \cdot \color{blue}{\left(\frac{5}{72} \cdot \frac{1}{{s}^{2}}\right)} + \left(\mathsf{neg}\left(\frac{1}{6} \cdot \frac{1}{s}\right)\right), \frac{1}{4}\right)}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} \]

   9. lower-fma.f32 N/A

      \[\leadsto \frac{\mathsf{fma}\left(r, \color{blue}{\mathsf{fma}\left(r, \frac{5}{72} \cdot \frac{1}{{s}^{2}}, \mathsf{neg}\left(\frac{1}{6} \cdot \frac{1}{s}\right)\right)}, \frac{1}{4}\right)}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} \]

   10. associate-*r/ N/A

       \[\leadsto \frac{\mathsf{fma}\left(r, \mathsf{fma}\left(r, \color{blue}{\frac{\frac{5}{72} \cdot 1}{{s}^{2}}}, \mathsf{neg}\left(\frac{1}{6} \cdot \frac{1}{s}\right)\right), \frac{1}{4}\right)}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} \]

   11. metadata-eval N/A

       \[\leadsto \frac{\mathsf{fma}\left(r, \mathsf{fma}\left(r, \frac{\color{blue}{\frac{5}{72}}}{{s}^{2}}, \mathsf{neg}\left(\frac{1}{6} \cdot \frac{1}{s}\right)\right), \frac{1}{4}\right)}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} \]

   12. lower-/.f32 N/A

       \[\leadsto \frac{\mathsf{fma}\left(r, \mathsf{fma}\left(r, \color{blue}{\frac{\frac{5}{72}}{{s}^{2}}}, \mathsf{neg}\left(\frac{1}{6} \cdot \frac{1}{s}\right)\right), \frac{1}{4}\right)}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} \]

   13. unpow2 N/A

       \[\leadsto \frac{\mathsf{fma}\left(r, \mathsf{fma}\left(r, \frac{\frac{5}{72}}{\color{blue}{s \cdot s}}, \mathsf{neg}\left(\frac{1}{6} \cdot \frac{1}{s}\right)\right), \frac{1}{4}\right)}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} \]

   14. lower-*.f32 N/A

       \[\leadsto \frac{\mathsf{fma}\left(r, \mathsf{fma}\left(r, \frac{\frac{5}{72}}{\color{blue}{s \cdot s}}, \mathsf{neg}\left(\frac{1}{6} \cdot \frac{1}{s}\right)\right), \frac{1}{4}\right)}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} \]

   15. associate-*r/ N/A

       \[\leadsto \frac{\mathsf{fma}\left(r, \mathsf{fma}\left(r, \frac{\frac{5}{72}}{s \cdot s}, \mathsf{neg}\left(\color{blue}{\frac{\frac{1}{6} \cdot 1}{s}}\right)\right), \frac{1}{4}\right)}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} \]

   16. metadata-eval N/A

       \[\leadsto \frac{\mathsf{fma}\left(r, \mathsf{fma}\left(r, \frac{\frac{5}{72}}{s \cdot s}, \mathsf{neg}\left(\frac{\color{blue}{\frac{1}{6}}}{s}\right)\right), \frac{1}{4}\right)}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} \]

   17. distribute-neg-frac N/A

       \[\leadsto \frac{\mathsf{fma}\left(r, \mathsf{fma}\left(r, \frac{\frac{5}{72}}{s \cdot s}, \color{blue}{\frac{\mathsf{neg}\left(\frac{1}{6}\right)}{s}}\right), \frac{1}{4}\right)}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} \]

   18. metadata-eval N/A

       \[\leadsto \frac{\mathsf{fma}\left(r, \mathsf{fma}\left(r, \frac{\frac{5}{72}}{s \cdot s}, \frac{\color{blue}{\frac{-1}{6}}}{s}\right), \frac{1}{4}\right)}{r \cdot \left(s \cdot \mathsf{PI}\left(\right)\right)} \]

   19. lower-/.f32 8.4

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

10. Applied rewrites 8.4%

    \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(r, \mathsf{fma}\left(r, \frac{0.06944444444444445}{s \cdot s}, \frac{-0.16666666666666666}{s}\right), 0.25\right)}}{r \cdot \left(s \cdot \pi\right)} \]
11. Add Preprocessing

Alternative 9: 9.0% accurate, 6.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
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 8.0

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

5. Applied rewrites 8.0%

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

   1. lift-PI.f32 N/A

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

   2. associate-*r* N/A

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

   3. lift-PI.f32 N/A

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

   4. add-sqr-sqrt N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f32 N/A

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

   11. lift-PI.f32 N/A

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

   12. lower-sqrt.f32 N/A

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

   13. lift-PI.f32 N/A

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

   14. lower-sqrt.f32 8.1

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

7. Applied rewrites 8.1%

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

8. Final simplification 8.1%

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

Alternative 10: 9.0% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{\pi}{\frac{0.25}{r \cdot s}}} \end{array} \]

FPCore

(FPCore (s r) :precision binary32 (/ 1.0 (/ PI (/ 0.25 (* r s)))))

C

float code(float s, float r) {
	return 1.0f / (((float) M_PI) / (0.25f / (r * s)));
}

Julia

function code(s, r)
	return Float32(Float32(1.0) / Float32(Float32(pi) / Float32(Float32(0.25) / Float32(r * s))))
end

MATLAB

function tmp = code(s, r)
	tmp = single(1.0) / (single(pi) / (single(0.25) / (r * s)));
end

Derivation

1. Initial program 99.7%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
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 8.0

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

5. Applied rewrites 8.0%

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

   1. lift-PI.f32 N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 8.1

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

7. Applied rewrites 8.1%

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

   1. lift-*.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. associate-/r* N/A

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

   4. clear-num N/A

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

   5. lower-/.f32 N/A

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

   6. lower-/.f32 N/A

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

   7. lower-/.f32 8.1

      \[\leadsto \frac{1}{\frac{\pi}{\color{blue}{\frac{0.25}{r \cdot s}}}} \]

9. Applied rewrites 8.1%

   \[\leadsto \color{blue}{\frac{1}{\frac{\pi}{\frac{0.25}{r \cdot s}}}} \]
10. Add Preprocessing

Alternative 11: 9.0% accurate, 10.6× speedup

Math

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

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

MATLAB

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

Derivation

1. Initial program 99.7%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
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 8.0

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

5. Applied rewrites 8.0%

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

   1. lift-PI.f32 N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 8.1

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

7. Applied rewrites 8.1%

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

   1. lift-*.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-/r* N/A

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

   5. lower-/.f32 N/A

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

   6. lower-/.f32 8.1

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

9. Applied rewrites 8.1%

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

Alternative 12: 9.0% accurate, 13.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
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 8.0

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

5. Applied rewrites 8.0%

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

   1. lift-PI.f32 N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 8.1

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

7. Applied rewrites 8.1%

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

Alternative 13: 9.0% accurate, 13.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
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 8.0

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

5. Applied rewrites 8.0%

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

Reproduce

herbie shell --seed 2024221 
(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))))