Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.6%

Time: 5.8s

Alternatives: 12

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: 99.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.6%

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

3. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{e^{\frac{r \cdot -0.3333333333333333}{s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r}, 0.75, \frac{\frac{0.125}{\left(\pi \cdot s\right) \cdot e^{\frac{r}{s}}}}{r}\right)} \]
4. Step-by-step derivation

   1. lift-/.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-/l/ N/A

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

   4. lower-/.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. associate-*l* N/A

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

   7. lower-*.f32 N/A

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

   8. lower-*.f32 99.6

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

5. Applied rewrites 99.6%

   \[\leadsto \mathsf{fma}\left(\frac{e^{\frac{r \cdot -0.3333333333333333}{s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r}, 0.75, \color{blue}{\frac{0.125}{\left(\pi \cdot s\right) \cdot \left(e^{\frac{r}{s}} \cdot r\right)}}\right) \]
6. Add Preprocessing

Alternative 2: 99.6% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.6%

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

3. Applied rewrites 99.6%

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

Alternative 3: 99.6% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Applied rewrites 99.6%

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

   1. lift-fma.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f32 N/A

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

   4. mult-flip N/A

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

   5. distribute-lft-out N/A

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

   6. lower-*.f32 N/A

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

   7. lift-/.f32 N/A

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

   8. lift-*.f32 N/A

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

   9. *-commutative N/A

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

   10. associate-/r* N/A

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

5. Applied rewrites 99.6%

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

Alternative 4: 43.0% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = single(0.25) / (log(exp((single(pi) * r))) * s);
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. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 9.0

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

4. Applied rewrites 9.0%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f32 N/A

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

   6. lower-*.f32 9.0

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

6. Applied rewrites 9.0%

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

   1. lift-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4}}{\left(r \cdot \pi\right) \cdot s} \]

   2. lift-PI.f32 N/A

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

   3. add-log-exp N/A

      \[\leadsto \frac{\frac{1}{4}}{\left(r \cdot \log \left(e^{\mathsf{PI}\left(\right)}\right)\right) \cdot s} \]

   4. log-pow-rev N/A

      \[\leadsto \frac{\frac{1}{4}}{\log \left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{r}\right) \cdot s} \]

   5. lower-log.f32 N/A

      \[\leadsto \frac{\frac{1}{4}}{\log \left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{r}\right) \cdot s} \]

   6. lift-PI.f32 N/A

      \[\leadsto \frac{\frac{1}{4}}{\log \left({\left(e^{\pi}\right)}^{r}\right) \cdot s} \]

   7. pow-exp N/A

      \[\leadsto \frac{\frac{1}{4}}{\log \left(e^{\pi \cdot r}\right) \cdot s} \]

   8. *-commutative N/A

      \[\leadsto \frac{\frac{1}{4}}{\log \left(e^{r \cdot \pi}\right) \cdot s} \]

   9. lift-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4}}{\log \left(e^{r \cdot \pi}\right) \cdot s} \]

   10. lower-exp.f32 43.0

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

   11. lift-*.f32 N/A

       \[\leadsto \frac{\frac{1}{4}}{\log \left(e^{r \cdot \pi}\right) \cdot s} \]

   12. *-commutative N/A

       \[\leadsto \frac{\frac{1}{4}}{\log \left(e^{\pi \cdot r}\right) \cdot s} \]

   13. lower-*.f32 43.0

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

8. Applied rewrites 43.0%

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

Alternative 5: 10.4% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 9.0

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

4. Applied rewrites 9.0%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

      \[\leadsto \frac{\frac{1}{4}}{\left(s \cdot r\right) \cdot \pi} \]

   5. lift-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4}}{\left(s \cdot r\right) \cdot \pi} \]

   6. lift-PI.f32 N/A

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

   7. add-log-exp N/A

      \[\leadsto \frac{\frac{1}{4}}{\left(s \cdot r\right) \cdot \log \left(e^{\mathsf{PI}\left(\right)}\right)} \]

   8. log-pow-rev N/A

      \[\leadsto \frac{\frac{1}{4}}{\log \left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{\left(s \cdot r\right)}\right)} \]

   9. lower-log.f32 N/A

      \[\leadsto \frac{\frac{1}{4}}{\log \left({\left(e^{\mathsf{PI}\left(\right)}\right)}^{\left(s \cdot r\right)}\right)} \]

   10. lift-PI.f32 N/A

       \[\leadsto \frac{\frac{1}{4}}{\log \left({\left(e^{\pi}\right)}^{\left(s \cdot r\right)}\right)} \]

   11. pow-exp N/A

       \[\leadsto \frac{\frac{1}{4}}{\log \left(e^{\pi \cdot \left(s \cdot r\right)}\right)} \]

   12. lift-*.f32 N/A

       \[\leadsto \frac{\frac{1}{4}}{\log \left(e^{\pi \cdot \left(s \cdot r\right)}\right)} \]

   13. associate-*l* N/A

       \[\leadsto \frac{\frac{1}{4}}{\log \left(e^{\left(\pi \cdot s\right) \cdot r}\right)} \]

   14. *-commutative N/A

       \[\leadsto \frac{\frac{1}{4}}{\log \left(e^{\left(s \cdot \pi\right) \cdot r}\right)} \]

   15. lift-*.f32 N/A

       \[\leadsto \frac{\frac{1}{4}}{\log \left(e^{\left(s \cdot \pi\right) \cdot r}\right)} \]

   16. *-commutative N/A

       \[\leadsto \frac{\frac{1}{4}}{\log \left(e^{r \cdot \left(s \cdot \pi\right)}\right)} \]

   17. lift-*.f32 N/A

       \[\leadsto \frac{\frac{1}{4}}{\log \left(e^{r \cdot \left(s \cdot \pi\right)}\right)} \]

   18. lower-exp.f32 10.4

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

   19. lift-*.f32 N/A

       \[\leadsto \frac{\frac{1}{4}}{\log \left(e^{r \cdot \left(s \cdot \pi\right)}\right)} \]

   20. *-commutative N/A

       \[\leadsto \frac{\frac{1}{4}}{\log \left(e^{\left(s \cdot \pi\right) \cdot r}\right)} \]

   21. lift-*.f32 N/A

       \[\leadsto \frac{\frac{1}{4}}{\log \left(e^{\left(s \cdot \pi\right) \cdot r}\right)} \]

   22. *-commutative N/A

       \[\leadsto \frac{\frac{1}{4}}{\log \left(e^{\left(\pi \cdot s\right) \cdot r}\right)} \]

   23. lift-*.f32 N/A

       \[\leadsto \frac{\frac{1}{4}}{\log \left(e^{\left(\pi \cdot s\right) \cdot r}\right)} \]

   24. lower-*.f32 10.4

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

6. Applied rewrites 10.4%

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

Alternative 6: 9.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\frac{1}{\pi}, \frac{0.125}{s}, \frac{0.125}{\left(s + r\right) \cdot \pi}\right)}{r} \end{array} \]

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.6%

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

3. Applied rewrites 99.6%

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

4. Taylor expanded in r around 0

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

   1. lower-fma.f32 N/A

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

   2. lower-PI.f32 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{e^{\frac{r \cdot \frac{-1}{3}}{s}}}{\pi \cdot s}, \frac{1}{8}, \frac{\frac{1}{8}}{\mathsf{fma}\left(r, \pi, s \cdot \mathsf{PI}\left(\right)\right)}\right)}{r} \]

   3. lower-*.f32 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{e^{\frac{r \cdot \frac{-1}{3}}{s}}}{\pi \cdot s}, \frac{1}{8}, \frac{\frac{1}{8}}{\mathsf{fma}\left(r, \pi, s \cdot \mathsf{PI}\left(\right)\right)}\right)}{r} \]

   4. lower-PI.f32 12.3

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

6. Applied rewrites 12.3%

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

   1. lift-fma.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-*l/ N/A

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

   4. lift-*.f32 N/A

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

   5. times-frac N/A

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

   6. lift-exp.f32 N/A

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

   7. lift-/.f32 N/A

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

   8. lift-*.f32 N/A

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

   9. lift-PI.f32 N/A

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

   10. lower-fma.f32 N/A

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

8. Applied rewrites 12.3%

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

9. Taylor expanded in s around inf

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

    1. lower-/.f32 N/A

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

    2. lower-PI.f32 9.4

       \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{\pi}, \frac{0.125}{s}, \frac{0.125}{\left(s + r\right) \cdot \pi}\right)}{r} \]

11. Applied rewrites 9.4%

    \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{1}{\pi}}, \frac{0.125}{s}, \frac{0.125}{\left(s + r\right) \cdot \pi}\right)}{r} \]
12. Add Preprocessing

Alternative 7: 9.0% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 9.0

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

4. Applied rewrites 9.0%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

      \[\leadsto \frac{\frac{1}{4}}{\left(s \cdot r\right) \cdot \pi} \]

   5. lift-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4}}{\left(s \cdot r\right) \cdot \pi} \]

   6. lower-*.f32 9.0

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

6. Applied rewrites 9.0%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-/r* N/A

      \[\leadsto \frac{\frac{\frac{1}{4}}{s \cdot r}}{\color{blue}{\pi}} \]

   4. mult-flip N/A

      \[\leadsto \frac{\frac{1}{4}}{s \cdot r} \cdot \color{blue}{\frac{1}{\pi}} \]

   5. lower-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4}}{s \cdot r} \cdot \color{blue}{\frac{1}{\pi}} \]

   6. lower-/.f32 N/A

      \[\leadsto \frac{\frac{1}{4}}{s \cdot r} \cdot \frac{\color{blue}{1}}{\pi} \]

   7. lower-/.f32 9.0

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

8. Applied rewrites 9.0%

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

Alternative 8: 9.0% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.25) / r) / single(pi)) / s;
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. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 9.0

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

4. Applied rewrites 9.0%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-/r* N/A

      \[\leadsto \frac{\frac{\frac{1}{4}}{r}}{\color{blue}{s \cdot \pi}} \]

   4. lift-*.f32 N/A

      \[\leadsto \frac{\frac{\frac{1}{4}}{r}}{s \cdot \color{blue}{\pi}} \]

   5. *-commutative N/A

      \[\leadsto \frac{\frac{\frac{1}{4}}{r}}{\pi \cdot \color{blue}{s}} \]

   6. associate-/r* N/A

      \[\leadsto \frac{\frac{\frac{\frac{1}{4}}{r}}{\pi}}{\color{blue}{s}} \]

   7. lower-/.f32 N/A

      \[\leadsto \frac{\frac{\frac{\frac{1}{4}}{r}}{\pi}}{\color{blue}{s}} \]

   8. lower-/.f32 N/A

      \[\leadsto \frac{\frac{\frac{\frac{1}{4}}{r}}{\pi}}{s} \]

   9. lower-/.f32 9.0

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

6. Applied rewrites 9.0%

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

Alternative 9: 9.0% accurate, 6.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Applied rewrites 99.6%

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

4. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

   2. lower-PI.f32 9.0

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

6. Applied rewrites 9.0%

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

Alternative 10: 9.0% accurate, 6.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 9.0

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

4. Applied rewrites 9.0%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

      \[\leadsto \frac{\frac{1}{4}}{\left(s \cdot r\right) \cdot \pi} \]

   6. lift-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4}}{\left(s \cdot r\right) \cdot \pi} \]

   7. associate-/r* N/A

      \[\leadsto \frac{\frac{\frac{1}{4}}{s \cdot r}}{\color{blue}{\pi}} \]

   8. lower-/.f32 N/A

      \[\leadsto \frac{\frac{\frac{1}{4}}{s \cdot r}}{\color{blue}{\pi}} \]

   9. lower-/.f32 9.0

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

6. Applied rewrites 9.0%

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

Alternative 11: 9.0% accurate, 6.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 9.0

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

4. Applied rewrites 9.0%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

      \[\leadsto \frac{\frac{1}{4}}{\left(s \cdot r\right) \cdot \pi} \]

   5. lift-*.f32 N/A

      \[\leadsto \frac{\frac{1}{4}}{\left(s \cdot r\right) \cdot \pi} \]

   6. lower-*.f32 9.0

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

6. Applied rewrites 9.0%

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

Alternative 12: 9.0% accurate, 6.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in s around inf

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

   1. lower-/.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 9.0

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

4. Applied rewrites 9.0%

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

Reproduce

herbie shell --seed 2025154 
(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))))