Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.5%

Time: 5.8s

Alternatives: 9

Speedup: N/A×

Specification

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

Math

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

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

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

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, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Applied rewrites 99.6%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f32 N/A

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

   5. mult-flip N/A

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

   6. lower-*.f32 N/A

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

   7. metadata-eval 99.5%

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

5. Applied rewrites 99.5%

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

   1. lift-/.f32 N/A

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

   2. lift-fma.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. distribute-rgt-out N/A

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

   5. lift-*.f32 N/A

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

   6. times-frac N/A

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

   7. lower-*.f32 N/A

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

   8. lower-/.f32 N/A

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

   9. lower-/.f32 N/A

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

7. Applied rewrites 99.5%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f32 N/A

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

   4. lift-/.f32 N/A

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

   5. associate-/l/ N/A

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

   6. lift-/.f32 N/A

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

   7. frac-times N/A

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

   8. associate-*r* N/A

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

   9. *-commutative N/A

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

   10. lift-*.f32 N/A

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

   11. *-commutative N/A

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

   12. lift-*.f32 N/A

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

   13. lower-/.f32 N/A

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

9. Applied rewrites 99.5%

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

Alternative 2: 43.7% accurate, 2.5× speedup

Math

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

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

   \[\leadsto \color{blue}{\frac{\frac{1}{4}}{r \cdot \left(s \cdot \pi\right)}} \]
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 9.2%

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

4. Applied rewrites 9.2%

   \[\leadsto \color{blue}{\frac{0.25}{r \cdot \left(s \cdot \pi\right)}} \]
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 9.2%

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

6. Applied rewrites 9.2%

   \[\leadsto \frac{0.25}{\left(r \cdot \pi\right) \cdot \color{blue}{s}} \]
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 43.7%

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

   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 43.7%

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

8. Applied rewrites 43.7%

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

Alternative 3: 9.9% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = single(0.25) / log(exp(((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

   \[\leadsto \color{blue}{\frac{\frac{1}{4}}{r \cdot \left(s \cdot \pi\right)}} \]
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 9.2%

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

4. Applied rewrites 9.2%

   \[\leadsto \color{blue}{\frac{0.25}{r \cdot \left(s \cdot \pi\right)}} \]
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. *-commutative N/A

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

   13. lift-*.f32 N/A

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

   14. *-commutative N/A

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

   15. associate-*r* N/A

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

   16. lift-*.f32 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 9.9%

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

   19. lift-*.f32 N/A

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

   20. lift-*.f32 N/A

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

   21. associate-*r* N/A

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

   22. *-commutative N/A

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

   23. lift-*.f32 N/A

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

   24. lower-*.f32 9.9%

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

6. Applied rewrites 9.9%

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

Alternative 4: 9.2% accurate, 4.5× speedup

Math

\[\frac{0.125}{s} \cdot \frac{\frac{2}{\pi}}{r} \]

FPCore

(FPCore (s r) :precision binary32 (* (/ 0.125 s) (/ (/ 2.0 PI) r)))

C

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

Julia

function code(s, r)
	return Float32(Float32(Float32(0.125) / s) * Float32(Float32(Float32(2.0) / Float32(pi)) / r))
end

MATLAB

function tmp = code(s, r)
	tmp = (single(0.125) / s) * ((single(2.0) / single(pi)) / r);
end

Derivation

1. Initial program 99.6%

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

3. Applied rewrites 99.6%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f32 N/A

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

   5. mult-flip N/A

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

   6. lower-*.f32 N/A

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

   7. metadata-eval 99.5%

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

5. Applied rewrites 99.5%

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

   1. lift-/.f32 N/A

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

   2. lift-fma.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. distribute-rgt-out N/A

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

   5. lift-*.f32 N/A

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

   6. times-frac N/A

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

   7. lower-*.f32 N/A

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

   8. lower-/.f32 N/A

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

   9. lower-/.f32 N/A

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

7. Applied rewrites 99.5%

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

8. Taylor expanded in s around inf

   \[\leadsto \frac{0.125}{s} \cdot \frac{\frac{\color{blue}{2}}{\pi}}{r} \]
9. Step-by-step derivation

10. Applied rewrites 9.2%

    \[\leadsto \frac{0.125}{s} \cdot \frac{\frac{\color{blue}{2}}{\pi}}{r} \]
11. Add Preprocessing

Alternative 5: 9.2% accurate, 6.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in s around inf

   \[\leadsto \color{blue}{\frac{\frac{1}{4}}{r \cdot \left(s \cdot \pi\right)}} \]
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 9.2%

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

4. Applied rewrites 9.2%

   \[\leadsto \color{blue}{\frac{0.25}{r \cdot \left(s \cdot \pi\right)}} \]
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 9.2%

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

6. Applied rewrites 9.2%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-/r* N/A

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

   5. lower-/.f32 N/A

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

   6. lower-/.f32 9.2%

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 9.2%

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

8. Applied rewrites 9.2%

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

Alternative 6: 9.2% accurate, 6.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in s around inf

   \[\leadsto \color{blue}{\frac{\frac{1}{4}}{r \cdot \left(s \cdot \pi\right)}} \]
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 9.2%

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

4. Applied rewrites 9.2%

   \[\leadsto \color{blue}{\frac{0.25}{r \cdot \left(s \cdot \pi\right)}} \]
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 9.2%

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

6. Applied rewrites 9.2%

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

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f32 N/A

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

   5. lower-/.f32 9.2%

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

   6. lift-*.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 9.2%

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

8. Applied rewrites 9.2%

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

Alternative 7: 9.2% accurate, 6.0× speedup

Math

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

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

   \[\leadsto \color{blue}{\frac{\frac{1}{4}}{r \cdot \left(s \cdot \pi\right)}} \]
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 9.2%

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

4. Applied rewrites 9.2%

   \[\leadsto \color{blue}{\frac{0.25}{r \cdot \left(s \cdot \pi\right)}} \]
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 9.2%

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

6. Applied rewrites 9.2%

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

Alternative 8: 9.2% accurate, 6.4× speedup

Math

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

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(0.25) / Float32(Float32(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

   \[\leadsto \color{blue}{\frac{\frac{1}{4}}{r \cdot \left(s \cdot \pi\right)}} \]
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 9.2%

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

4. Applied rewrites 9.2%

   \[\leadsto \color{blue}{\frac{0.25}{r \cdot \left(s \cdot \pi\right)}} \]
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 9.2%

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

6. Applied rewrites 9.2%

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

Alternative 9: 9.2% accurate, 6.4× speedup

Math

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

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

   \[\leadsto \color{blue}{\frac{\frac{1}{4}}{r \cdot \left(s \cdot \pi\right)}} \]
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 9.2%

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

4. Applied rewrites 9.2%

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

Reproduce

herbie shell --seed 2025183 
(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))))