Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.6%

Time: 6.1s

Alternatives: 14

Speedup: 1.1×

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: 99.6% accurate, 1.0× speedup

Math

\[\mathsf{fma}\left(\frac{e^{\frac{r}{-3 \cdot s}} \cdot 0.75}{s}, \frac{1}{\left(6 \cdot \pi\right) \cdot r}, \frac{\frac{0.125}{\left(\pi \cdot s\right) \cdot e^{\frac{r}{s}}}}{r}\right) \]

FPCore

(FPCore (s r)
 :precision binary32
 (fma
  (/ (* (exp (/ r (* -3.0 s))) 0.75) s)
  (/ 1.0 (* (* 6.0 PI) r))
  (/ (/ 0.125 (* (* PI s) (exp (/ r s)))) r)))

C

float code(float s, float r) {
	return fmaf(((expf((r / (-3.0f * s))) * 0.75f) / s), (1.0f / ((6.0f * ((float) M_PI)) * r)), ((0.125f / ((((float) M_PI) * s) * expf((r / s)))) / r));
}

Julia

function code(s, r)
	return fma(Float32(Float32(exp(Float32(r / Float32(Float32(-3.0) * s))) * Float32(0.75)) / s), Float32(Float32(1.0) / Float32(Float32(Float32(6.0) * Float32(pi)) * r)), Float32(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%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{e^{\frac{r}{-3 \cdot s}} \cdot 0.75}{s}, \frac{1}{\left(6 \cdot \pi\right) \cdot r}, \frac{\frac{0.125}{\left(\pi \cdot s\right) \cdot e^{\frac{r}{s}}}}{r}\right)} \]
4. Add Preprocessing

Alternative 2: 99.6% accurate, 1.1× speedup

Math

\[\mathsf{fma}\left(\frac{0.75 \cdot e^{\frac{r}{s \cdot -3}}}{r}, \frac{0.053051647563049226}{s}, \frac{0.125}{\left(\pi \cdot s\right) \cdot \left(e^{\frac{r}{s}} \cdot r\right)}\right) \]

FPCore

(FPCore (s r)
 :precision binary32
 (fma
  (/ (* 0.75 (exp (/ r (* s -3.0)))) r)
  (/ 0.053051647563049226 s)
  (/ 0.125 (* (* PI s) (* (exp (/ r s)) r)))))

C

float code(float s, float r) {
	return fmaf(((0.75f * expf((r / (s * -3.0f)))) / r), (0.053051647563049226f / s), (0.125f / ((((float) M_PI) * s) * (expf((r / s)) * r))));
}

Julia

function code(s, r)
	return fma(Float32(Float32(Float32(0.75) * exp(Float32(r / Float32(s * Float32(-3.0))))) / r), Float32(Float32(0.053051647563049226) / s), 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%

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

4. Evaluated real constant 99.6%

   \[\leadsto \mathsf{fma}\left(\frac{e^{\frac{r}{-3 \cdot s}}}{\color{blue}{18.84955596923828} \cdot s}, \frac{0.75}{r}, \frac{\frac{0.125}{\left(\pi \cdot s\right) \cdot e^{\frac{r}{s}}}}{r}\right) \]

6. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0.75 \cdot e^{\frac{r}{s \cdot -3}}}{r}, \frac{0.053051647563049226}{s}, \frac{0.125}{\left(\pi \cdot s\right) \cdot \left(e^{\frac{r}{s}} \cdot r\right)}\right)} \]
7. Add Preprocessing

Alternative 3: 99.5% accurate, 1.2× speedup

Math

\[\frac{\mathsf{fma}\left(\frac{e^{\frac{r}{s \cdot -3}}}{18.84955596923828 \cdot s}, 0.75, \frac{0.125}{\left(e^{\frac{r}{s}} \cdot \pi\right) \cdot s}\right)}{r} \]

FPCore

(FPCore (s r)
 :precision binary32
 (/
  (fma
   (/ (exp (/ r (* s -3.0))) (* 18.84955596923828 s))
   0.75
   (/ 0.125 (* (* (exp (/ r s)) PI) s)))
  r))

C

float code(float s, float r) {
	return fmaf((expf((r / (s * -3.0f))) / (18.84955596923828f * s)), 0.75f, (0.125f / ((expf((r / s)) * ((float) M_PI)) * s))) / r;
}

Julia

function code(s, r)
	return Float32(fma(Float32(exp(Float32(r / Float32(s * Float32(-3.0)))) / Float32(Float32(18.84955596923828) * s)), Float32(0.75), Float32(Float32(0.125) / Float32(Float32(exp(Float32(r / s)) * Float32(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%

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

4. Evaluated real constant 99.6%

   \[\leadsto \mathsf{fma}\left(\frac{e^{\frac{r}{-3 \cdot s}}}{\color{blue}{18.84955596923828} \cdot s}, \frac{0.75}{r}, \frac{\frac{0.125}{\left(\pi \cdot s\right) \cdot e^{\frac{r}{s}}}}{r}\right) \]
5. Step-by-step derivation

   1. lift-fma.f32 N/A

      \[\leadsto \color{blue}{\frac{e^{\frac{r}{-3 \cdot s}}}{\frac{2470649}{131072} \cdot s} \cdot \frac{\frac{3}{4}}{r} + \frac{\frac{\frac{1}{8}}{\left(\pi \cdot s\right) \cdot e^{\frac{r}{s}}}}{r}} \]

   2. lift-/.f32 N/A

      \[\leadsto \frac{e^{\frac{r}{-3 \cdot s}}}{\frac{2470649}{131072} \cdot s} \cdot \color{blue}{\frac{\frac{3}{4}}{r}} + \frac{\frac{\frac{1}{8}}{\left(\pi \cdot s\right) \cdot e^{\frac{r}{s}}}}{r} \]

   3. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{\frac{e^{\frac{r}{-3 \cdot s}}}{\frac{2470649}{131072} \cdot s} \cdot \frac{3}{4}}{r}} + \frac{\frac{\frac{1}{8}}{\left(\pi \cdot s\right) \cdot e^{\frac{r}{s}}}}{r} \]

   4. lift-/.f32 N/A

      \[\leadsto \frac{\frac{e^{\frac{r}{-3 \cdot s}}}{\frac{2470649}{131072} \cdot s} \cdot \frac{3}{4}}{r} + \color{blue}{\frac{\frac{\frac{1}{8}}{\left(\pi \cdot s\right) \cdot e^{\frac{r}{s}}}}{r}} \]

   5. div-add-rev N/A

      \[\leadsto \color{blue}{\frac{\frac{e^{\frac{r}{-3 \cdot s}}}{\frac{2470649}{131072} \cdot s} \cdot \frac{3}{4} + \frac{\frac{1}{8}}{\left(\pi \cdot s\right) \cdot e^{\frac{r}{s}}}}{r}} \]

   6. lower-/.f32 N/A

      \[\leadsto \color{blue}{\frac{\frac{e^{\frac{r}{-3 \cdot s}}}{\frac{2470649}{131072} \cdot s} \cdot \frac{3}{4} + \frac{\frac{1}{8}}{\left(\pi \cdot s\right) \cdot e^{\frac{r}{s}}}}{r}} \]

6. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{e^{\frac{r}{s \cdot -3}}}{18.84955596923828 \cdot s}, 0.75, \frac{0.125}{\left(e^{\frac{r}{s}} \cdot \pi\right) \cdot s}\right)}{r}} \]
7. Add Preprocessing

Alternative 4: 99.5% accurate, 1.2× speedup

Math

\[\frac{\mathsf{fma}\left(\frac{e^{\frac{r}{-3 \cdot s}}}{\pi \cdot s}, 0.125, \frac{0.125}{\left(\pi \cdot s\right) \cdot e^{\frac{r}{s}}}\right)}{r} \]

FPCore

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

C

float code(float s, float r) {
	return fmaf((expf((r / (-3.0f * 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(r / Float32(Float32(-3.0) * 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 99.5%

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

Alternative 5: 99.5% accurate, 1.2× speedup

Math

\[\frac{\frac{\mathsf{fma}\left(\frac{e^{\frac{-r}{s}}}{\pi}, 0.125, \frac{e^{\frac{r}{-3 \cdot s}}}{\pi} \cdot 0.125\right)}{s}}{r} \]

FPCore

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

C

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

Julia

function code(s, r)
	return Float32(Float32(fma(Float32(exp(Float32(Float32(-r) / s)) / Float32(pi)), Float32(0.125), Float32(Float32(exp(Float32(r / Float32(Float32(-3.0) * s))) / Float32(pi)) * Float32(0.125))) / 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 99.5%

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

Alternative 6: 99.5% accurate, 1.2× speedup

Math

\[\frac{\mathsf{fma}\left(\frac{e^{\frac{-r}{s}}}{\pi}, 0.125, \frac{e^{\frac{r}{-3 \cdot s}}}{\pi} \cdot 0.125\right)}{s \cdot r} \]

FPCore

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

C

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

Julia

function code(s, r)
	return Float32(fma(Float32(exp(Float32(Float32(-r) / s)) / Float32(pi)), Float32(0.125), Float32(Float32(exp(Float32(r / Float32(Float32(-3.0) * s))) / Float32(pi)) * Float32(0.125))) / Float32(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 99.5%

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

Alternative 7: 44.4% 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 9.0%

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

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. lift-*.f32 N/A

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

   6. lower-*.f32 9.0%

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 9.0%

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

6. Applied rewrites 9.0%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. rem-log-exp N/A

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

   4. lift-exp.f32 N/A

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

   5. log-pow-rev N/A

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

   6. lower-log.f32 N/A

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

   7. lift-exp.f32 N/A

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

   8. pow-exp N/A

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

   9. lift-*.f32 N/A

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

   10. lower-exp.f32 44.4%

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

8. Applied rewrites 44.4%

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

Alternative 8: 10.0% accurate, 2.5× speedup

Math

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

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) / log(exp(Float32(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 9.0%

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

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. lift-PI.f32 N/A

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

   5. add-log-exp N/A

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

   6. log-pow-rev N/A

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

   7. lower-log.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-pow.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. lower-exp.f32 N/A

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

   12. lower-*.f32 10.0%

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

6. Applied rewrites 10.0%

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

   1. lift-pow.f32 N/A

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

   2. lift-exp.f32 N/A

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

   3. pow-exp N/A

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

   4. lift-*.f32 N/A

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

   7. lift-*.f32 N/A

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

   8. lift-*.f32 N/A

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

   9. lower-exp.f32 10.0%

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

8. Applied rewrites 10.0%

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

Alternative 9: 9.0% accurate, 2.6× speedup

Math

\[\frac{\frac{\mathsf{fma}\left(-0.16666666666666666, \frac{r}{s \cdot \pi}, 0.25 \cdot \frac{1}{\pi}\right)}{r}}{s} \]

FPCore

(FPCore (s r)
 :precision binary32
 (/ (/ (fma -0.16666666666666666 (/ r (* s PI)) (* 0.25 (/ 1.0 PI))) r) s))

C

float code(float s, float r) {
	return (fmaf(-0.16666666666666666f, (r / (s * ((float) M_PI))), (0.25f * (1.0f / ((float) M_PI)))) / r) / s;
}

Julia

function code(s, r)
	return Float32(Float32(fma(Float32(-0.16666666666666666), Float32(r / Float32(s * Float32(pi))), Float32(Float32(0.25) * Float32(Float32(1.0) / Float32(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

   \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{\frac{-1}{16} \cdot \frac{r}{\pi} + \frac{-1}{144} \cdot \frac{r}{\pi}}{s} - \frac{1}{6} \cdot \frac{1}{\pi}}{s} - \frac{1}{4} \cdot \frac{1}{r \cdot \pi}}{s}} \]
3. Step-by-step derivation

   1. lower-*.f32 N/A

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

   2. lower-/.f32 N/A

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

4. Applied rewrites 9.9%

   \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{\mathsf{fma}\left(-0.0625, \frac{r}{\pi}, -0.006944444444444444 \cdot \frac{r}{\pi}\right)}{s} - 0.16666666666666666 \cdot \frac{1}{\pi}}{s} - 0.25 \cdot \frac{1}{r \cdot \pi}}{s}} \]

6. Applied rewrites 9.9%

   \[\leadsto \color{blue}{\frac{\frac{0.25}{\pi \cdot r} - \frac{\frac{0.16666666666666666}{\pi} - \frac{\frac{r}{\pi} \cdot 0.06944444444444445}{s}}{s}}{s}} \]

7. Taylor expanded in r around 0

   \[\leadsto \frac{\frac{\frac{-1}{6} \cdot \frac{r}{s \cdot \pi} + \frac{1}{4} \cdot \frac{1}{\pi}}{r}}{s} \]
8. Step-by-step derivation

   1. lower-/.f32 N/A

      \[\leadsto \frac{\frac{\frac{-1}{6} \cdot \frac{r}{s \cdot \mathsf{PI}\left(\right)} + \frac{1}{4} \cdot \frac{1}{\mathsf{PI}\left(\right)}}{r}}{s} \]

   2. lower-fma.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-PI.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower-/.f32 N/A

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

   8. lower-PI.f32 8.9%

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

9. Applied rewrites 8.9%

   \[\leadsto \frac{\frac{\mathsf{fma}\left(-0.16666666666666666, \frac{r}{s \cdot \pi}, 0.25 \cdot \frac{1}{\pi}\right)}{r}}{s} \]
10. Add Preprocessing

Alternative 10: 9.0% accurate, 3.2× speedup

Math

\[\frac{\frac{0.25}{\pi \cdot r} - \frac{\frac{0.16666666666666666}{\pi}}{s}}{s} \]

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.25) / (single(pi) * r)) - ((single(0.16666666666666666) / single(pi)) / s)) / 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

   \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{\frac{-1}{16} \cdot \frac{r}{\pi} + \frac{-1}{144} \cdot \frac{r}{\pi}}{s} - \frac{1}{6} \cdot \frac{1}{\pi}}{s} - \frac{1}{4} \cdot \frac{1}{r \cdot \pi}}{s}} \]
3. Step-by-step derivation

   1. lower-*.f32 N/A

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

   2. lower-/.f32 N/A

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

4. Applied rewrites 9.9%

   \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{\mathsf{fma}\left(-0.0625, \frac{r}{\pi}, -0.006944444444444444 \cdot \frac{r}{\pi}\right)}{s} - 0.16666666666666666 \cdot \frac{1}{\pi}}{s} - 0.25 \cdot \frac{1}{r \cdot \pi}}{s}} \]

6. Applied rewrites 9.9%

   \[\leadsto \color{blue}{\frac{\frac{0.25}{\pi \cdot r} - \frac{\frac{0.16666666666666666}{\pi} - \frac{\frac{r}{\pi} \cdot 0.06944444444444445}{s}}{s}}{s}} \]

7. Taylor expanded in s around inf

   \[\leadsto \frac{\frac{0.25}{\pi \cdot r} - \frac{\frac{\frac{1}{6}}{\pi}}{s}}{s} \]
8. Step-by-step derivation

   1. lower-/.f32 N/A

      \[\leadsto \frac{\frac{\frac{1}{4}}{\pi \cdot r} - \frac{\frac{\frac{1}{6}}{\mathsf{PI}\left(\right)}}{s}}{s} \]

   2. lower-PI.f32 8.9%

      \[\leadsto \frac{\frac{0.25}{\pi \cdot r} - \frac{\frac{0.16666666666666666}{\pi}}{s}}{s} \]

9. Applied rewrites 8.9%

   \[\leadsto \frac{\frac{0.25}{\pi \cdot r} - \frac{\frac{0.16666666666666666}{\pi}}{s}}{s} \]
10. Add Preprocessing

Alternative 11: 9.0% accurate, 4.8× speedup

Math

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

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

   \[\leadsto \color{blue}{\frac{\frac{1}{4}}{r \cdot \left(s \cdot \pi\right)}} \]
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 9.0%

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

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. metadata-eval N/A

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

   3. lift-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. associate-*r* N/A

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

   6. times-frac N/A

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

   7. lift-/.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-/.f32 N/A

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

   11. lower-*.f32 9.0%

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

6. Applied rewrites 9.0%

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

Alternative 12: 9.0% accurate, 5.7× speedup

Math

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

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(Float32(Float32(0.25) / r) / s) / Float32(pi))
end

MATLAB

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

Derivation

1. Initial program 99.6%

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

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 9.0%

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

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. associate-/r* N/A

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

   6. lower-/.f32 N/A

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

   7. lower-/.f32 N/A

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

   8. lower-/.f32 9.0%

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

6. Applied rewrites 9.0%

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

Alternative 13: 8.9% 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 9.0%

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

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. associate-/r* N/A

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

   6. lower-/.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-/.f32 N/A

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

   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 14: 8.9% 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 9.0%

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

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 2025181 
(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))))