Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.6%

Time: 5.4s

Alternatives: 13

Speedup: N/A×

Specification

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

Math

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

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, 1.1× speedup

Math

\[\mathsf{fma}\left(e^{\frac{r}{-3 \cdot s}} \cdot \frac{0.039788734167814255}{s}, \frac{1}{r}, \frac{\frac{0.125}{\left(\pi \cdot s\right) \cdot e^{\frac{r}{s}}}}{r}\right) \]

FPCore

(FPCore (s r)
 :precision binary32
 (fma
  (* (exp (/ r (* -3.0 s))) (/ 0.039788734167814255 s))
  (/ 1.0 r)
  (/ (/ 0.125 (* (* PI s) (exp (/ r s)))) r)))

C

float code(float s, float r) {
	return fmaf((expf((r / (-3.0f * s))) * (0.039788734167814255f / s)), (1.0f / r), ((0.125f / ((((float) M_PI) * s) * expf((r / s)))) / r));
}

Julia

function code(s, r)
	return fma(Float32(exp(Float32(r / Float32(Float32(-3.0) * s))) * Float32(Float32(0.039788734167814255) / s)), Float32(Float32(1.0) / 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%

   \[\leadsto \color{blue}{\mathsf{fma}\left(e^{\frac{r}{-3 \cdot s}} \cdot \frac{0.125}{\pi \cdot s}, \frac{1}{r}, \frac{\frac{0.125}{\left(\pi \cdot s\right) \cdot e^{\frac{r}{s}}}}{r}\right)} \]
4. Step-by-step derivation

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f32 N/A

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

   5. lower-/.f32 99.6%

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

5. Applied rewrites 99.6%

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

6. Evaluated real constant 99.6%

   \[\leadsto \mathsf{fma}\left(e^{\frac{r}{-3 \cdot s}} \cdot \frac{\color{blue}{0.039788734167814255}}{s}, \frac{1}{r}, \frac{\frac{0.125}{\left(\pi \cdot s\right) \cdot e^{\frac{r}{s}}}}{r}\right) \]
7. Add Preprocessing

Alternative 2: 99.6% 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 3: 99.5% accurate, 1.2× speedup

Math

\[\frac{\mathsf{fma}\left(e^{\frac{r}{s \cdot -3}}, \frac{0.039788734167814255}{s}, \frac{0.125}{e^{\frac{r}{s}} \cdot \left(\pi \cdot s\right)}\right)}{r} \]

FPCore

(FPCore (s r)
 :precision binary32
 (/
  (fma
   (exp (/ r (* s -3.0)))
   (/ 0.039788734167814255 s)
   (/ 0.125 (* (exp (/ r s)) (* PI s))))
  r))

C

float code(float s, float r) {
	return fmaf(expf((r / (s * -3.0f))), (0.039788734167814255f / s), (0.125f / (expf((r / s)) * (((float) M_PI) * s)))) / r;
}

Julia

function code(s, r)
	return Float32(fma(exp(Float32(r / Float32(s * Float32(-3.0)))), Float32(Float32(0.039788734167814255) / s), Float32(Float32(0.125) / Float32(exp(Float32(r / s)) * Float32(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%

   \[\leadsto \color{blue}{\mathsf{fma}\left(e^{\frac{r}{-3 \cdot s}} \cdot \frac{0.125}{\pi \cdot s}, \frac{1}{r}, \frac{\frac{0.125}{\left(\pi \cdot s\right) \cdot e^{\frac{r}{s}}}}{r}\right)} \]
4. Step-by-step derivation

   1. lift-/.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f32 N/A

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

   5. lower-/.f32 99.6%

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

5. Applied rewrites 99.6%

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

6. Evaluated real constant 99.6%

   \[\leadsto \mathsf{fma}\left(e^{\frac{r}{-3 \cdot s}} \cdot \frac{\color{blue}{0.039788734167814255}}{s}, \frac{1}{r}, \frac{\frac{0.125}{\left(\pi \cdot s\right) \cdot e^{\frac{r}{s}}}}{r}\right) \]

8. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(e^{\frac{r}{s \cdot -3}}, \frac{0.039788734167814255}{s}, \frac{0.125}{e^{\frac{r}{s}} \cdot \left(\pi \cdot s\right)}\right)}{r}} \]
9. Add Preprocessing

Alternative 4: 45.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} t_0 := e^{\pi \cdot r}\\ \frac{0.25}{\log \left(\sqrt{t\_0 \cdot t\_0}\right) \cdot s} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (let* ((t_0 (exp (* PI r)))) (/ 0.25 (* (log (sqrt (* t_0 t_0))) s))))

C

float code(float s, float r) {
	float t_0 = expf((((float) M_PI) * r));
	return 0.25f / (logf(sqrtf((t_0 * t_0))) * s);
}

Julia

function code(s, r)
	t_0 = exp(Float32(Float32(pi) * r))
	return Float32(Float32(0.25) / Float32(log(sqrt(Float32(t_0 * t_0))) * s))
end

MATLAB

function tmp = code(s, r)
	t_0 = exp((single(pi) * r));
	tmp = single(0.25) / (log(sqrt((t_0 * t_0))) * 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 8.9%

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

4. Applied rewrites 8.9%

   \[\leadsto \color{blue}{\frac{0.25}{r \cdot \left(s \cdot \pi\right)}} \]
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 8.9%

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

6. Applied rewrites 8.9%

   \[\leadsto \frac{0.25}{\left(r \cdot \pi\right) \cdot \color{blue}{s}} \]
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. lower-pow.f32 N/A

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

   7. lift-PI.f32 N/A

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

   8. lower-exp.f32 43.3%

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

8. Applied rewrites 43.3%

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

   1. lift-pow.f32 N/A

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

   2. lift-exp.f32 N/A

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

   3. pow-exp N/A

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

   4. *-commutative N/A

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

   5. lift-*.f32 N/A

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

   6. exp-fabs N/A

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. pow-exp N/A

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

   10. lift-exp.f32 N/A

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

   11. lift-pow.f32 N/A

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

   12. rem-sqrt-square-rev N/A

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

   13. lower-sqrt.f32 N/A

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

   14. lower-*.f32 45.2%

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

   15. lift-pow.f32 N/A

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

   16. lift-exp.f32 N/A

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

   17. pow-exp N/A

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

   18. *-commutative N/A

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

   19. lift-*.f32 N/A

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

   20. lower-exp.f32 45.2%

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

   21. lift-*.f32 N/A

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

   22. *-commutative N/A

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

   23. lift-*.f32 45.2%

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

   24. lift-pow.f32 N/A

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

   25. lift-exp.f32 N/A

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

   26. pow-exp N/A

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

10. Applied rewrites 45.2%

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

Alternative 5: 43.3% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

function code(s, r)
	return Float32(Float32(0.25) / Float32(log((exp(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 8.9%

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

4. Applied rewrites 8.9%

   \[\leadsto \color{blue}{\frac{0.25}{r \cdot \left(s \cdot \pi\right)}} \]
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 8.9%

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

6. Applied rewrites 8.9%

   \[\leadsto \frac{0.25}{\left(r \cdot \pi\right) \cdot \color{blue}{s}} \]
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. lower-pow.f32 N/A

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

   7. lift-PI.f32 N/A

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

   8. lower-exp.f32 43.3%

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

8. Applied rewrites 43.3%

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

Alternative 6: 43.3% 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 8.9%

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

4. Applied rewrites 8.9%

   \[\leadsto \color{blue}{\frac{0.25}{r \cdot \left(s \cdot \pi\right)}} \]
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 8.9%

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

6. Applied rewrites 8.9%

   \[\leadsto \frac{0.25}{\left(r \cdot \pi\right) \cdot \color{blue}{s}} \]
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. lower-pow.f32 N/A

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

   7. lift-PI.f32 N/A

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

   8. lower-exp.f32 43.3%

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

8. Applied rewrites 43.3%

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

   1. lift-pow.f32 N/A

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

   2. lift-exp.f32 N/A

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

   3. pow-exp N/A

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

   4. *-commutative N/A

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

   5. lift-*.f32 N/A

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

   6. lower-exp.f32 43.3%

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. lift-*.f32 43.3%

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

10. Applied rewrites 43.3%

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

Alternative 7: 10.0% accurate, 2.5× speedup

Math

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

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

   \[\leadsto \color{blue}{\frac{\frac{1}{4}}{r \cdot \left(s \cdot \pi\right)}} \]
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 8.9%

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

4. Applied rewrites 8.9%

   \[\leadsto \color{blue}{\frac{0.25}{r \cdot \left(s \cdot \pi\right)}} \]
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. lift-PI.f32 N/A

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

   9. *-commutative N/A

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

   10. pow-exp N/A

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

   11. associate-*l* N/A

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

   12. *-commutative N/A

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

   13. lift-*.f32 N/A

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

   14. *-commutative N/A

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

   15. lift-*.f32 N/A

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

   16. lower-exp.f32 10.0%

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

   17. lift-*.f32 N/A

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

   18. *-commutative N/A

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

   19. lift-*.f32 N/A

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

   20. *-commutative N/A

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

   21. lift-*.f32 N/A

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

   22. lower-*.f32 10.0%

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

6. Applied rewrites 10.0%

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

Alternative 8: 8.9% accurate, 4.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = single(1.0) / (r / (single(0.25) / (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 8.9%

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

4. Applied rewrites 8.9%

   \[\leadsto \color{blue}{\frac{0.25}{r \cdot \left(s \cdot \pi\right)}} \]
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 8.9%

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

6. Applied rewrites 8.9%

   \[\leadsto \frac{0.25}{\left(r \cdot \pi\right) \cdot \color{blue}{s}} \]
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. lift-*.f32 N/A

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

   4. associate-*l* N/A

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

   5. lift-*.f32 N/A

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

   6. *-commutative N/A

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

   7. associate-/l/ N/A

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

   8. lift-/.f32 N/A

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

   9. div-flip N/A

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

   10. lower-unsound-/.f32 N/A

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

   11. lower-unsound-/.f32 8.9%

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

8. Applied rewrites 8.9%

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

Alternative 9: 8.9% accurate, 6.0× speedup

Math

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

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%

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

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 8.9%

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

4. Applied rewrites 8.9%

   \[\leadsto \color{blue}{\frac{0.25}{r \cdot \left(s \cdot \pi\right)}} \]
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 8.9%

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

6. Applied rewrites 8.9%

   \[\leadsto \frac{0.25}{\left(r \cdot \pi\right) \cdot \color{blue}{s}} \]
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. lift-*.f32 N/A

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

   4. *-commutative N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

   7. lift-*.f32 N/A

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

   8. associate-/r* N/A

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

   9. lower-/.f32 N/A

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

   10. lower-/.f32 8.9%

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

8. Applied rewrites 8.9%

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

Alternative 10: 8.9% 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 8.9%

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

4. Applied rewrites 8.9%

   \[\leadsto \color{blue}{\frac{0.25}{r \cdot \left(s \cdot \pi\right)}} \]
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. lower-*.f32 N/A

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

   6. lower-*.f32 8.9%

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

6. Applied rewrites 8.9%

   \[\leadsto \frac{0.25}{\left(s \cdot r\right) \cdot \color{blue}{\pi}} \]
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. lift-*.f32 N/A

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

   4. associate-*l* N/A

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

   5. lift-*.f32 N/A

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

   6. associate-/r* N/A

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

   7. lower-/.f32 N/A

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

   8. lower-/.f32 8.9%

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

   9. lift-*.f32 N/A

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

   10. *-commutative N/A

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

   11. lift-*.f32 8.9%

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

8. Applied rewrites 8.9%

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

Alternative 11: 8.9% accurate, 6.0× speedup

Math

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

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(Float32(0.25) / r) / Float32(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 8.9%

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

4. Applied rewrites 8.9%

   \[\leadsto \color{blue}{\frac{0.25}{r \cdot \left(s \cdot \pi\right)}} \]
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. lift-*.f32 N/A

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

   7. lower-/.f32 N/A

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

   8. lower-/.f32 8.9%

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

6. Applied rewrites 8.9%

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

Alternative 12: 8.9% accurate, 6.5× 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 8.9%

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

4. Applied rewrites 8.9%

   \[\leadsto \color{blue}{\frac{0.25}{r \cdot \left(s \cdot \pi\right)}} \]
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 8.9%

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

6. Applied rewrites 8.9%

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

Alternative 13: 8.9% accurate, 6.5× 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 8.9%

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

4. Applied rewrites 8.9%

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

Reproduce

herbie shell --seed 2025207 
(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))))