Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.6%

Time: 16.8s

Alternatives: 15

Speedup: N/A×

Specification

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

Math

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

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/ (* 0.25 (exp (/ (- r) s))) (* (* (* 2.0 PI) s) r))
  (/ (* 0.75 (exp (/ (- r) (* 3.0 s)))) (* (* (* 6.0 PI) s) r))))

C

float code(float s, float r) {
	return ((0.25f * expf((-r / s))) / (((2.0f * ((float) M_PI)) * s) * r)) + ((0.75f * expf((-r / (3.0f * s)))) / (((6.0f * ((float) M_PI)) * s) * r));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.25) * exp(Float32(Float32(-r) / s))) / Float32(Float32(Float32(Float32(2.0) * Float32(pi)) * s) * r)) + Float32(Float32(Float32(0.75) * exp(Float32(Float32(-r) / Float32(Float32(3.0) * s)))) / Float32(Float32(Float32(Float32(6.0) * Float32(pi)) * s) * r)))
end

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.25) * exp((-r / s))) / (((single(2.0) * single(pi)) * s) * r)) + ((single(0.75) * exp((-r / (single(3.0) * s)))) / (((single(6.0) * single(pi)) * s) * r));
end

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/ (* 0.25 (exp (/ (- r) s))) (* (* (* 2.0 PI) s) r))
  (/ (* 0.75 (exp (/ (- r) (* 3.0 s)))) (* (* (* 6.0 PI) s) r))))

C

float code(float s, float r) {
	return ((0.25f * expf((-r / s))) / (((2.0f * ((float) M_PI)) * s) * r)) + ((0.75f * expf((-r / (3.0f * s)))) / (((6.0f * ((float) M_PI)) * s) * r));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.25) * exp(Float32(Float32(-r) / s))) / Float32(Float32(Float32(Float32(2.0) * Float32(pi)) * s) * r)) + Float32(Float32(Float32(0.75) * exp(Float32(Float32(-r) / Float32(Float32(3.0) * s)))) / Float32(Float32(Float32(Float32(6.0) * Float32(pi)) * s) * r)))
end

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.25) * exp((-r / s))) / (((single(2.0) * single(pi)) * s) * r)) + ((single(0.75) * exp((-r / (single(3.0) * s)))) / (((single(6.0) * single(pi)) * s) * r));
end

Alternative 1: 99.6% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/ (* 0.25 (exp (/ r (- s)))) (* s (* r (* PI 2.0))))
  (/ (* 0.75 (exp (/ r (* s (- 3.0))))) (* r (* s (* PI 6.0))))))

C

float code(float s, float r) {
	return ((0.25f * expf((r / -s))) / (s * (r * (((float) M_PI) * 2.0f)))) + ((0.75f * expf((r / (s * -3.0f)))) / (r * (s * (((float) M_PI) * 6.0f))));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.25) * exp(Float32(r / Float32(-s)))) / Float32(s * Float32(r * Float32(Float32(pi) * Float32(2.0))))) + Float32(Float32(Float32(0.75) * exp(Float32(r / Float32(s * Float32(-Float32(3.0)))))) / Float32(r * Float32(s * Float32(Float32(pi) * Float32(6.0))))))
end

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.25) * exp((r / -s))) / (s * (r * (single(pi) * single(2.0))))) + ((single(0.75) * exp((r / (s * -single(3.0))))) / (r * (s * (single(pi) * single(6.0)))));
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. Add Preprocessing
3. Step-by-step derivation

   1. pow1 99.6%

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\color{blue}{{\left(\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r\right)}^{1}}} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   2. *-commutative 99.6%

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{{\left(\color{blue}{\left(s \cdot \left(2 \cdot \pi\right)\right)} \cdot r\right)}^{1}} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   3. associate-*l* 99.7%

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{{\color{blue}{\left(s \cdot \left(\left(2 \cdot \pi\right) \cdot r\right)\right)}}^{1}} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   4. *-commutative 99.7%

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{{\left(s \cdot \left(\color{blue}{\left(\pi \cdot 2\right)} \cdot r\right)\right)}^{1}} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

4. Applied egg-rr 99.7%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\color{blue}{{\left(s \cdot \left(\left(\pi \cdot 2\right) \cdot r\right)\right)}^{1}}} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

5. Final simplification 99.7%

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

Alternative 2: 99.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.125}{s \cdot \pi}\\ \mathsf{fma}\left(t\_0, \frac{e^{\frac{r}{s \cdot -3}}}{r}, t\_0 \cdot \frac{e^{\frac{r}{-s}}}{r}\right) \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (let* ((t_0 (/ 0.125 (* s PI))))
   (fma t_0 (/ (exp (/ r (* s -3.0))) r) (* t_0 (/ (exp (/ r (- s))) r)))))

C

float code(float s, float r) {
	float t_0 = 0.125f / (s * ((float) M_PI));
	return fmaf(t_0, (expf((r / (s * -3.0f))) / r), (t_0 * (expf((r / -s)) / r)));
}

Julia

function code(s, r)
	t_0 = Float32(Float32(0.125) / Float32(s * Float32(pi)))
	return fma(t_0, Float32(exp(Float32(r / Float32(s * Float32(-3.0)))) / r), Float32(t_0 * Float32(exp(Float32(r / Float32(-s))) / r)))
end

Derivation

1. Initial program 99.6%

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

   1. +-commutative 99.6%

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

   2. times-frac 99.7%

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

   3. fma-define 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0.75}{\left(6 \cdot \pi\right) \cdot s}, \frac{e^{\frac{-r}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right)} \]

   4. associate-*l* 99.6%

      \[\leadsto \mathsf{fma}\left(\frac{0.75}{\color{blue}{6 \cdot \left(\pi \cdot s\right)}}, \frac{e^{\frac{-r}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   5. associate-/r* 99.7%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\frac{0.75}{6}}{\pi \cdot s}}, \frac{e^{\frac{-r}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   6. metadata-eval 99.7%

      \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{0.125}}{\pi \cdot s}, \frac{e^{\frac{-r}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   7. *-commutative 99.7%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{\color{blue}{s \cdot \pi}}, \frac{e^{\frac{-r}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   8. neg-mul-1 99.7%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\frac{\color{blue}{-1 \cdot r}}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   9. times-frac 99.6%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\color{blue}{\frac{-1}{3} \cdot \frac{r}{s}}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   10. metadata-eval 99.6%

       \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\color{blue}{-0.3333333333333333} \cdot \frac{r}{s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   11. times-frac 99.6%

       \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r}, \color{blue}{\frac{0.25}{\left(2 \cdot \pi\right) \cdot s} \cdot \frac{e^{\frac{-r}{s}}}{r}}\right) \]

3. Simplified 99.6%

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

   1. metadata-eval 99.6%

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

   2. times-frac 99.7%

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

   3. neg-mul-1 99.7%

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

   4. frac-2neg 99.7%

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

   5. remove-double-neg 99.7%

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

   6. *-commutative 99.7%

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

   7. distribute-rgt-neg-in 99.7%

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

   8. metadata-eval 99.7%

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

6. Applied egg-rr 99.7%

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

7. Final simplification 99.7%

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

Alternative 3: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{r}{-s}}}{r} + 0.75 \cdot \frac{e^{\frac{r}{s \cdot \left(-3\right)}}}{r \cdot \left(s \cdot \left(\pi \cdot 6\right)\right)} \end{array} \]

FPCore

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

C

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

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.125) / Float32(s * Float32(pi))) * Float32(exp(Float32(r / Float32(-s))) / r)) + Float32(Float32(0.75) * Float32(exp(Float32(r / Float32(s * Float32(-Float32(3.0))))) / Float32(r * Float32(s * Float32(Float32(pi) * Float32(6.0)))))))
end

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.125) / (s * single(pi))) * (exp((r / -s)) / r)) + (single(0.75) * (exp((r / (s * -single(3.0)))) / (r * (s * (single(pi) * single(6.0))))));
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. Step-by-step derivation

   1. times-frac 99.7%

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

   2. *-commutative 99.7%

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

   3. distribute-frac-neg 99.7%

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

   4. associate-/l* 99.6%

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

   5. *-commutative 99.6%

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

   6. *-commutative 99.6%

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

   7. associate-*l* 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in s around 0 99.6%

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

6. Taylor expanded in s around 0 99.6%

   \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \frac{e^{-\frac{r}{s}}}{r} + 0.75 \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r \cdot \color{blue}{\left(6 \cdot \left(s \cdot \pi\right)\right)}} \]
7. Step-by-step derivation

   1. *-commutative 99.6%

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

   2. associate-*r* 99.6%

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

8. Simplified 99.6%

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

9. Final simplification 99.6%

   \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{r}{-s}}}{r} + 0.75 \cdot \frac{e^{\frac{r}{s \cdot \left(-3\right)}}}{r \cdot \left(s \cdot \left(\pi \cdot 6\right)\right)} \]
10. Add Preprocessing

Alternative 4: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = (single(0.125) / (s * single(pi))) * ((exp((r / -s)) / r) + (exp(((r / s) * single(-0.3333333333333333))) / 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. Simplified 99.4%

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

5. Taylor expanded in r around inf 99.6%

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

6. Final simplification 99.6%

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

Alternative 5: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = (single(0.125) / (s * single(pi))) * ((exp((r / -s)) / r) + (exp(((r * single(-0.3333333333333333)) / 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. Simplified 99.4%

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

   1. pow-to-exp 99.4%

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

   2. associate-*r/ 99.4%

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

   3. *-commutative 99.4%

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

   4. rem-log-exp 99.6%

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

6. Applied egg-rr 99.6%

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

7. Final simplification 99.6%

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

Alternative 6: 12.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{0.25}{\mathsf{log1p}\left(\mathsf{expm1}\left(r \cdot \left(s \cdot \pi\right)\right)\right)} \end{array} \]

FPCore

(FPCore (s r) :precision binary32 (/ 0.25 (log1p (expm1 (* r (* s PI))))))

C

float code(float s, float r) {
	return 0.25f / log1pf(expm1f((r * (s * ((float) M_PI)))));
}

Julia

function code(s, r)
	return Float32(Float32(0.25) / log1p(expm1(Float32(r * Float32(s * Float32(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. Simplified 99.4%

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

5. Taylor expanded in r around 0 8.6%

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

6. Taylor expanded in s around inf 8.2%

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

   1. log1p-expm1-u 12.6%

      \[\leadsto \frac{0.25}{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(r \cdot \left(s \cdot \pi\right)\right)\right)}} \]

8. Applied egg-rr 12.6%

   \[\leadsto \frac{0.25}{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(r \cdot \left(s \cdot \pi\right)\right)\right)}} \]

9. Final simplification 12.6%

   \[\leadsto \frac{0.25}{\mathsf{log1p}\left(\mathsf{expm1}\left(r \cdot \left(s \cdot \pi\right)\right)\right)} \]
10. Add Preprocessing

Alternative 7: 9.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ 0.125 \cdot \frac{\frac{e^{\frac{r}{-s}}}{r} + \frac{1}{r}}{s \cdot \pi} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = single(0.125) * (((exp((r / -s)) / r) + (single(1.0) / 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. Simplified 99.4%

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

5. Taylor expanded in r around 0 8.6%

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

6. Taylor expanded in s around 0 8.6%

   \[\leadsto \color{blue}{0.125 \cdot \frac{\frac{1}{r} + \frac{e^{-1 \cdot \frac{r}{s}}}{r}}{s \cdot \pi}} \]
7. Step-by-step derivation

   1. associate-*r/ 8.6%

      \[\leadsto 0.125 \cdot \frac{\frac{1}{r} + \frac{e^{\color{blue}{\frac{-1 \cdot r}{s}}}}{r}}{s \cdot \pi} \]

   2. neg-mul-1 8.6%

      \[\leadsto 0.125 \cdot \frac{\frac{1}{r} + \frac{e^{\frac{\color{blue}{-r}}{s}}}{r}}{s \cdot \pi} \]

   3. *-commutative 8.6%

      \[\leadsto 0.125 \cdot \frac{\frac{1}{r} + \frac{e^{\frac{-r}{s}}}{r}}{\color{blue}{\pi \cdot s}} \]

8. Simplified 8.6%

   \[\leadsto \color{blue}{0.125 \cdot \frac{\frac{1}{r} + \frac{e^{\frac{-r}{s}}}{r}}{\pi \cdot s}} \]

9. Final simplification 8.6%

   \[\leadsto 0.125 \cdot \frac{\frac{e^{\frac{r}{-s}}}{r} + \frac{1}{r}}{s \cdot \pi} \]
10. Add Preprocessing

Alternative 8: 9.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{1}{r}\right) \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = (single(0.125) / (s * single(pi))) * ((exp((r / -s)) / r) + (single(1.0) / 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. Simplified 99.4%

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

5. Taylor expanded in r around 0 8.6%

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

6. Final simplification 8.6%

   \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{1}{r}\right) \]
7. Add Preprocessing

Alternative 9: 9.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ 0.125 \cdot \frac{e^{\frac{r}{-s}} + 1}{r \cdot \left(s \cdot \pi\right)} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = single(0.125) * ((exp((r / -s)) + single(1.0)) / (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. Simplified 99.4%

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

5. Taylor expanded in r around 0 8.6%

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

6. Taylor expanded in r around inf 8.6%

   \[\leadsto \color{blue}{0.125 \cdot \frac{1 + e^{-1 \cdot \frac{r}{s}}}{r \cdot \left(s \cdot \pi\right)}} \]
7. Step-by-step derivation

   1. associate-*r/ 8.6%

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

   2. neg-mul-1 8.6%

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

   3. *-commutative 8.6%

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

8. Simplified 8.6%

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

9. Final simplification 8.6%

   \[\leadsto 0.125 \cdot \frac{e^{\frac{r}{-s}} + 1}{r \cdot \left(s \cdot \pi\right)} \]
10. Add Preprocessing

Alternative 10: 9.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{r}{-s}} + 1}{r} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = (single(0.125) / (s * single(pi))) * ((exp((r / -s)) + single(1.0)) / 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. Simplified 99.4%

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

5. Taylor expanded in r around 0 8.6%

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

6. Taylor expanded in r around inf 8.6%

   \[\leadsto \color{blue}{0.125 \cdot \frac{1 + e^{-1 \cdot \frac{r}{s}}}{r \cdot \left(s \cdot \pi\right)}} \]
7. Step-by-step derivation

   1. associate-*r/ 8.6%

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

   2. *-commutative 8.6%

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

   3. *-commutative 8.6%

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

   4. times-frac 8.6%

      \[\leadsto \color{blue}{\frac{0.125}{\pi \cdot s} \cdot \frac{1 + e^{-1 \cdot \frac{r}{s}}}{r}} \]

   5. associate-*r/ 8.6%

      \[\leadsto \frac{0.125}{\pi \cdot s} \cdot \frac{1 + e^{\color{blue}{\frac{-1 \cdot r}{s}}}}{r} \]

   6. neg-mul-1 8.6%

      \[\leadsto \frac{0.125}{\pi \cdot s} \cdot \frac{1 + e^{\frac{\color{blue}{-r}}{s}}}{r} \]

8. Simplified 8.6%

   \[\leadsto \color{blue}{\frac{0.125}{\pi \cdot s} \cdot \frac{1 + e^{\frac{-r}{s}}}{r}} \]

9. Final simplification 8.6%

   \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{r}{-s}} + 1}{r} \]
10. Add Preprocessing

Alternative 11: 9.4% accurate, 25.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = (single(0.25) / r) * (single(1.0) / (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. Simplified 99.4%

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

5. Taylor expanded in r around 0 8.6%

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

6. Taylor expanded in s around inf 8.2%

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

   1. associate-/r* 8.2%

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

   2. div-inv 8.2%

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

8. Applied egg-rr 8.2%

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

9. Final simplification 8.2%

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

Alternative 12: 7.2% accurate, 33.0× speedup

Math

\[\begin{array}{l} \\ \frac{0.125}{r \cdot \left(s \cdot \pi\right)} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = 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. Simplified 99.4%

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

5. Taylor expanded in r around 0 8.6%

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

6. Taylor expanded in r around 0 7.4%

   \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{\color{blue}{1 + -1 \cdot \frac{r}{s}}}{r} + \frac{1}{r}\right) \]
7. Step-by-step derivation

   1. mul-1-neg 7.4%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{1 + \color{blue}{\left(-\frac{r}{s}\right)}}{r} + \frac{1}{r}\right) \]

   2. unsub-neg 7.4%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{\color{blue}{1 - \frac{r}{s}}}{r} + \frac{1}{r}\right) \]

8. Simplified 7.4%

   \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{\color{blue}{1 - \frac{r}{s}}}{r} + \frac{1}{r}\right) \]
9. Step-by-step derivation

   1. add-sqr-sqrt 3.4%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\color{blue}{\sqrt{\frac{1 - \frac{r}{s}}{r}} \cdot \sqrt{\frac{1 - \frac{r}{s}}{r}}} + \frac{1}{r}\right) \]

   2. sqrt-unprod 8.0%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\color{blue}{\sqrt{\frac{1 - \frac{r}{s}}{r} \cdot \frac{1 - \frac{r}{s}}{r}}} + \frac{1}{r}\right) \]

   3. pow2 8.0%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\sqrt{\color{blue}{{\left(\frac{1 - \frac{r}{s}}{r}\right)}^{2}}} + \frac{1}{r}\right) \]

10. Applied egg-rr 8.0%

    \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\color{blue}{\sqrt{{\left(\frac{1 - \frac{r}{s}}{r}\right)}^{2}}} + \frac{1}{r}\right) \]
11. Step-by-step derivation

    1. unpow2 8.0%

       \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\sqrt{\color{blue}{\frac{1 - \frac{r}{s}}{r} \cdot \frac{1 - \frac{r}{s}}{r}}} + \frac{1}{r}\right) \]

    2. rem-sqrt-square 8.0%

       \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\color{blue}{\left|\frac{1 - \frac{r}{s}}{r}\right|} + \frac{1}{r}\right) \]

12. Simplified 8.0%

    \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\color{blue}{\left|\frac{1 - \frac{r}{s}}{r}\right|} + \frac{1}{r}\right) \]

13. Taylor expanded in r around 0 7.1%

    \[\leadsto \color{blue}{\frac{0.125}{r \cdot \left(s \cdot \pi\right)}} \]

14. Final simplification 7.1%

    \[\leadsto \frac{0.125}{r \cdot \left(s \cdot \pi\right)} \]
15. Add Preprocessing

Alternative 13: 9.4% accurate, 33.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Simplified 99.4%

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

5. Taylor expanded in r around 0 8.6%

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

6. Taylor expanded in s around inf 8.2%

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

7. Final simplification 8.2%

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

Alternative 14: 9.4% accurate, 33.0× speedup

Math

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

FPCore

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

C

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

Julia

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

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

3. Simplified 99.4%

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

5. Taylor expanded in r around 0 8.6%

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

6. Taylor expanded in s around inf 8.2%

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

   1. associate-/r* 8.2%

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

   2. *-commutative 8.2%

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

8. Simplified 8.2%

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

9. Final simplification 8.2%

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

Alternative 15: 9.4% accurate, 33.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(s, r)
	return Float32(Float32(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%

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

3. Simplified 99.4%

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

5. Taylor expanded in r around 0 8.6%

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

6. Taylor expanded in r around 0 7.4%

   \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{\color{blue}{1 + -1 \cdot \frac{r}{s}}}{r} + \frac{1}{r}\right) \]
7. Step-by-step derivation

   1. mul-1-neg 7.4%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{1 + \color{blue}{\left(-\frac{r}{s}\right)}}{r} + \frac{1}{r}\right) \]

   2. unsub-neg 7.4%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{\color{blue}{1 - \frac{r}{s}}}{r} + \frac{1}{r}\right) \]

8. Simplified 7.4%

   \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{\color{blue}{1 - \frac{r}{s}}}{r} + \frac{1}{r}\right) \]
9. Step-by-step derivation

   1. associate-/r* 7.4%

      \[\leadsto \color{blue}{\frac{\frac{0.125}{s}}{\pi}} \cdot \left(\frac{1 - \frac{r}{s}}{r} + \frac{1}{r}\right) \]

   2. associate-*l/ 7.4%

      \[\leadsto \color{blue}{\frac{\frac{0.125}{s} \cdot \left(\frac{1 - \frac{r}{s}}{r} + \frac{1}{r}\right)}{\pi}} \]

   3. +-commutative 7.4%

      \[\leadsto \frac{\frac{0.125}{s} \cdot \color{blue}{\left(\frac{1}{r} + \frac{1 - \frac{r}{s}}{r}\right)}}{\pi} \]

   4. *-un-lft-identity 7.4%

      \[\leadsto \frac{\frac{0.125}{s} \cdot \left(\color{blue}{1 \cdot \frac{1}{r}} + \frac{1 - \frac{r}{s}}{r}\right)}{\pi} \]

   5. div-inv 7.4%

      \[\leadsto \frac{\frac{0.125}{s} \cdot \left(1 \cdot \frac{1}{r} + \color{blue}{\left(1 - \frac{r}{s}\right) \cdot \frac{1}{r}}\right)}{\pi} \]

   6. distribute-rgt-out 7.4%

      \[\leadsto \frac{\frac{0.125}{s} \cdot \color{blue}{\left(\frac{1}{r} \cdot \left(1 + \left(1 - \frac{r}{s}\right)\right)\right)}}{\pi} \]

10. Applied egg-rr 7.4%

    \[\leadsto \color{blue}{\frac{\frac{0.125}{s} \cdot \left(\frac{1}{r} \cdot \left(1 + \left(1 - \frac{r}{s}\right)\right)\right)}{\pi}} \]

11. Taylor expanded in s around inf 8.2%

    \[\leadsto \frac{\color{blue}{\frac{0.25}{r \cdot s}}}{\pi} \]
12. Step-by-step derivation

    1. associate-/r* 8.2%

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

13. Simplified 8.2%

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

14. Final simplification 8.2%

    \[\leadsto \frac{\frac{\frac{0.25}{r}}{s}}{\pi} \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024060 
(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))))