Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.6%

Time: 16.2s

Alternatives: 12

Speedup: N/A×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(s, r)
	return Float32(Float32(0.125) * Float32(Float32(Float32(exp(Float32(-Float32(r / s))) / r) + Float32((Float32(exp(1)) ^ Float32(r * Float32(Float32(-0.3333333333333333) / s))) / r)) / Float32(s * Float32(pi))))
end

MATLAB

function tmp = code(s, r)
	tmp = single(0.125) * (((exp(-(r / s)) / r) + ((single(2.71828182845904523536) ^ (r * (single(-0.3333333333333333) / s))) / 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 98.9%

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

5. Taylor expanded in s around 0 99.5%

   \[\leadsto \color{blue}{0.125 \cdot \frac{\frac{e^{-1 \cdot \frac{r}{s}}}{r} + \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r}}{s \cdot \pi}} \]
6. Step-by-step derivation

   1. *-un-lft-identity 99.5%

      \[\leadsto 0.125 \cdot \frac{\frac{e^{-1 \cdot \frac{r}{s}}}{r} + \frac{e^{\color{blue}{1 \cdot \left(-0.3333333333333333 \cdot \frac{r}{s}\right)}}}{r}}{s \cdot \pi} \]

   2. exp-prod 99.6%

      \[\leadsto 0.125 \cdot \frac{\frac{e^{-1 \cdot \frac{r}{s}}}{r} + \frac{\color{blue}{{\left(e^{1}\right)}^{\left(-0.3333333333333333 \cdot \frac{r}{s}\right)}}}{r}}{s \cdot \pi} \]

   3. associate-*r/ 99.6%

      \[\leadsto 0.125 \cdot \frac{\frac{e^{-1 \cdot \frac{r}{s}}}{r} + \frac{{\left(e^{1}\right)}^{\color{blue}{\left(\frac{-0.3333333333333333 \cdot r}{s}\right)}}}{r}}{s \cdot \pi} \]

   4. associate-*l/ 99.6%

      \[\leadsto 0.125 \cdot \frac{\frac{e^{-1 \cdot \frac{r}{s}}}{r} + \frac{{\left(e^{1}\right)}^{\color{blue}{\left(\frac{-0.3333333333333333}{s} \cdot r\right)}}}{r}}{s \cdot \pi} \]

   5. *-commutative 99.6%

      \[\leadsto 0.125 \cdot \frac{\frac{e^{-1 \cdot \frac{r}{s}}}{r} + \frac{{\left(e^{1}\right)}^{\color{blue}{\left(r \cdot \frac{-0.3333333333333333}{s}\right)}}}{r}}{s \cdot \pi} \]

7. Applied egg-rr 99.6%

   \[\leadsto 0.125 \cdot \frac{\frac{e^{-1 \cdot \frac{r}{s}}}{r} + \frac{\color{blue}{{\left(e^{1}\right)}^{\left(r \cdot \frac{-0.3333333333333333}{s}\right)}}}{r}}{s \cdot \pi} \]

8. Final simplification 99.6%

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

Alternative 2: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

float code(float s, float r) {
	return (0.125f / (s * ((float) M_PI))) * ((expf(-(r / s)) + 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(-Float32(r / s))) + exp(Float32(r * Float32(Float32(-0.3333333333333333) / s)))) / r))
end

MATLAB

function tmp = code(s, r)
	tmp = (single(0.125) / (s * single(pi))) * ((exp(-(r / s)) + 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 98.9%

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

5. Taylor expanded in r around inf 99.5%

   \[\leadsto \color{blue}{0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}} + e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r \cdot \left(s \cdot \pi\right)}} \]
6. Step-by-step derivation

   1. associate-*r/ 99.5%

      \[\leadsto \color{blue}{\frac{0.125 \cdot \left(e^{-1 \cdot \frac{r}{s}} + e^{-0.3333333333333333 \cdot \frac{r}{s}}\right)}{r \cdot \left(s \cdot \pi\right)}} \]

   2. *-commutative 99.5%

      \[\leadsto \frac{0.125 \cdot \left(e^{-1 \cdot \frac{r}{s}} + e^{-0.3333333333333333 \cdot \frac{r}{s}}\right)}{\color{blue}{\left(s \cdot \pi\right) \cdot r}} \]

   3. times-frac 99.1%

      \[\leadsto \color{blue}{\frac{0.125}{s \cdot \pi} \cdot \frac{e^{-1 \cdot \frac{r}{s}} + e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r}} \]

   4. associate-/r* 99.1%

      \[\leadsto \color{blue}{\frac{\frac{0.125}{s}}{\pi}} \cdot \frac{e^{-1 \cdot \frac{r}{s}} + e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r} \]

   5. mul-1-neg 99.1%

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

   6. distribute-neg-frac2 99.1%

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

   7. associate-*r/ 99.1%

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

   8. associate-*l/ 99.1%

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

7. Simplified 99.1%

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

8. Taylor expanded in s around 0 99.1%

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

9. Final simplification 99.1%

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

Alternative 3: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = single(0.125) * ((exp(-(r / s)) + exp(((r / s) * single(-0.3333333333333333)))) / (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 98.9%

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

5. Taylor expanded in r around inf 99.5%

   \[\leadsto \color{blue}{0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}} + e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r \cdot \left(s \cdot \pi\right)}} \]

6. Final simplification 99.5%

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

Alternative 4: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = single(0.125) * ((exp(-(r / s)) + exp((single(-0.3333333333333333) / (s / r)))) / (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 98.9%

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

5. Taylor expanded in r around inf 99.5%

   \[\leadsto \color{blue}{0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}} + e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r \cdot \left(s \cdot \pi\right)}} \]
6. Step-by-step derivation

   1. rem-log-exp 99.3%

      \[\leadsto 0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}} + e^{\color{blue}{\log \left(e^{-0.3333333333333333}\right)} \cdot \frac{r}{s}}}{r \cdot \left(s \cdot \pi\right)} \]

   2. clear-num 99.4%

      \[\leadsto 0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}} + e^{\log \left(e^{-0.3333333333333333}\right) \cdot \color{blue}{\frac{1}{\frac{s}{r}}}}}{r \cdot \left(s \cdot \pi\right)} \]

   3. un-div-inv 99.3%

      \[\leadsto 0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}} + e^{\color{blue}{\frac{\log \left(e^{-0.3333333333333333}\right)}{\frac{s}{r}}}}}{r \cdot \left(s \cdot \pi\right)} \]

   4. rem-log-exp 99.6%

      \[\leadsto 0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}} + e^{\frac{\color{blue}{-0.3333333333333333}}{\frac{s}{r}}}}{r \cdot \left(s \cdot \pi\right)} \]

7. Applied egg-rr 99.6%

   \[\leadsto 0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}} + e^{\color{blue}{\frac{-0.3333333333333333}{\frac{s}{r}}}}}{r \cdot \left(s \cdot \pi\right)} \]

8. Final simplification 99.6%

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

Alternative 5: 11.8% 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 98.9%

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

5. Taylor expanded in s around inf 11.1%

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

   1. log1p-expm1-u 13.4%

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

7. Applied egg-rr 13.4%

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

8. Final simplification 13.4%

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

Alternative 6: 43.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

function code(s, r)
	return Float32(Float32(0.25) / Float32(s * log1p(expm1(Float32(r * 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 98.9%

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

5. Taylor expanded in s around 0 99.5%

   \[\leadsto \color{blue}{0.125 \cdot \frac{\frac{e^{-1 \cdot \frac{r}{s}}}{r} + \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r}}{s \cdot \pi}} \]
6. Step-by-step derivation

   1. associate-*r/ 99.5%

      \[\leadsto \color{blue}{\frac{0.125 \cdot \left(\frac{e^{-1 \cdot \frac{r}{s}}}{r} + \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r}\right)}{s \cdot \pi}} \]

   2. times-frac 99.0%

      \[\leadsto \color{blue}{\frac{0.125}{s} \cdot \frac{\frac{e^{-1 \cdot \frac{r}{s}}}{r} + \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r}}{\pi}} \]

   3. +-commutative 99.0%

      \[\leadsto \frac{0.125}{s} \cdot \frac{\color{blue}{\frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r} + \frac{e^{-1 \cdot \frac{r}{s}}}{r}}}{\pi} \]

   4. associate-*r/ 99.1%

      \[\leadsto \frac{0.125}{s} \cdot \frac{\frac{e^{\color{blue}{\frac{-0.3333333333333333 \cdot r}{s}}}}{r} + \frac{e^{-1 \cdot \frac{r}{s}}}{r}}{\pi} \]

   5. associate-*l/ 99.0%

      \[\leadsto \frac{0.125}{s} \cdot \frac{\frac{e^{\color{blue}{\frac{-0.3333333333333333}{s} \cdot r}}}{r} + \frac{e^{-1 \cdot \frac{r}{s}}}{r}}{\pi} \]

   6. mul-1-neg 99.0%

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

   7. distribute-neg-frac2 99.0%

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

7. Simplified 99.0%

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

8. Taylor expanded in s around inf 11.1%

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

   1. associate-/r* 11.1%

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

10. Simplified 11.1%

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

    1. *-un-lft-identity 11.1%

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

    2. *-commutative 11.1%

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

    3. associate-/l/ 11.1%

       \[\leadsto \color{blue}{\frac{0.25}{\left(s \cdot \pi\right) \cdot r}} \cdot 1 \]

    4. associate-*l* 11.1%

       \[\leadsto \frac{0.25}{\color{blue}{s \cdot \left(\pi \cdot r\right)}} \cdot 1 \]

12. Applied egg-rr 11.1%

    \[\leadsto \color{blue}{\frac{0.25}{s \cdot \left(\pi \cdot r\right)} \cdot 1} \]
13. Step-by-step derivation

    1. log1p-expm1-u 46.9%

       \[\leadsto \frac{0.25}{s \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\pi \cdot r\right)\right)}} \cdot 1 \]

    2. *-commutative 46.9%

       \[\leadsto \frac{0.25}{s \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{r \cdot \pi}\right)\right)} \cdot 1 \]

14. Applied egg-rr 46.9%

    \[\leadsto \frac{0.25}{s \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(r \cdot \pi\right)\right)}} \cdot 1 \]

15. Final simplification 46.9%

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

Alternative 7: 9.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ 0.125 \cdot \frac{\frac{e^{-\frac{r}{s}}}{r} + \frac{1 + \frac{r \cdot -0.3333333333333333}{s}}{r}}{s \cdot \pi} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (*
  0.125
  (/
   (+ (/ (exp (- (/ r s))) r) (/ (+ 1.0 (/ (* r -0.3333333333333333) s)) r))
   (* s PI))))

C

float code(float s, float r) {
	return 0.125f * (((expf(-(r / s)) / r) + ((1.0f + ((r * -0.3333333333333333f) / s)) / r)) / (s * ((float) M_PI)));
}

Julia

function code(s, r)
	return Float32(Float32(0.125) * Float32(Float32(Float32(exp(Float32(-Float32(r / s))) / r) + Float32(Float32(Float32(1.0) + Float32(Float32(r * Float32(-0.3333333333333333)) / s)) / r)) / Float32(s * Float32(pi))))
end

MATLAB

function tmp = code(s, r)
	tmp = single(0.125) * (((exp(-(r / s)) / r) + ((single(1.0) + ((r * single(-0.3333333333333333)) / s)) / 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 98.9%

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

5. Taylor expanded in s around 0 99.5%

   \[\leadsto \color{blue}{0.125 \cdot \frac{\frac{e^{-1 \cdot \frac{r}{s}}}{r} + \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r}}{s \cdot \pi}} \]

6. Taylor expanded in r around 0 12.1%

   \[\leadsto 0.125 \cdot \frac{\frac{e^{-1 \cdot \frac{r}{s}}}{r} + \frac{\color{blue}{1 + -0.3333333333333333 \cdot \frac{r}{s}}}{r}}{s \cdot \pi} \]
7. Step-by-step derivation

   1. associate-*r/ 11.3%

      \[\leadsto 0.125 \cdot \frac{\left(1 - \frac{r}{s}\right) + \left(1 + \color{blue}{\frac{-0.3333333333333333 \cdot r}{s}}\right)}{r \cdot \left(s \cdot \pi\right)} \]

8. Simplified 12.1%

   \[\leadsto 0.125 \cdot \frac{\frac{e^{-1 \cdot \frac{r}{s}}}{r} + \frac{\color{blue}{1 + \frac{-0.3333333333333333 \cdot r}{s}}}{r}}{s \cdot \pi} \]

9. Final simplification 12.1%

   \[\leadsto 0.125 \cdot \frac{\frac{e^{-\frac{r}{s}}}{r} + \frac{1 + \frac{r \cdot -0.3333333333333333}{s}}{r}}{s \cdot \pi} \]
10. Add Preprocessing

Alternative 8: 9.5% 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(-Float32(r / 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 98.9%

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

5. Taylor expanded in s around 0 99.5%

   \[\leadsto \color{blue}{0.125 \cdot \frac{\frac{e^{-1 \cdot \frac{r}{s}}}{r} + \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r}}{s \cdot \pi}} \]

6. Taylor expanded in r around 0 11.8%

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

7. Final simplification 11.8%

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

Alternative 9: 9.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ 0.125 \cdot \frac{1 + e^{\frac{r}{s} \cdot -0.3333333333333333}}{r \cdot \left(s \cdot \pi\right)} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (* 0.125 (/ (+ 1.0 (exp (* (/ r s) -0.3333333333333333))) (* r (* s PI)))))

C

float code(float s, float r) {
	return 0.125f * ((1.0f + expf(((r / s) * -0.3333333333333333f))) / (r * (s * ((float) M_PI))));
}

Julia

function code(s, r)
	return Float32(Float32(0.125) * Float32(Float32(Float32(1.0) + exp(Float32(Float32(r / s) * Float32(-0.3333333333333333)))) / Float32(r * Float32(s * Float32(pi)))))
end

MATLAB

function tmp = code(s, r)
	tmp = single(0.125) * ((single(1.0) + exp(((r / s) * single(-0.3333333333333333)))) / (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 98.9%

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

5. Taylor expanded in r around inf 99.5%

   \[\leadsto \color{blue}{0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}} + e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r \cdot \left(s \cdot \pi\right)}} \]

6. Taylor expanded in r around 0 11.3%

   \[\leadsto 0.125 \cdot \frac{\color{blue}{1} + e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r \cdot \left(s \cdot \pi\right)} \]

7. Final simplification 11.3%

   \[\leadsto 0.125 \cdot \frac{1 + e^{\frac{r}{s} \cdot -0.3333333333333333}}{r \cdot \left(s \cdot \pi\right)} \]
8. Add Preprocessing

Alternative 10: 9.0% accurate, 11.0× speedup

Math

\[\begin{array}{l} \\ 0.125 \cdot \frac{\left(1 + \frac{r \cdot -0.3333333333333333}{s}\right) + \left(1 - \frac{r}{s}\right)}{r \cdot \left(s \cdot \pi\right)} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (*
  0.125
  (/
   (+ (+ 1.0 (/ (* r -0.3333333333333333) s)) (- 1.0 (/ r s)))
   (* r (* s PI)))))

C

float code(float s, float r) {
	return 0.125f * (((1.0f + ((r * -0.3333333333333333f) / s)) + (1.0f - (r / s))) / (r * (s * ((float) M_PI))));
}

Julia

function code(s, r)
	return Float32(Float32(0.125) * Float32(Float32(Float32(Float32(1.0) + Float32(Float32(r * Float32(-0.3333333333333333)) / s)) + Float32(Float32(1.0) - Float32(r / s))) / Float32(r * Float32(s * Float32(pi)))))
end

MATLAB

function tmp = code(s, r)
	tmp = single(0.125) * (((single(1.0) + ((r * single(-0.3333333333333333)) / s)) + (single(1.0) - (r / s))) / (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 98.9%

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

5. Taylor expanded in r around inf 99.5%

   \[\leadsto \color{blue}{0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}} + e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r \cdot \left(s \cdot \pi\right)}} \]

6. Taylor expanded in r around 0 11.4%

   \[\leadsto 0.125 \cdot \frac{\color{blue}{\left(1 + -1 \cdot \frac{r}{s}\right)} + e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r \cdot \left(s \cdot \pi\right)} \]
7. Step-by-step derivation

   1. mul-1-neg 11.4%

      \[\leadsto 0.125 \cdot \frac{\left(1 + \color{blue}{\left(-\frac{r}{s}\right)}\right) + e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r \cdot \left(s \cdot \pi\right)} \]

   2. unsub-neg 11.4%

      \[\leadsto 0.125 \cdot \frac{\color{blue}{\left(1 - \frac{r}{s}\right)} + e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r \cdot \left(s \cdot \pi\right)} \]

8. Simplified 11.4%

   \[\leadsto 0.125 \cdot \frac{\color{blue}{\left(1 - \frac{r}{s}\right)} + e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r \cdot \left(s \cdot \pi\right)} \]

9. Taylor expanded in r around 0 11.3%

   \[\leadsto 0.125 \cdot \frac{\left(1 - \frac{r}{s}\right) + \color{blue}{\left(1 + -0.3333333333333333 \cdot \frac{r}{s}\right)}}{r \cdot \left(s \cdot \pi\right)} \]
10. Step-by-step derivation

    1. associate-*r/ 11.3%

       \[\leadsto 0.125 \cdot \frac{\left(1 - \frac{r}{s}\right) + \left(1 + \color{blue}{\frac{-0.3333333333333333 \cdot r}{s}}\right)}{r \cdot \left(s \cdot \pi\right)} \]

11. Simplified 11.3%

    \[\leadsto 0.125 \cdot \frac{\left(1 - \frac{r}{s}\right) + \color{blue}{\left(1 + \frac{-0.3333333333333333 \cdot r}{s}\right)}}{r \cdot \left(s \cdot \pi\right)} \]

12. Final simplification 11.3%

    \[\leadsto 0.125 \cdot \frac{\left(1 + \frac{r \cdot -0.3333333333333333}{s}\right) + \left(1 - \frac{r}{s}\right)}{r \cdot \left(s \cdot \pi\right)} \]
13. Add Preprocessing

Alternative 11: 9.0% 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 98.9%

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

5. Taylor expanded in s around inf 11.1%

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

6. Final simplification 11.1%

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

Alternative 12: 9.0% 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 98.9%

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

5. Taylor expanded in s around 0 99.5%

   \[\leadsto \color{blue}{0.125 \cdot \frac{\frac{e^{-1 \cdot \frac{r}{s}}}{r} + \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r}}{s \cdot \pi}} \]
6. Step-by-step derivation

   1. associate-*r/ 99.5%

      \[\leadsto \color{blue}{\frac{0.125 \cdot \left(\frac{e^{-1 \cdot \frac{r}{s}}}{r} + \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r}\right)}{s \cdot \pi}} \]

   2. times-frac 99.0%

      \[\leadsto \color{blue}{\frac{0.125}{s} \cdot \frac{\frac{e^{-1 \cdot \frac{r}{s}}}{r} + \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r}}{\pi}} \]

   3. +-commutative 99.0%

      \[\leadsto \frac{0.125}{s} \cdot \frac{\color{blue}{\frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r} + \frac{e^{-1 \cdot \frac{r}{s}}}{r}}}{\pi} \]

   4. associate-*r/ 99.1%

      \[\leadsto \frac{0.125}{s} \cdot \frac{\frac{e^{\color{blue}{\frac{-0.3333333333333333 \cdot r}{s}}}}{r} + \frac{e^{-1 \cdot \frac{r}{s}}}{r}}{\pi} \]

   5. associate-*l/ 99.0%

      \[\leadsto \frac{0.125}{s} \cdot \frac{\frac{e^{\color{blue}{\frac{-0.3333333333333333}{s} \cdot r}}}{r} + \frac{e^{-1 \cdot \frac{r}{s}}}{r}}{\pi} \]

   6. mul-1-neg 99.0%

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

   7. distribute-neg-frac2 99.0%

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

7. Simplified 99.0%

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

8. Taylor expanded in s around inf 11.1%

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

   1. associate-/r* 11.1%

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

10. Simplified 11.1%

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

11. Final simplification 11.1%

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

Reproduce

herbie shell --seed 2024040 
(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))))