Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.5% → 99.5%

Time: 18.6s

Alternatives: 10

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: 99.5% 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: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{0.25 \cdot e^{\frac{-r}{s}}}{r \cdot \left(s \cdot \left(2 \cdot \pi\right)\right)} + \frac{0.75 \cdot e^{\frac{-0.3333333333333333}{\frac{s}{r}}}}{s \cdot \left(r \cdot \left(\pi \cdot 6\right)\right)} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/ (* 0.25 (exp (/ (- r) s))) (* r (* s (* 2.0 PI))))
  (/ (* 0.75 (exp (/ -0.3333333333333333 (/ s r)))) (* s (* r (* PI 6.0))))))

C

float code(float s, float r) {
	return ((0.25f * expf((-r / s))) / (r * (s * (2.0f * ((float) M_PI))))) + ((0.75f * expf((-0.3333333333333333f / (s / r)))) / (s * (r * (((float) M_PI) * 6.0f))));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.25) * exp(Float32(Float32(-r) / s))) / Float32(r * Float32(s * Float32(Float32(2.0) * Float32(pi))))) + Float32(Float32(Float32(0.75) * exp(Float32(Float32(-0.3333333333333333) / Float32(s / r)))) / Float32(s * Float32(r * Float32(Float32(pi) * Float32(6.0))))))
end

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.25) * exp((-r / s))) / (r * (s * (single(2.0) * single(pi))))) + ((single(0.75) * exp((single(-0.3333333333333333) / (s / r)))) / (s * (r * (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. Taylor expanded in r around 0 99.6%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\color{blue}{-0.3333333333333333 \cdot \frac{r}{s}}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
3. Step-by-step derivation

   1. associate-*r/ 99.6%

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\color{blue}{\frac{-0.3333333333333333 \cdot r}{s}}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   2. associate-/l* 99.7%

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\color{blue}{\frac{-0.3333333333333333}{\frac{s}{r}}}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

4. Simplified 99.7%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\color{blue}{\frac{-0.3333333333333333}{\frac{s}{r}}}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

5. Taylor expanded in s around 0 99.6%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-0.3333333333333333}{\frac{s}{r}}}}{\color{blue}{6 \cdot \left(r \cdot \left(s \cdot \pi\right)\right)}} \]
6. Step-by-step derivation

   1. *-commutative 99.6%

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-0.3333333333333333}{\frac{s}{r}}}}{\color{blue}{\left(r \cdot \left(s \cdot \pi\right)\right) \cdot 6}} \]

   2. associate-*r* 99.6%

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-0.3333333333333333}{\frac{s}{r}}}}{\color{blue}{\left(\left(r \cdot s\right) \cdot \pi\right)} \cdot 6} \]

   3. *-commutative 99.6%

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-0.3333333333333333}{\frac{s}{r}}}}{\left(\color{blue}{\left(s \cdot r\right)} \cdot \pi\right) \cdot 6} \]

   4. associate-*r* 99.7%

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-0.3333333333333333}{\frac{s}{r}}}}{\color{blue}{\left(s \cdot r\right) \cdot \left(\pi \cdot 6\right)}} \]

   5. associate-*l* 99.7%

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-0.3333333333333333}{\frac{s}{r}}}}{\color{blue}{s \cdot \left(r \cdot \left(\pi \cdot 6\right)\right)}} \]

7. Simplified 99.7%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-0.3333333333333333}{\frac{s}{r}}}}{\color{blue}{s \cdot \left(r \cdot \left(\pi \cdot 6\right)\right)}} \]

8. Final simplification 99.7%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{r \cdot \left(s \cdot \left(2 \cdot \pi\right)\right)} + \frac{0.75 \cdot e^{\frac{-0.3333333333333333}{\frac{s}{r}}}}{s \cdot \left(r \cdot \left(\pi \cdot 6\right)\right)} \]

Alternative 2: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{0.25 \cdot e^{\frac{-r}{s}}}{r \cdot \left(s \cdot \left(2 \cdot \pi\right)\right)} + \frac{0.75 \cdot e^{r \cdot \frac{-0.3333333333333333}{s}}}{r \cdot \left(s \cdot \left(\pi \cdot 6\right)\right)} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/ (* 0.25 (exp (/ (- r) s))) (* r (* s (* 2.0 PI))))
  (/ (* 0.75 (exp (* r (/ -0.3333333333333333 s)))) (* r (* s (* PI 6.0))))))

C

float code(float s, float r) {
	return ((0.25f * expf((-r / s))) / (r * (s * (2.0f * ((float) M_PI))))) + ((0.75f * expf((r * (-0.3333333333333333f / s)))) / (r * (s * (((float) M_PI) * 6.0f))));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.25) * exp(Float32(Float32(-r) / s))) / Float32(r * Float32(s * Float32(Float32(2.0) * Float32(pi))))) + Float32(Float32(Float32(0.75) * exp(Float32(r * Float32(Float32(-0.3333333333333333) / s)))) / Float32(r * Float32(s * Float32(Float32(pi) * Float32(6.0))))))
end

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.25) * exp((-r / s))) / (r * (s * (single(2.0) * single(pi))))) + ((single(0.75) * exp((r * (single(-0.3333333333333333) / s)))) / (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. Taylor expanded in r around 0 99.6%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\color{blue}{-0.3333333333333333 \cdot \frac{r}{s}}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
3. Step-by-step derivation

   1. associate-*r/ 99.6%

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\color{blue}{\frac{-0.3333333333333333 \cdot r}{s}}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   2. associate-/l* 99.7%

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\color{blue}{\frac{-0.3333333333333333}{\frac{s}{r}}}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

4. Simplified 99.7%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\color{blue}{\frac{-0.3333333333333333}{\frac{s}{r}}}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
5. Step-by-step derivation

   1. associate-/r/ 99.6%

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\color{blue}{\frac{-0.3333333333333333}{s} \cdot r}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

6. Applied egg-rr 99.6%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\color{blue}{\frac{-0.3333333333333333}{s} \cdot r}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

7. Final simplification 99.6%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{r \cdot \left(s \cdot \left(2 \cdot \pi\right)\right)} + \frac{0.75 \cdot e^{r \cdot \frac{-0.3333333333333333}{s}}}{r \cdot \left(s \cdot \left(\pi \cdot 6\right)\right)} \]

Alternative 3: 99.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \left(0.75 \cdot \frac{1}{s \cdot \left(\pi \cdot 6\right)}\right) \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{e^{\frac{-0.3333333333333333}{\frac{s}{r}}}}{r}\right) \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (*
  (* 0.75 (/ 1.0 (* s (* PI 6.0))))
  (+ (/ (exp (/ r (- s))) r) (/ (exp (/ -0.3333333333333333 (/ s r))) r))))

C

float code(float s, float r) {
	return (0.75f * (1.0f / (s * (((float) M_PI) * 6.0f)))) * ((expf((r / -s)) / r) + (expf((-0.3333333333333333f / (s / r))) / r));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(0.75) * Float32(Float32(1.0) / Float32(s * Float32(Float32(pi) * Float32(6.0))))) * Float32(Float32(exp(Float32(r / Float32(-s))) / r) + Float32(exp(Float32(Float32(-0.3333333333333333) / Float32(s / r))) / r)))
end

MATLAB

function tmp = code(s, r)
	tmp = (single(0.75) * (single(1.0) / (s * (single(pi) * single(6.0))))) * ((exp((r / -s)) / r) + (exp((single(-0.3333333333333333) / (s / r))) / 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%

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

   1. associate-/l/ 99.4%

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

   2. metadata-eval 99.4%

      \[\leadsto \frac{\color{blue}{\frac{0.75}{6}}}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{{\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}}{r}\right) \]

   3. *-commutative 99.4%

      \[\leadsto \frac{\frac{0.75}{6}}{\color{blue}{\pi \cdot s}} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{{\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}}{r}\right) \]

   4. associate-/r* 99.3%

      \[\leadsto \color{blue}{\frac{0.75}{6 \cdot \left(\pi \cdot s\right)}} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{{\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}}{r}\right) \]

   5. div-inv 99.4%

      \[\leadsto \color{blue}{\left(0.75 \cdot \frac{1}{6 \cdot \left(\pi \cdot s\right)}\right)} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{{\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}}{r}\right) \]

   6. associate-*r* 99.4%

      \[\leadsto \left(0.75 \cdot \frac{1}{\color{blue}{\left(6 \cdot \pi\right) \cdot s}}\right) \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{{\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}}{r}\right) \]

   7. *-commutative 99.4%

      \[\leadsto \left(0.75 \cdot \frac{1}{\color{blue}{s \cdot \left(6 \cdot \pi\right)}}\right) \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{{\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}}{r}\right) \]

   8. *-commutative 99.4%

      \[\leadsto \left(0.75 \cdot \frac{1}{s \cdot \color{blue}{\left(\pi \cdot 6\right)}}\right) \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{{\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}}{r}\right) \]

5. Applied egg-rr 99.4%

   \[\leadsto \color{blue}{\left(0.75 \cdot \frac{1}{s \cdot \left(\pi \cdot 6\right)}\right)} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{{\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}}{r}\right) \]

6. Taylor expanded in r around inf 99.6%

   \[\leadsto \left(0.75 \cdot \frac{1}{s \cdot \left(\pi \cdot 6\right)}\right) \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\color{blue}{e^{-0.3333333333333333 \cdot \frac{r}{s}}}}{r}\right) \]
7. Step-by-step derivation

   1. associate-*r/ 99.6%

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

   2. associate-/l* 99.6%

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

8. Simplified 99.6%

   \[\leadsto \left(0.75 \cdot \frac{1}{s \cdot \left(\pi \cdot 6\right)}\right) \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\color{blue}{e^{\frac{-0.3333333333333333}{\frac{s}{r}}}}}{r}\right) \]

9. Final simplification 99.6%

   \[\leadsto \left(0.75 \cdot \frac{1}{s \cdot \left(\pi \cdot 6\right)}\right) \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{e^{\frac{-0.3333333333333333}{\frac{s}{r}}}}{r}\right) \]

Alternative 4: 99.4% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Simplified 99.4%

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

4. Taylor expanded in r around inf 99.5%

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

   1. associate-*r/ 99.6%

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

   2. associate-/l* 99.6%

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

6. Simplified 99.6%

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

7. Final simplification 99.6%

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

Alternative 5: 11.8% accurate, 1.4× 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%

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

4. Taylor expanded in r around 0 9.2%

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

5. Taylor expanded in s around inf 8.8%

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

   1. log1p-expm1-u 13.7%

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

7. Applied egg-rr 13.7%

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

8. Final simplification 13.7%

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

Alternative 6: 9.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = (single(0.125) / single(pi)) * ((exp((-r / s)) + single(1.0)) / (r * s));
end

Derivation

1. Initial program 99.6%

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

3. Simplified 99.4%

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

4. Taylor expanded in r around 0 9.2%

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

5. Taylor expanded in s around 0 9.2%

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

   1. associate-*r/ 9.2%

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

   2. *-commutative 9.2%

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

   3. times-frac 9.2%

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

   4. metadata-eval 9.2%

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

   5. times-frac 9.2%

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

   6. *-lft-identity 9.2%

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

   7. neg-mul-1 9.2%

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

7. Simplified 9.2%

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

8. Taylor expanded in r around inf 9.2%

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

   1. associate-*r/ 9.2%

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

   2. *-commutative 9.2%

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

   3. *-commutative 9.2%

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

   4. associate-*r* 9.2%

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

   5. times-frac 9.2%

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

   6. *-commutative 9.2%

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

   7. associate-*r/ 9.2%

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

   8. neg-mul-1 9.2%

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

10. Simplified 9.2%

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

11. Final simplification 9.2%

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

Alternative 7: 9.5% accurate, 2.0× speedup

Math

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

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(Float32(0.125) * Float32(exp(Float32(Float32(-r) / 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%

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

4. Taylor expanded in r around 0 9.2%

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

5. Taylor expanded in r around inf 9.2%

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

   1. associate-*r/ 9.2%

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

   2. associate-*r/ 9.2%

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

   3. neg-mul-1 9.2%

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

7. Simplified 9.2%

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

8. Final simplification 9.2%

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

Alternative 8: 9.1% accurate, 4.0× 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%

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

4. Taylor expanded in r around 0 9.2%

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

5. Taylor expanded in s around inf 8.8%

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

6. Final simplification 8.8%

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

Alternative 9: 9.1% accurate, 4.0× 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%

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

4. Taylor expanded in r around 0 9.2%

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

5. Taylor expanded in s around inf 8.8%

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

   1. associate-/r* 8.8%

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

7. Simplified 8.8%

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

8. Final simplification 8.8%

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

Alternative 10: 9.1% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Simplified 99.4%

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

4. Taylor expanded in r around 0 9.2%

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

5. Taylor expanded in s around inf 8.8%

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

   1. associate-*r* 8.8%

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

   2. *-commutative 8.8%

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

7. Simplified 8.8%

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

8. Taylor expanded in r around 0 8.8%

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

   1. *-commutative 8.8%

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

   2. associate-/r* 8.8%

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

   3. associate-/l/ 8.8%

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

   4. associate-/l/ 8.8%

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

10. Simplified 8.8%

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

11. Final simplification 8.8%

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

Reproduce

herbie shell --seed 2023287 
(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))))