Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.6%

Time: 16.5s

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, 1.3× speedup

Math

\[\begin{array}{l} \\ \left(\frac{0.16666666666666666}{\pi \cdot s} \cdot 0.75\right) \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{e^{\frac{r}{s \cdot -3}}}{r}\right) \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (*
  (* (/ 0.16666666666666666 (* PI s)) 0.75)
  (+ (/ (exp (/ r (- s))) r) (/ (exp (/ r (* s -3.0))) r))))

C

float code(float s, float r) {
	return ((0.16666666666666666f / (((float) M_PI) * s)) * 0.75f) * ((expf((r / -s)) / r) + (expf((r / (s * -3.0f))) / r));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.16666666666666666) / Float32(Float32(pi) * s)) * Float32(0.75)) * Float32(Float32(exp(Float32(r / Float32(-s))) / r) + Float32(exp(Float32(r / Float32(s * Float32(-3.0)))) / r)))
end

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.16666666666666666) / (single(pi) * s)) * single(0.75)) * ((exp((r / -s)) / r) + (exp((r / (s * single(-3.0)))) / r));
end

Derivation

1. Initial program 99.5%

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

3. Simplified 99.1%

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

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. metadata-eval 99.5%

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

   2. times-frac 99.5%

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

   3. neg-mul-1 99.5%

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

   4. frac-2neg 99.5%

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

   5. remove-double-neg 99.5%

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

   6. *-commutative 99.5%

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

   7. distribute-rgt-neg-in 99.5%

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

   8. metadata-eval 99.5%

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

6. Applied egg-rr 99.5%

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

   1. associate-/r* 99.6%

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

   2. metadata-eval 99.6%

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

   3. associate-/r* 99.6%

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

   4. div-inv 99.5%

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

   5. *-commutative 99.5%

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

   6. associate-/r* 99.6%

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

   7. metadata-eval 99.6%

      \[\leadsto \left(\frac{\color{blue}{0.16666666666666666}}{\pi \cdot s} \cdot 0.75\right) \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{e^{\frac{r}{s \cdot -3}}}{r}\right) \]

8. Applied egg-rr 99.6%

   \[\leadsto \color{blue}{\left(\frac{0.16666666666666666}{\pi \cdot s} \cdot 0.75\right)} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{e^{\frac{r}{s \cdot -3}}}{r}\right) \]

9. Final simplification 99.6%

   \[\leadsto \left(\frac{0.16666666666666666}{\pi \cdot s} \cdot 0.75\right) \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{e^{\frac{r}{s \cdot -3}}}{r}\right) \]

Alternative 2: 99.5% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Simplified 99.1%

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

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. Taylor expanded in s around 0 99.5%

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

6. Final simplification 99.5%

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

Alternative 3: 99.6% accurate, 1.3× speedup

Math

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

FPCore

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

C

float code(float s, float r) {
	return ((expf((r / -s)) / r) + (expf((r / (s * -3.0f))) / 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(r / Float32(s * Float32(-3.0)))) / r)) * Float32(Float32(0.125) / Float32(Float32(pi) * s)))
end

MATLAB

function tmp = code(s, r)
	tmp = ((exp((r / -s)) / r) + (exp((r / (s * single(-3.0)))) / r)) * (single(0.125) / (single(pi) * s));
end

Derivation

1. Initial program 99.5%

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

3. Simplified 99.1%

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

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. metadata-eval 99.5%

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

   2. times-frac 99.5%

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

   3. neg-mul-1 99.5%

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

   4. frac-2neg 99.5%

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

   5. remove-double-neg 99.5%

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

   6. *-commutative 99.5%

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

   7. distribute-rgt-neg-in 99.5%

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

   8. metadata-eval 99.5%

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

6. Applied egg-rr 99.5%

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

7. Taylor expanded in s around 0 99.6%

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

8. Final simplification 99.6%

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

Alternative 4: 12.1% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

function code(s, r)
	return Float32(Float32(0.25) / log1p(expm1(Float32(r * Float32(Float32(pi) * s)))))
end

Derivation

1. Initial program 99.5%

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

3. Simplified 99.1%

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

4. Taylor expanded in r around 0 8.5%

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

5. Taylor expanded in s around inf 8.1%

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

   1. log1p-expm1-u 14.1%

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

7. Applied egg-rr 14.1%

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

8. Final simplification 14.1%

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

Alternative 5: 9.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Simplified 99.1%

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

4. Taylor expanded in r around 0 8.5%

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

   1. associate-/r* 8.5%

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

   2. metadata-eval 8.5%

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

   3. associate-/r* 8.5%

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

   4. frac-2neg 8.5%

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

   5. div-inv 8.5%

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

   6. metadata-eval 8.5%

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

   7. *-commutative 8.5%

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

   8. distribute-rgt-neg-in 8.5%

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

   9. metadata-eval 8.5%

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

6. Applied egg-rr 8.5%

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

7. Final simplification 8.5%

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

Alternative 6: 9.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Simplified 99.1%

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

4. Taylor expanded in r around 0 8.5%

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

5. Taylor expanded in s around 0 8.5%

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

   1. associate-*r/ 8.5%

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

   2. neg-mul-1 8.5%

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

7. Simplified 8.5%

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

8. Taylor expanded in r around inf 8.5%

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

   1. associate-*r/ 8.5%

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

   2. mul-1-neg 8.5%

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

   3. *-commutative 8.5%

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

   4. *-commutative 8.5%

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

   5. associate-*l* 8.5%

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

   6. *-commutative 8.5%

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

10. Simplified 8.5%

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

11. Final simplification 8.5%

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

Alternative 7: 9.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Simplified 99.1%

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

4. Taylor expanded in r around 0 8.5%

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

5. Taylor expanded in s around 0 8.5%

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

   1. associate-*r/ 8.5%

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

   2. *-commutative 8.5%

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

   3. times-frac 8.5%

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

   4. associate-*r/ 8.5%

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

   5. neg-mul-1 8.5%

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

7. Simplified 8.5%

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

8. Taylor expanded in r around inf 8.5%

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

   1. mul-1-neg 8.5%

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

   2. distribute-neg-frac 8.5%

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

10. Simplified 8.5%

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

11. Final simplification 8.5%

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

Alternative 8: 9.0% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \frac{0.125}{\pi} \cdot \frac{2}{s \cdot r} \end{array} \]

FPCore

(FPCore (s r) :precision binary32 (* (/ 0.125 PI) (/ 2.0 (* s r))))

C

float code(float s, float r) {
	return (0.125f / ((float) M_PI)) * (2.0f / (s * r));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(0.125) / Float32(pi)) * Float32(Float32(2.0) / Float32(s * r)))
end

MATLAB

function tmp = code(s, r)
	tmp = (single(0.125) / single(pi)) * (single(2.0) / (s * r));
end

Derivation

1. Initial program 99.5%

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

3. Simplified 99.1%

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

4. Taylor expanded in r around 0 8.5%

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

5. Taylor expanded in s around 0 8.5%

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

   1. associate-*r/ 8.5%

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

   2. *-commutative 8.5%

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

   3. times-frac 8.5%

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

   4. associate-*r/ 8.5%

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

   5. neg-mul-1 8.5%

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

7. Simplified 8.5%

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

8. Taylor expanded in r around 0 8.1%

   \[\leadsto \frac{0.125}{\pi} \cdot \color{blue}{\frac{2}{r \cdot s}} \]

9. Final simplification 8.1%

   \[\leadsto \frac{0.125}{\pi} \cdot \frac{2}{s \cdot r} \]

Alternative 9: 9.0% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = (single(0.25) / single(pi)) * ((single(1.0) / r) / s);
end

Derivation

1. Initial program 99.5%

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

3. Simplified 99.1%

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

4. Taylor expanded in r around 0 8.5%

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

5. Taylor expanded in s around inf 8.1%

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

   1. associate-/r* 8.1%

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

7. Simplified 8.1%

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

   1. div-inv 8.1%

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

   2. *-commutative 8.1%

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

   3. times-frac 8.1%

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

9. Applied egg-rr 8.1%

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

10. Final simplification 8.1%

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

Alternative 10: 9.0% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = (single(1.0) / single(pi)) * (single(0.25) / (s * r));
end

Derivation

1. Initial program 99.5%

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

3. Simplified 99.1%

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

4. Taylor expanded in r around 0 8.5%

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

5. Taylor expanded in s around inf 8.1%

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

   1. associate-/r* 8.1%

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

7. Simplified 8.1%

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

   1. *-un-lft-identity 8.1%

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

   2. *-commutative 8.1%

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

   3. times-frac 8.1%

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

   4. associate-/l/ 8.1%

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

9. Applied egg-rr 8.1%

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

10. Final simplification 8.1%

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

Alternative 11: 9.0% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Simplified 99.1%

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

4. Taylor expanded in r around 0 8.5%

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

5. Taylor expanded in s around inf 8.1%

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

6. Final simplification 8.1%

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

Alternative 12: 9.0% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Simplified 99.1%

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

4. Taylor expanded in r around 0 8.5%

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

5. Taylor expanded in s around inf 8.1%

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

   1. associate-/r* 8.1%

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

7. Simplified 8.1%

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

8. Taylor expanded in r around 0 8.1%

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

   1. *-commutative 8.1%

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

   2. *-commutative 8.1%

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

   3. associate-*l* 8.1%

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

   4. *-commutative 8.1%

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

10. Simplified 8.1%

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

11. Final simplification 8.1%

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

Reproduce

herbie shell --seed 2023313 
(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))))