Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.5% → 99.5%

Time: 20.4s

Alternatives: 18

Speedup: 1.0×

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

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

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/ (* 0.25 (exp (/ (- r) s))) (* r (* s (* 2.0 PI))))
  (/ (* 0.75 (exp (/ (- r) (* s 3.0)))) (* 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 / (s * 3.0f)))) / (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(Float32(-r) / Float32(s * Float32(3.0))))) / Float32(r * Float32(s * Float32(Float32(pi) * Float32(6.0))))))
end

MATLAB

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

Derivation

1. Initial program 99.6%

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

2. Final simplification 99.6%

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

Alternative 2: 99.5% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/ (* 0.25 (exp (/ (- r) s))) (* r (* s (* 2.0 PI))))
  (/ (* 0.75 (exp (/ (- r) (* s 3.0)))) (* 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((-r / (s * 3.0f)))) / (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(-r) / Float32(s * Float32(3.0))))) / 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((-r / (s * single(3.0))))) / (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 s around 0 99.5%

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

   1. *-commutative 99.5%

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

   2. associate-*r* 99.5%

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

   3. associate-*r* 99.6%

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

   4. associate-*l* 99.5%

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

4. Simplified 99.5%

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

5. Final simplification 99.5%

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

Alternative 3: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.25) / (single(2.0) * (s * single(pi)))) * (exp((-r / s)) / r)) + ((single(0.75) / (single(6.0) * (s * single(pi)))) * (exp((single(-0.3333333333333333) * (r / s))) / r));
end

Derivation

1. Initial program 99.6%

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

   1. times-frac 99.5%

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

   2. fma-def 99.5%

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

   3. /-rgt-identity 99.5%

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

   4. fma-def 99.5%

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

   5. /-rgt-identity 99.5%

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

   6. associate-*l* 99.5%

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

   7. times-frac 99.5%

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

3. Simplified 99.5%

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

   1. neg-mul-1 99.5%

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

   2. *-commutative 99.5%

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

   3. times-frac 99.5%

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

   4. metadata-eval 99.5%

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

   5. *-commutative 99.5%

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

5. Applied egg-rr 99.5%

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

6. Final simplification 99.5%

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

Alternative 4: 99.5% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/ (* 0.25 (exp (/ (- r) s))) (* r (* s (* 2.0 PI))))
  (/ (* 0.75 (exp (* r (/ -0.3333333333333333 s)))) (* 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((r * (-0.3333333333333333f / s)))) / (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(r * Float32(Float32(-0.3333333333333333) / s)))) / 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((r * (single(-0.3333333333333333) / s)))) / (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 s around 0 99.5%

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

   1. *-commutative 99.5%

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

   2. associate-*r* 99.5%

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

   3. associate-*r* 99.6%

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

   4. associate-*l* 99.5%

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

4. Simplified 99.5%

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

5. Taylor expanded in r around 0 99.5%

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

   1. *-commutative 99.5%

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

   2. metadata-eval 99.5%

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

   3. distribute-rgt-neg-in 99.5%

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

   4. metadata-eval 99.5%

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

   5. times-frac 99.5%

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

   6. associate-*r/ 99.5%

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

   7. distribute-rgt-neg-in 99.5%

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

   8. *-commutative 99.5%

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

   9. associate-/r* 99.5%

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

   10. metadata-eval 99.5%

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

   11. distribute-neg-frac 99.5%

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

   12. metadata-eval 99.5%

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

7. Simplified 99.5%

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

8. Final simplification 99.5%

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

Alternative 5: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.125}{\pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s \cdot 2}\right)}}{r}\right)}{s} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (/
  (*
   (/ 0.125 PI)
   (+
    (/ (exp (/ r (- s))) r)
    (/ (pow (exp -0.6666666666666666) (/ r (* s 2.0))) r)))
  s))

C

float code(float s, float r) {
	return ((0.125f / ((float) M_PI)) * ((expf((r / -s)) / r) + (powf(expf(-0.6666666666666666f), (r / (s * 2.0f))) / r))) / s;
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.125) / Float32(pi)) * Float32(Float32(exp(Float32(r / Float32(-s))) / r) + Float32((exp(Float32(-0.6666666666666666)) ^ Float32(r / Float32(s * Float32(2.0)))) / r))) / s)
end

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.125) / single(pi)) * ((exp((r / -s)) / r) + ((exp(single(-0.6666666666666666)) ^ (r / (s * single(2.0)))) / 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 99.1%

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

   1. add-sqr-sqrt 99.1%

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

   2. sqrt-unprod 98.7%

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

   3. pow-prod-down 98.7%

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

   4. prod-exp 98.9%

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

   5. metadata-eval 98.9%

      \[\leadsto \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\sqrt{{\left(e^{\color{blue}{-0.6666666666666666}}\right)}^{\left(\frac{r}{s}\right)}}}{r}\right) \]

5. Applied egg-rr 98.9%

   \[\leadsto \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\color{blue}{\sqrt{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s}\right)}}}}{r}\right) \]
6. Step-by-step derivation

   1. sqrt-pow1 99.4%

      \[\leadsto \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\color{blue}{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{\frac{r}{s}}{2}\right)}}}{r}\right) \]

7. Applied egg-rr 99.4%

   \[\leadsto \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\color{blue}{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{\frac{r}{s}}{2}\right)}}}{r}\right) \]
8. Step-by-step derivation

   1. associate-*l/ 99.4%

      \[\leadsto \color{blue}{\frac{\frac{0.125}{\pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{\frac{r}{s}}{2}\right)}}{r}\right)}{s}} \]

   2. associate-/l/ 99.4%

      \[\leadsto \frac{\frac{0.125}{\pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{{\left(e^{-0.6666666666666666}\right)}^{\color{blue}{\left(\frac{r}{2 \cdot s}\right)}}}{r}\right)}{s} \]

9. Applied egg-rr 99.4%

   \[\leadsto \color{blue}{\frac{\frac{0.125}{\pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{2 \cdot s}\right)}}{r}\right)}{s}} \]

10. Final simplification 99.4%

    \[\leadsto \frac{\frac{0.125}{\pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s \cdot 2}\right)}}{r}\right)}{s} \]

Alternative 6: 99.5% accurate, 1.3× speedup

Math

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

FPCore

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

C

float code(float s, float r) {
	return 0.125f * (((expf((-r / s)) / r) + (expf(((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(exp(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) + (exp(((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 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. Step-by-step derivation

   1. add-sqr-sqrt 99.1%

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

   2. sqrt-unprod 98.7%

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

   3. pow-prod-down 98.7%

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

   4. prod-exp 98.9%

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

   5. metadata-eval 98.9%

      \[\leadsto \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\sqrt{{\left(e^{\color{blue}{-0.6666666666666666}}\right)}^{\left(\frac{r}{s}\right)}}}{r}\right) \]

5. Applied egg-rr 98.9%

   \[\leadsto \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\color{blue}{\sqrt{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s}\right)}}}}{r}\right) \]
6. Step-by-step derivation

   1. sqrt-pow1 99.4%

      \[\leadsto \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\color{blue}{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{\frac{r}{s}}{2}\right)}}}{r}\right) \]

7. Applied egg-rr 99.4%

   \[\leadsto \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\color{blue}{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{\frac{r}{s}}{2}\right)}}}{r}\right) \]

8. Taylor expanded in s around 0 99.5%

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

   1. mul-1-neg 99.5%

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

   2. distribute-frac-neg 99.5%

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

   3. *-commutative 99.5%

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

   4. associate-*l/ 99.4%

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

10. Simplified 99.4%

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

11. Final simplification 99.4%

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

Alternative 7: 99.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = single(0.125) * ((exp((-r / s)) + exp(((r * single(-0.3333333333333333)) / s))) / (s * (r * 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 99.1%

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

   1. add-sqr-sqrt 99.1%

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

   2. sqrt-unprod 98.7%

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

   3. pow-prod-down 98.7%

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

   4. prod-exp 98.9%

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

   5. metadata-eval 98.9%

      \[\leadsto \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\sqrt{{\left(e^{\color{blue}{-0.6666666666666666}}\right)}^{\left(\frac{r}{s}\right)}}}{r}\right) \]

5. Applied egg-rr 98.9%

   \[\leadsto \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\color{blue}{\sqrt{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s}\right)}}}}{r}\right) \]
6. Step-by-step derivation

   1. sqrt-pow1 99.4%

      \[\leadsto \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\color{blue}{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{\frac{r}{s}}{2}\right)}}}{r}\right) \]

7. Applied egg-rr 99.4%

   \[\leadsto \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\color{blue}{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{\frac{r}{s}}{2}\right)}}}{r}\right) \]

8. Taylor expanded in r around inf 99.4%

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

   1. mul-1-neg 99.4%

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

   2. distribute-frac-neg 99.4%

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

   3. *-commutative 99.4%

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

   4. associate-*l/ 99.4%

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

   5. *-commutative 99.4%

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

10. Simplified 99.4%

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

11. Final simplification 99.4%

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

Alternative 8: 99.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

MATLAB

function tmp = code(s, r)
	tmp = single(0.125) * ((exp((-r / s)) + exp((single(-0.3333333333333333) * (r / s)))) / (s * (r * 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 99.1%

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

   1. add-sqr-sqrt 99.1%

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

   2. sqrt-unprod 98.7%

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

   3. pow-prod-down 98.7%

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

   4. prod-exp 98.9%

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

   5. metadata-eval 98.9%

      \[\leadsto \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\sqrt{{\left(e^{\color{blue}{-0.6666666666666666}}\right)}^{\left(\frac{r}{s}\right)}}}{r}\right) \]

5. Applied egg-rr 98.9%

   \[\leadsto \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\color{blue}{\sqrt{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s}\right)}}}}{r}\right) \]
6. Step-by-step derivation

   1. sqrt-pow1 99.4%

      \[\leadsto \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\color{blue}{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{\frac{r}{s}}{2}\right)}}}{r}\right) \]

7. Applied egg-rr 99.4%

   \[\leadsto \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\color{blue}{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{\frac{r}{s}}{2}\right)}}}{r}\right) \]

8. Taylor expanded in r around inf 99.4%

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

9. Final simplification 99.4%

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

Alternative 9: 44.4% accurate, 1.4× 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 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 10.0%

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

5. Taylor expanded in s around inf 9.5%

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

   1. log1p-expm1-u 40.8%

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

7. Applied egg-rr 40.8%

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

8. Final simplification 40.8%

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

Alternative 10: 10.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{1 + \frac{r}{s} \cdot \left(-0.3333333333333333 + \frac{r}{s} \cdot 0.05555555555555555\right)}{r}\right) \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (*
  (/ (/ 0.125 PI) s)
  (+
   (/ (exp (/ r (- s))) r)
   (/
    (+ 1.0 (* (/ r s) (+ -0.3333333333333333 (* (/ r s) 0.05555555555555555))))
    r))))

C

float code(float s, float r) {
	return ((0.125f / ((float) M_PI)) / s) * ((expf((r / -s)) / r) + ((1.0f + ((r / s) * (-0.3333333333333333f + ((r / s) * 0.05555555555555555f)))) / r));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.125) / Float32(pi)) / s) * Float32(Float32(exp(Float32(r / Float32(-s))) / r) + Float32(Float32(Float32(1.0) + Float32(Float32(r / s) * Float32(Float32(-0.3333333333333333) + Float32(Float32(r / s) * Float32(0.05555555555555555))))) / r)))
end

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.125) / single(pi)) / s) * ((exp((r / -s)) / r) + ((single(1.0) + ((r / s) * (single(-0.3333333333333333) + ((r / s) * single(0.05555555555555555))))) / 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 99.1%

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

4. Taylor expanded in r around 0 11.3%

   \[\leadsto \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\color{blue}{1 + \left(0.05555555555555555 \cdot \frac{{r}^{2}}{{s}^{2}} + -0.3333333333333333 \cdot \frac{r}{s}\right)}}{r}\right) \]
5. Step-by-step derivation

   1. fma-def 11.3%

      \[\leadsto \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{1 + \color{blue}{\mathsf{fma}\left(0.05555555555555555, \frac{{r}^{2}}{{s}^{2}}, -0.3333333333333333 \cdot \frac{r}{s}\right)}}{r}\right) \]

   2. unpow2 11.3%

      \[\leadsto \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{1 + \mathsf{fma}\left(0.05555555555555555, \frac{\color{blue}{r \cdot r}}{{s}^{2}}, -0.3333333333333333 \cdot \frac{r}{s}\right)}{r}\right) \]

   3. unpow2 11.3%

      \[\leadsto \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{1 + \mathsf{fma}\left(0.05555555555555555, \frac{r \cdot r}{\color{blue}{s \cdot s}}, -0.3333333333333333 \cdot \frac{r}{s}\right)}{r}\right) \]

6. Simplified 11.3%

   \[\leadsto \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\color{blue}{1 + \mathsf{fma}\left(0.05555555555555555, \frac{r \cdot r}{s \cdot s}, -0.3333333333333333 \cdot \frac{r}{s}\right)}}{r}\right) \]

7. Taylor expanded in r around 0 11.3%

   \[\leadsto \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{1 + \color{blue}{\left(0.05555555555555555 \cdot \frac{{r}^{2}}{{s}^{2}} + -0.3333333333333333 \cdot \frac{r}{s}\right)}}{r}\right) \]
8. Step-by-step derivation

   1. fma-def 11.3%

      \[\leadsto \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{1 + \color{blue}{\mathsf{fma}\left(0.05555555555555555, \frac{{r}^{2}}{{s}^{2}}, -0.3333333333333333 \cdot \frac{r}{s}\right)}}{r}\right) \]

   2. unpow2 11.3%

      \[\leadsto \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{1 + \mathsf{fma}\left(0.05555555555555555, \frac{\color{blue}{r \cdot r}}{{s}^{2}}, -0.3333333333333333 \cdot \frac{r}{s}\right)}{r}\right) \]

   3. unpow2 11.3%

      \[\leadsto \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{1 + \mathsf{fma}\left(0.05555555555555555, \frac{r \cdot r}{\color{blue}{s \cdot s}}, -0.3333333333333333 \cdot \frac{r}{s}\right)}{r}\right) \]

   4. times-frac 11.3%

      \[\leadsto \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{1 + \mathsf{fma}\left(0.05555555555555555, \color{blue}{\frac{r}{s} \cdot \frac{r}{s}}, -0.3333333333333333 \cdot \frac{r}{s}\right)}{r}\right) \]

   5. unpow2 11.3%

      \[\leadsto \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{1 + \mathsf{fma}\left(0.05555555555555555, \color{blue}{{\left(\frac{r}{s}\right)}^{2}}, -0.3333333333333333 \cdot \frac{r}{s}\right)}{r}\right) \]

   6. fma-def 11.3%

      \[\leadsto \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{1 + \color{blue}{\left(0.05555555555555555 \cdot {\left(\frac{r}{s}\right)}^{2} + -0.3333333333333333 \cdot \frac{r}{s}\right)}}{r}\right) \]

   7. +-commutative 11.3%

      \[\leadsto \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{1 + \color{blue}{\left(-0.3333333333333333 \cdot \frac{r}{s} + 0.05555555555555555 \cdot {\left(\frac{r}{s}\right)}^{2}\right)}}{r}\right) \]

   8. unpow2 11.3%

      \[\leadsto \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{1 + \left(-0.3333333333333333 \cdot \frac{r}{s} + 0.05555555555555555 \cdot \color{blue}{\left(\frac{r}{s} \cdot \frac{r}{s}\right)}\right)}{r}\right) \]

   9. associate-*r* 11.3%

      \[\leadsto \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{1 + \left(-0.3333333333333333 \cdot \frac{r}{s} + \color{blue}{\left(0.05555555555555555 \cdot \frac{r}{s}\right) \cdot \frac{r}{s}}\right)}{r}\right) \]

   10. distribute-rgt-out 11.3%

       \[\leadsto \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{1 + \color{blue}{\frac{r}{s} \cdot \left(-0.3333333333333333 + 0.05555555555555555 \cdot \frac{r}{s}\right)}}{r}\right) \]

9. Simplified 11.3%

   \[\leadsto \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{1 + \color{blue}{\frac{r}{s} \cdot \left(-0.3333333333333333 + 0.05555555555555555 \cdot \frac{r}{s}\right)}}{r}\right) \]

10. Final simplification 11.3%

    \[\leadsto \frac{\frac{0.125}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{1 + \frac{r}{s} \cdot \left(-0.3333333333333333 + \frac{r}{s} \cdot 0.05555555555555555\right)}{r}\right) \]

Alternative 11: 9.7% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.125) / single(pi)) / s) * ((exp((r / -s)) / r) + (((single(-0.3333333333333333) * (r / s)) + single(1.0)) / r));
end

Derivation

1. Initial program 99.6%

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

3. Simplified 99.1%

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

4. Taylor expanded in r around 0 10.4%

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

5. Final simplification 10.4%

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

Alternative 12: 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 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 10.0%

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

5. Taylor expanded in s around 0 10.0%

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

   1. associate-*r/ 10.0%

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

   2. neg-mul-1 10.0%

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

7. Simplified 10.0%

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

8. Final simplification 10.0%

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

Alternative 13: 9.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Simplified 99.1%

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

4. Taylor expanded in r around 0 10.0%

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

5. Taylor expanded in s around 0 10.0%

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

   1. associate-*r/ 10.0%

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

   2. neg-mul-1 10.0%

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

7. Simplified 10.0%

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

8. Taylor expanded in s around 0 10.0%

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

   1. associate-*r/ 10.0%

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

   2. mul-1-neg 10.0%

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

   3. distribute-frac-neg 10.0%

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

   4. associate-/l* 10.0%

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

10. Simplified 10.0%

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

11. Final simplification 10.0%

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

Alternative 14: 9.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.125) / single(pi)) * ((exp((r / -s)) / r) + (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 99.1%

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

4. Taylor expanded in r around 0 10.0%

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

   1. associate-*l/ 10.0%

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

6. Applied egg-rr 10.0%

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

7. Final simplification 10.0%

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

Alternative 15: 9.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = single(0.125) * ((exp((-r / s)) + single(1.0)) / (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 99.1%

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

4. Taylor expanded in r around 0 10.0%

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

5. Taylor expanded in s around 0 10.0%

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

   1. associate-*r/ 10.0%

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

   2. neg-mul-1 10.0%

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

7. Simplified 10.0%

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

8. Taylor expanded in r around inf 10.0%

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

   1. +-commutative 10.0%

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

   2. mul-1-neg 10.0%

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

   3. distribute-frac-neg 10.0%

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

   4. associate-*r* 10.0%

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

10. Simplified 10.0%

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

11. Final simplification 10.0%

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

Alternative 16: 9.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = (exp((-r / s)) + single(1.0)) * (((single(0.125) / 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 99.1%

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

4. Taylor expanded in r around 0 10.0%

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

5. Taylor expanded in s around 0 10.0%

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

7. Simplified 10.0%

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

8. Final simplification 10.0%

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

Alternative 17: 9.0% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Simplified 99.1%

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

4. Taylor expanded in r around 0 10.0%

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

5. Taylor expanded in s around 0 10.0%

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

   1. associate-*r/ 10.0%

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

   2. neg-mul-1 10.0%

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

7. Simplified 10.0%

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

8. Taylor expanded in r around 0 9.5%

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

9. Final simplification 9.5%

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

Alternative 18: 9.0% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = single(0.25) / (s * (r * 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 99.1%

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

4. Taylor expanded in r around 0 10.0%

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

5. Taylor expanded in s around inf 9.5%

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

6. Final simplification 9.5%

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

Reproduce

herbie shell --seed 2023274 
(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))))