Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.6%

Time: 12.1s

Alternatives: 15

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: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.25) * exp((r / -s))) / (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 99.7%

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

3. Final simplification 99.7%

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

Alternative 2: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

   1. times-frac 99.7%

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

   2. *-commutative 99.7%

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

   3. distribute-frac-neg 99.7%

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

   4. associate-/l* 99.7%

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

   5. *-commutative 99.7%

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

   6. *-commutative 99.7%

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

   7. associate-*l* 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in s around 0 99.6%

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

   1. *-commutative 99.6%

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

   2. associate-*r* 99.7%

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

7. Simplified 99.7%

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

8. Final simplification 99.7%

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

Alternative 3: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in r around inf 99.7%

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

   1. neg-mul-1 99.7%

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

   2. rec-exp 99.6%

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

   3. associate-*r/ 99.6%

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

   4. metadata-eval 99.6%

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

5. Simplified 99.6%

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

6. Final simplification 99.6%

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

Alternative 4: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.125) / exp((r / s))) / (r * (s * single(pi)))) + (single(0.75) * (exp((r / (s * -single(3.0)))) / (r * (single(6.0) * (s * single(pi))))));
end

Derivation

1. Initial program 99.7%

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

   1. times-frac 99.7%

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

   2. *-commutative 99.7%

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

   3. distribute-frac-neg 99.7%

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

   4. associate-/l* 99.7%

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

   5. *-commutative 99.7%

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

   6. *-commutative 99.7%

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

   7. associate-*l* 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in s around 0 99.7%

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

   1. associate-*r/ 99.7%

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

   2. rec-exp 99.6%

      \[\leadsto \frac{0.125 \cdot \color{blue}{\frac{1}{e^{\frac{r}{s}}}}}{r \cdot \left(s \cdot \pi\right)} + 0.75 \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r \cdot \left(6 \cdot \left(\pi \cdot s\right)\right)} \]

   3. associate-*r/ 99.6%

      \[\leadsto \frac{\color{blue}{\frac{0.125 \cdot 1}{e^{\frac{r}{s}}}}}{r \cdot \left(s \cdot \pi\right)} + 0.75 \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r \cdot \left(6 \cdot \left(\pi \cdot s\right)\right)} \]

   4. metadata-eval 99.6%

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

7. Simplified 99.6%

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

8. Final simplification 99.6%

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

Alternative 5: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

float code(float s, float r) {
	return ((0.125f / expf((r / s))) / (r * (s * ((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(0.125) / exp(Float32(r / s))) / Float32(r * Float32(s * Float32(pi)))) + Float32(Float32(0.75) * Float32(exp(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(0.125) / exp((r / s))) / (r * (s * single(pi)))) + (single(0.75) * (exp((r / (s * single(-3.0)))) / (r * (s * (single(pi) * single(6.0))))));
end

Derivation

1. Initial program 99.7%

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

   1. times-frac 99.7%

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

   2. *-commutative 99.7%

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

   3. distribute-frac-neg 99.7%

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

   4. associate-/l* 99.7%

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

   5. *-commutative 99.7%

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

   6. *-commutative 99.7%

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

   7. associate-*l* 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in s around 0 99.6%

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

   1. *-commutative 99.6%

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

   2. associate-*r* 99.7%

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

7. Simplified 99.7%

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

8. Taylor expanded in s around 0 99.6%

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

   1. associate-*r/ 99.7%

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

   2. rec-exp 99.6%

      \[\leadsto \frac{0.125 \cdot \color{blue}{\frac{1}{e^{\frac{r}{s}}}}}{r \cdot \left(s \cdot \pi\right)} + 0.75 \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r \cdot \left(6 \cdot \left(\pi \cdot s\right)\right)} \]

   3. associate-*r/ 99.6%

      \[\leadsto \frac{\color{blue}{\frac{0.125 \cdot 1}{e^{\frac{r}{s}}}}}{r \cdot \left(s \cdot \pi\right)} + 0.75 \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r \cdot \left(6 \cdot \left(\pi \cdot s\right)\right)} \]

   4. metadata-eval 99.6%

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

10. Simplified 99.6%

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

    1. distribute-frac-neg 99.6%

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

12. Applied egg-rr 99.6%

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

    1. distribute-neg-frac2 99.6%

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

    2. distribute-rgt-neg-in 99.6%

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

    3. metadata-eval 99.6%

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

14. Simplified 99.6%

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

Alternative 6: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 0.125 \cdot \frac{\frac{e^{\frac{r}{-s}}}{r} + \frac{\frac{1}{e^{\frac{r}{s \cdot 3}}}}{r}}{s \cdot \pi} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (*
  0.125
  (/
   (+ (/ (exp (/ r (- s))) r) (/ (/ 1.0 (exp (/ r (* s 3.0)))) r))
   (* s PI))))

C

float code(float s, float r) {
	return 0.125f * (((expf((r / -s)) / r) + ((1.0f / expf((r / (s * 3.0f)))) / r)) / (s * ((float) M_PI)));
}

Julia

function code(s, r)
	return Float32(Float32(0.125) * Float32(Float32(Float32(exp(Float32(r / Float32(-s))) / r) + Float32(Float32(Float32(1.0) / exp(Float32(r / Float32(s * Float32(3.0))))) / r)) / Float32(s * Float32(pi))))
end

MATLAB

function tmp = code(s, r)
	tmp = single(0.125) * (((exp((r / -s)) / r) + ((single(1.0) / exp((r / (s * single(3.0))))) / r)) / (s * single(pi)));
end

Derivation

1. Initial program 99.7%

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

3. Simplified 99.3%

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

5. Taylor expanded in s around 0 99.6%

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

   1. metadata-eval 99.6%

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

   2. times-frac 99.6%

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

   3. neg-mul-1 99.6%

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

   4. distribute-frac-neg 99.6%

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

   5. exp-neg 99.6%

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

   6. *-commutative 99.6%

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

7. Applied egg-rr 99.6%

   \[\leadsto 0.125 \cdot \frac{\frac{e^{-1 \cdot \frac{r}{s}}}{r} + \frac{\color{blue}{\frac{1}{e^{\frac{r}{s \cdot 3}}}}}{r}}{s \cdot \pi} \]
8. Step-by-step derivation

   1. associate-*r/ 99.6%

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

   2. neg-mul-1 99.6%

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

   3. neg-sub0 99.6%

      \[\leadsto 0.125 \cdot \frac{\frac{e^{\frac{\color{blue}{0 - r}}{s}}}{r} + \frac{\frac{1}{e^{\frac{r}{s \cdot 3}}}}{r}}{s \cdot \pi} \]

   4. div-sub 99.6%

      \[\leadsto 0.125 \cdot \frac{\frac{e^{\color{blue}{\frac{0}{s} - \frac{r}{s}}}}{r} + \frac{\frac{1}{e^{\frac{r}{s \cdot 3}}}}{r}}{s \cdot \pi} \]

9. Applied egg-rr 99.6%

   \[\leadsto 0.125 \cdot \frac{\frac{e^{\color{blue}{\frac{0}{s} - \frac{r}{s}}}}{r} + \frac{\frac{1}{e^{\frac{r}{s \cdot 3}}}}{r}}{s \cdot \pi} \]
10. Step-by-step derivation

    1. div0 99.6%

       \[\leadsto 0.125 \cdot \frac{\frac{e^{\color{blue}{0} - \frac{r}{s}}}{r} + \frac{\frac{1}{e^{\frac{r}{s \cdot 3}}}}{r}}{s \cdot \pi} \]

    2. neg-sub0 99.6%

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

    3. distribute-neg-frac2 99.6%

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

11. Simplified 99.6%

    \[\leadsto 0.125 \cdot \frac{\frac{e^{\color{blue}{\frac{r}{-s}}}}{r} + \frac{\frac{1}{e^{\frac{r}{s \cdot 3}}}}{r}}{s \cdot \pi} \]
12. Add Preprocessing

Alternative 7: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Simplified 99.3%

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

   1. add-sqr-sqrt 99.3%

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

   2. sqrt-unprod 98.8%

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

   3. pow-prod-down 98.8%

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

   4. prod-exp 99.0%

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

   5. metadata-eval 99.0%

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

6. Applied egg-rr 99.0%

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

   1. sqrt-pow1 99.6%

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

8. Applied egg-rr 99.6%

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

   1. pow-to-exp 99.4%

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

   2. clear-num 99.4%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{e^{\log \left(e^{-0.6666666666666666}\right) \cdot \color{blue}{\frac{1}{\frac{2}{\frac{r}{s}}}}}}{r}\right) \]

   3. un-div-inv 99.5%

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

   4. rem-log-exp 99.6%

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

   5. div-inv 99.6%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{e^{\frac{-0.6666666666666666}{\color{blue}{2 \cdot \frac{1}{\frac{r}{s}}}}}}{r}\right) \]

   6. clear-num 99.6%

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

10. Applied egg-rr 99.6%

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

    1. associate-/r* 99.6%

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

    2. metadata-eval 99.6%

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

12. Simplified 99.6%

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

Alternative 8: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Simplified 99.3%

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

   1. add-sqr-sqrt 99.3%

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

   2. sqrt-unprod 98.8%

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

   3. pow-prod-down 98.8%

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

   4. prod-exp 99.0%

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

   5. metadata-eval 99.0%

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

6. Applied egg-rr 99.0%

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

7. Taylor expanded in s around 0 99.0%

   \[\leadsto \color{blue}{0.125 \cdot \frac{\frac{1}{r} \cdot \sqrt{e^{-0.6666666666666666 \cdot \frac{r}{s}}} + \frac{e^{-1 \cdot \frac{r}{s}}}{r}}{s \cdot \pi}} \]
8. Step-by-step derivation

9. Simplified 99.6%

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

10. Final simplification 99.6%

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

Alternative 9: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Simplified 99.3%

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

   1. add-sqr-sqrt 99.3%

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

   2. sqrt-unprod 98.8%

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

   3. pow-prod-down 98.8%

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

   4. prod-exp 99.0%

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

   5. metadata-eval 99.0%

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

6. Applied egg-rr 99.0%

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

   1. sqrt-pow1 99.6%

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

8. Applied egg-rr 99.6%

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

9. Taylor expanded in r around inf 99.5%

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

    1. associate-*r* 99.5%

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

    2. associate-*r/ 99.5%

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

    3. associate-*r* 99.5%

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

    4. times-frac 99.6%

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

11. Simplified 99.5%

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

12. Final simplification 99.5%

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

Alternative 10: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Simplified 99.3%

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

   1. add-sqr-sqrt 99.3%

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

   2. sqrt-unprod 98.8%

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

   3. pow-prod-down 98.8%

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

   4. prod-exp 99.0%

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

   5. metadata-eval 99.0%

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

6. Applied egg-rr 99.0%

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

7. Taylor expanded in r around inf 98.9%

   \[\leadsto \color{blue}{0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}} + \sqrt{e^{-0.6666666666666666 \cdot \frac{r}{s}}}}{r \cdot \left(s \cdot \pi\right)}} \]
8. Step-by-step derivation

   1. neg-mul-1 98.9%

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

   2. distribute-neg-frac2 98.9%

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

   3. *-commutative 98.9%

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

   4. exp-sqrt 99.5%

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

   5. associate-/l* 99.5%

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

   6. metadata-eval 99.5%

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

   7. *-commutative 99.5%

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

9. Simplified 99.5%

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

10. Final simplification 99.5%

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

Alternative 11: 45.0% accurate, 1.1× speedup

Math

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

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(Float32(0.25) / log1p(expm1(Float32(r * Float32(pi))))) / s)
end

Derivation

1. Initial program 99.7%

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

3. Simplified 99.3%

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

   1. add-sqr-sqrt 99.3%

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

   2. sqrt-unprod 98.8%

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

   3. pow-prod-down 98.8%

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

   4. prod-exp 99.0%

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

   5. metadata-eval 99.0%

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

6. Applied egg-rr 99.0%

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

7. Taylor expanded in s around inf 8.5%

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

   1. associate-/r* 8.5%

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

   2. *-commutative 8.5%

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

   3. associate-/r* 8.5%

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

   4. associate-/r* 8.5%

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

9. Simplified 8.5%

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

    1. log1p-expm1-u 49.1%

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

    2. *-commutative 49.1%

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

11. Applied egg-rr 49.1%

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

12. Final simplification 49.1%

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

Alternative 12: 15.5% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Simplified 99.3%

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

5. Taylor expanded in s around 0 99.6%

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

   1. metadata-eval 99.6%

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

   2. times-frac 99.6%

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

   3. neg-mul-1 99.6%

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

   4. distribute-frac-neg 99.6%

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

   5. exp-neg 99.6%

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

   6. *-commutative 99.6%

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

7. Applied egg-rr 99.6%

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

8. Taylor expanded in r around 0 14.0%

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

   1. associate-*r/ 14.0%

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

10. Simplified 14.0%

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

11. Final simplification 14.0%

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

Alternative 13: 16.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Simplified 99.3%

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

5. Taylor expanded in s around 0 99.6%

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

   1. metadata-eval 99.6%

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

   2. times-frac 99.6%

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

   3. neg-mul-1 99.6%

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

   4. distribute-frac-neg 99.6%

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

   5. exp-neg 99.6%

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

   6. *-commutative 99.6%

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

7. Applied egg-rr 99.6%

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

8. Taylor expanded in r around 0 14.0%

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

9. Final simplification 14.0%

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

Alternative 14: 10.3% accurate, 11.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\frac{0.25}{r}}{\pi} + \frac{\frac{\frac{r}{\pi} \cdot 0.06944444444444445}{s} + \frac{-0.16666666666666666}{\pi}}{s}}{s} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (/
  (+
   (/ (/ 0.25 r) PI)
   (/
    (+ (/ (* (/ r PI) 0.06944444444444445) s) (/ -0.16666666666666666 PI))
    s))
  s))

C

float code(float s, float r) {
	return (((0.25f / r) / ((float) M_PI)) + (((((r / ((float) M_PI)) * 0.06944444444444445f) / s) + (-0.16666666666666666f / ((float) M_PI))) / s)) / s;
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(Float32(0.25) / r) / Float32(pi)) + Float32(Float32(Float32(Float32(Float32(r / Float32(pi)) * Float32(0.06944444444444445)) / s) + Float32(Float32(-0.16666666666666666) / Float32(pi))) / s)) / s)
end

MATLAB

function tmp = code(s, r)
	tmp = (((single(0.25) / r) / single(pi)) + (((((r / single(pi)) * single(0.06944444444444445)) / s) + (single(-0.16666666666666666) / single(pi))) / s)) / s;
end

Derivation

1. Initial program 99.7%

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

3. Simplified 99.3%

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

5. Taylor expanded in s around 0 99.6%

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

6. Taylor expanded in s around -inf 9.5%

   \[\leadsto 0.125 \cdot \frac{\color{blue}{\left(-1 \cdot \frac{1 + -0.5 \cdot \frac{r}{s}}{s} + \frac{1}{r}\right)} + \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r}}{s \cdot \pi} \]

7. Taylor expanded in s around inf 9.5%

   \[\leadsto \color{blue}{\frac{\left(0.125 \cdot \frac{0.05555555555555555 \cdot \frac{r}{\pi} + 0.5 \cdot \frac{r}{\pi}}{{s}^{2}} + 0.25 \cdot \frac{1}{r \cdot \pi}\right) - \frac{0.16666666666666666}{s \cdot \pi}}{s}} \]

9. Simplified 9.5%

   \[\leadsto \color{blue}{\frac{\frac{\frac{0.25}{r}}{\pi} + \frac{\frac{\frac{r}{\pi} \cdot 0.06944444444444445}{s} + \frac{-0.16666666666666666}{\pi}}{s}}{s}} \]
10. Add Preprocessing

Alternative 15: 9.1% accurate, 33.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Simplified 99.3%

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

5. Taylor expanded in s around inf 8.5%

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

Reproduce

herbie shell --seed 2024157 
(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))))