Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.6%

Time: 14.3s

Alternatives: 19

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

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

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/ (* 0.25 (exp (/ r (- s)))) (* (* s 2.0) (* r 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))) / ((s * 2.0f) * (r * ((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(Float32(s * Float32(2.0)) * Float32(r * 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))) / ((s * single(2.0)) * (r * 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. Add Preprocessing

3. Taylor expanded in s around 0 99.6%

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

   1. *-commutative 99.6%

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

   2. *-commutative 99.6%

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

   3. associate-*r* 99.6%

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

   4. associate-*r* 99.6%

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

   5. *-commutative 99.6%

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

5. Simplified 99.6%

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

6. Final simplification 99.6%

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

Alternative 2: 99.6% accurate, 1.0× speedup

Math

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

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(r / Float32(-s)))) / Float32(r * Float32(s * Float32(Float32(2.0) * Float32(pi))))) + Float32(Float32(Float32(0.75) * 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(0.25) * 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. Add Preprocessing

3. Taylor expanded in r around 0 99.5%

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

   1. *-commutative 99.5%

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

   2. metadata-eval 99.5%

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

   3. times-frac 99.6%

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

   4. *-rgt-identity 99.6%

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

5. Simplified 99.6%

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

6. Final simplification 99.6%

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

Alternative 3: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.125) / Float32(s * 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(Float32(pi) * Float32(s * Float32(6.0)))))))
end

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.125) / (s * single(pi))) * (exp((r / -s)) / r)) + (single(0.75) * (exp((r / (s * -single(3.0)))) / (r * (single(pi) * (s * 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. Step-by-step derivation

   1. times-frac 99.6%

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

   2. *-commutative 99.6%

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

   3. distribute-frac-neg 99.6%

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

   4. associate-/l* 99.6%

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

   5. *-commutative 99.6%

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

   6. *-commutative 99.6%

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

   7. associate-*l* 99.5%

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

3. Simplified 99.5%

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

5. Taylor expanded in s around 0 99.5%

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

6. Taylor expanded in s around 0 99.5%

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

   1. associate-*r* 99.6%

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

8. Simplified 99.6%

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

9. Final simplification 99.6%

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

Alternative 4: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

float code(float s, float r) {
	return ((0.125f / (s * ((float) M_PI))) * (expf((r / -s)) / r)) + (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) / Float32(s * 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(Float32(6.0) * Float32(s * Float32(pi)))))))
end

MATLAB

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

Derivation

1. Initial program 99.6%

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

   1. times-frac 99.6%

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

   2. *-commutative 99.6%

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

   3. distribute-frac-neg 99.6%

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

   4. associate-/l* 99.6%

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

   5. *-commutative 99.6%

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

   6. *-commutative 99.6%

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

   7. associate-*l* 99.5%

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

3. Simplified 99.5%

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

5. Taylor expanded in s around 0 99.5%

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

6. Final simplification 99.5%

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

Alternative 5: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

MATLAB

function tmp = code(s, r)
	tmp = (single(0.125) / (s * single(pi))) * ((exp((r / -s)) / r) + (exp((r / (s * single(-3.0)))) / 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 99.2%

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

5. Taylor expanded in r around inf 99.4%

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

   1. *-commutative 99.4%

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

   2. metadata-eval 99.4%

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

   3. times-frac 99.5%

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

   4. *-rgt-identity 99.5%

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

7. Simplified 99.5%

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

Alternative 6: 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 99.6%

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

3. Simplified 99.2%

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

   1. add-sqr-sqrt 99.2%

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

   2. sqrt-unprod 98.6%

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

   3. pow-prod-down 98.6%

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

   4. prod-exp 98.8%

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

   5. metadata-eval 98.8%

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

6. Applied egg-rr 98.8%

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

7. Taylor expanded in r around inf 98.8%

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

   1. mul-1-neg 98.8%

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

   2. exp-prod 98.8%

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

   3. unpow1/2 98.8%

      \[\leadsto 0.125 \cdot \frac{e^{-\frac{r}{s}} + \color{blue}{{\left({\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s}\right)}\right)}^{0.5}}}{r \cdot \left(s \cdot \pi\right)} \]

   4. exp-prod 98.8%

      \[\leadsto 0.125 \cdot \frac{e^{-\frac{r}{s}} + {\color{blue}{\left(e^{-0.6666666666666666 \cdot \frac{r}{s}}\right)}}^{0.5}}{r \cdot \left(s \cdot \pi\right)} \]

   5. *-commutative 98.8%

      \[\leadsto 0.125 \cdot \frac{e^{-\frac{r}{s}} + {\left(e^{\color{blue}{\frac{r}{s} \cdot -0.6666666666666666}}\right)}^{0.5}}{r \cdot \left(s \cdot \pi\right)} \]

   6. exp-prod 99.4%

      \[\leadsto 0.125 \cdot \frac{e^{-\frac{r}{s}} + \color{blue}{e^{\left(\frac{r}{s} \cdot -0.6666666666666666\right) \cdot 0.5}}}{r \cdot \left(s \cdot \pi\right)} \]

   7. associate-*l* 99.4%

      \[\leadsto 0.125 \cdot \frac{e^{-\frac{r}{s}} + e^{\color{blue}{\frac{r}{s} \cdot \left(-0.6666666666666666 \cdot 0.5\right)}}}{r \cdot \left(s \cdot \pi\right)} \]

   8. metadata-eval 99.4%

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

9. Simplified 99.4%

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

10. Final simplification 99.4%

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

Alternative 7: 11.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

function code(s, r)
	return Float32(Float32(0.25) / log1p(expm1(Float32(Float32(pi) * Float32(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 99.2%

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

5. Taylor expanded in r around 0 9.8%

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

6. Taylor expanded in s around inf 9.3%

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

   1. associate-*r* 9.3%

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

   2. *-commutative 9.3%

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

   3. log1p-expm1-u 13.7%

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

8. Applied egg-rr 13.7%

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

Alternative 8: 10.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.25}{r \cdot \pi} - \frac{\frac{0.16666666666666666}{\pi} - 0.125 \cdot \frac{0.5 \cdot \left(\frac{r}{\pi} + \frac{{r}^{2} \cdot 0.1111111111111111}{r \cdot \pi}\right)}{s}}{s}}{s} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (/
  (-
   (/ 0.25 (* r PI))
   (/
    (-
     (/ 0.16666666666666666 PI)
     (*
      0.125
      (/
       (* 0.5 (+ (/ r PI) (/ (* (pow r 2.0) 0.1111111111111111) (* r PI))))
       s)))
    s))
  s))

C

float code(float s, float r) {
	return ((0.25f / (r * ((float) M_PI))) - (((0.16666666666666666f / ((float) M_PI)) - (0.125f * ((0.5f * ((r / ((float) M_PI)) + ((powf(r, 2.0f) * 0.1111111111111111f) / (r * ((float) M_PI))))) / s))) / s)) / s;
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.25) / Float32(r * Float32(pi))) - Float32(Float32(Float32(Float32(0.16666666666666666) / Float32(pi)) - Float32(Float32(0.125) * Float32(Float32(Float32(0.5) * Float32(Float32(r / Float32(pi)) + Float32(Float32((r ^ Float32(2.0)) * Float32(0.1111111111111111)) / Float32(r * Float32(pi))))) / s))) / s)) / s)
end

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.25) / (r * single(pi))) - (((single(0.16666666666666666) / single(pi)) - (single(0.125) * ((single(0.5) * ((r / single(pi)) + (((r ^ single(2.0)) * single(0.1111111111111111)) / (r * single(pi))))) / s))) / s)) / 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 99.2%

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

   1. add-sqr-sqrt 99.2%

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

   2. sqrt-unprod 98.6%

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

   3. pow-prod-down 98.6%

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

   4. prod-exp 98.8%

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

   5. metadata-eval 98.8%

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

6. Applied egg-rr 98.8%

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

7. Taylor expanded in s around -inf 10.5%

   \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{0.125 \cdot \frac{0.5 \cdot \frac{r}{\pi} + 0.5 \cdot \frac{0.2222222222222222 \cdot {r}^{2} - 0.1111111111111111 \cdot {r}^{2}}{r \cdot \pi}}{s} - 0.16666666666666666 \cdot \frac{1}{\pi}}{s} - 0.25 \cdot \frac{1}{r \cdot \pi}}{s}} \]

9. Simplified 10.5%

   \[\leadsto \color{blue}{-\frac{\left(-\frac{0.125 \cdot \frac{0.5 \cdot \left(\frac{r}{\pi} + \frac{{r}^{2} \cdot 0.1111111111111111}{r \cdot \pi}\right)}{s} - \frac{0.16666666666666666}{\pi}}{s}\right) - \frac{0.25}{r \cdot \pi}}{s}} \]

10. Final simplification 10.5%

    \[\leadsto \frac{\frac{0.25}{r \cdot \pi} - \frac{\frac{0.16666666666666666}{\pi} - 0.125 \cdot \frac{0.5 \cdot \left(\frac{r}{\pi} + \frac{{r}^{2} \cdot 0.1111111111111111}{r \cdot \pi}\right)}{s}}{s}}{s} \]
11. Add Preprocessing

Alternative 9: 9.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \frac{0.125}{s \cdot \pi} \cdot \left(\frac{1 - \frac{r + \frac{{r}^{2} \cdot -0.5}{s}}{s}}{r} + \frac{\frac{r}{s} \cdot -0.3333333333333333 + 1}{r}\right) \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (*
  (/ 0.125 (* s PI))
  (+
   (/ (- 1.0 (/ (+ r (/ (* (pow r 2.0) -0.5) s)) s)) r)
   (/ (+ (* (/ r s) -0.3333333333333333) 1.0) r))))

C

float code(float s, float r) {
	return (0.125f / (s * ((float) M_PI))) * (((1.0f - ((r + ((powf(r, 2.0f) * -0.5f) / s)) / s)) / r) + ((((r / s) * -0.3333333333333333f) + 1.0f) / r));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(0.125) / Float32(s * Float32(pi))) * Float32(Float32(Float32(Float32(1.0) - Float32(Float32(r + Float32(Float32((r ^ Float32(2.0)) * Float32(-0.5)) / s)) / s)) / r) + Float32(Float32(Float32(Float32(r / s) * Float32(-0.3333333333333333)) + Float32(1.0)) / r)))
end

MATLAB

function tmp = code(s, r)
	tmp = (single(0.125) / (s * single(pi))) * (((single(1.0) - ((r + (((r ^ single(2.0)) * single(-0.5)) / s)) / s)) / r) + ((((r / s) * single(-0.3333333333333333)) + 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 99.2%

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

   1. add-sqr-sqrt 99.2%

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

   2. sqrt-unprod 98.6%

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

   3. pow-prod-down 98.6%

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

   4. prod-exp 98.8%

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

   5. metadata-eval 98.8%

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

6. Applied egg-rr 98.8%

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

7. Taylor expanded in r around 0 10.2%

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

   1. *-commutative 10.2%

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

9. Simplified 10.2%

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

10. Taylor expanded in s around -inf 10.2%

    \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{\color{blue}{1 + -1 \cdot \frac{r + -0.5 \cdot \frac{{r}^{2}}{s}}{s}}}{r} + \frac{1 + \frac{r}{s} \cdot -0.3333333333333333}{r}\right) \]
11. Step-by-step derivation

    1. mul-1-neg 10.2%

       \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{1 + \color{blue}{\left(-\frac{r + -0.5 \cdot \frac{{r}^{2}}{s}}{s}\right)}}{r} + \frac{1 + \frac{r}{s} \cdot -0.3333333333333333}{r}\right) \]

    2. associate-*r/ 10.2%

       \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{1 + \left(-\frac{r + \color{blue}{\frac{-0.5 \cdot {r}^{2}}{s}}}{s}\right)}{r} + \frac{1 + \frac{r}{s} \cdot -0.3333333333333333}{r}\right) \]

12. Simplified 10.2%

    \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{\color{blue}{1 + \left(-\frac{r + \frac{-0.5 \cdot {r}^{2}}{s}}{s}\right)}}{r} + \frac{1 + \frac{r}{s} \cdot -0.3333333333333333}{r}\right) \]

13. Final simplification 10.2%

    \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{1 - \frac{r + \frac{{r}^{2} \cdot -0.5}{s}}{s}}{r} + \frac{\frac{r}{s} \cdot -0.3333333333333333 + 1}{r}\right) \]
14. Add Preprocessing

Alternative 10: 9.7% accurate, 1.9× speedup

Math

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

FPCore

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

C

float code(float s, float r) {
	return (0.125f / (s * ((float) M_PI))) * ((expf((r / -s)) / r) + ((((r / s) * -0.3333333333333333f) + 1.0f) / 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(Float32(Float32(Float32(r / s) * Float32(-0.3333333333333333)) + Float32(1.0)) / r)))
end

MATLAB

function tmp = code(s, r)
	tmp = (single(0.125) / (s * single(pi))) * ((exp((r / -s)) / r) + ((((r / s) * single(-0.3333333333333333)) + 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 99.2%

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

5. Taylor expanded in r around 0 10.2%

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

6. Final simplification 10.2%

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

Alternative 11: 9.7% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = (single(0.125) / (s * single(pi))) * ((exp((r / -s)) / r) + ((single(1.0) / r) - (single(0.3333333333333333) / 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 99.2%

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

   1. add-sqr-sqrt 99.2%

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

   2. sqrt-unprod 98.6%

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

   3. pow-prod-down 98.6%

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

   4. prod-exp 98.8%

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

   5. metadata-eval 98.8%

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

6. Applied egg-rr 98.8%

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

7. Taylor expanded in r around 0 10.2%

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

   1. *-commutative 10.2%

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

9. Simplified 10.2%

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

10. Taylor expanded in r around inf 10.1%

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

    1. associate-*r/ 10.1%

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

    2. metadata-eval 10.1%

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

12. Simplified 10.1%

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

Alternative 12: 9.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.125) / s) * (single(1.0) / 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 99.2%

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

5. Taylor expanded in r around 0 9.8%

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

   1. associate-/r* 9.8%

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

   2. div-inv 9.8%

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

7. Applied egg-rr 9.8%

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

Alternative 13: 9.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Simplified 99.2%

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

5. Taylor expanded in r around 0 9.8%

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

6. Taylor expanded in r around inf 9.8%

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

   1. associate-*r/ 9.8%

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

   2. *-commutative 9.8%

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

   3. *-commutative 9.8%

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

   4. associate-*l* 9.8%

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

   5. *-commutative 9.8%

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

   6. times-frac 9.8%

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

   7. mul-1-neg 9.8%

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

   8. distribute-neg-frac2 9.8%

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

8. Simplified 9.8%

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

9. Final simplification 9.8%

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

Alternative 14: 9.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Simplified 99.2%

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

5. Taylor expanded in r around 0 9.8%

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

6. Taylor expanded in r around inf 9.8%

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

   1. mul-1-neg 9.8%

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

8. Simplified 9.8%

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

9. Final simplification 9.8%

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

Alternative 15: 9.1% accurate, 17.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Simplified 99.2%

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

   1. add-sqr-sqrt 99.2%

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

   2. sqrt-unprod 98.6%

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

   3. pow-prod-down 98.6%

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

   4. prod-exp 98.8%

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

   5. metadata-eval 98.8%

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

6. Applied egg-rr 98.8%

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

7. Taylor expanded in s around inf 9.7%

   \[\leadsto \color{blue}{\frac{0.25 \cdot \frac{1}{r \cdot \pi} - 0.16666666666666666 \cdot \frac{1}{s \cdot \pi}}{s}} \]
8. Step-by-step derivation

   1. associate-*r/ 9.7%

      \[\leadsto \frac{0.25 \cdot \frac{1}{r \cdot \pi} - \color{blue}{\frac{0.16666666666666666 \cdot 1}{s \cdot \pi}}}{s} \]

   2. metadata-eval 9.7%

      \[\leadsto \frac{0.25 \cdot \frac{1}{r \cdot \pi} - \frac{\color{blue}{0.16666666666666666}}{s \cdot \pi}}{s} \]

   3. associate-*r/ 9.7%

      \[\leadsto \frac{\color{blue}{\frac{0.25 \cdot 1}{r \cdot \pi}} - \frac{0.16666666666666666}{s \cdot \pi}}{s} \]

   4. metadata-eval 9.7%

      \[\leadsto \frac{\frac{\color{blue}{0.25}}{r \cdot \pi} - \frac{0.16666666666666666}{s \cdot \pi}}{s} \]

9. Simplified 9.7%

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

Alternative 16: 9.0% accurate, 25.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = (single(1.0) / r) * ((single(0.25) / 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 99.2%

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

5. Taylor expanded in r around 0 9.8%

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

6. Taylor expanded in r around inf 9.8%

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

   1. associate-*r/ 9.8%

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

   2. *-commutative 9.8%

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

   3. *-commutative 9.8%

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

   4. associate-*l* 9.8%

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

   5. *-commutative 9.8%

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

   6. times-frac 9.8%

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

   7. mul-1-neg 9.8%

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

   8. distribute-neg-frac2 9.8%

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

8. Simplified 9.8%

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

9. Taylor expanded in r around 0 9.3%

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

    1. *-commutative 9.3%

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

    2. associate-/l/ 9.3%

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

    3. *-commutative 9.3%

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

    4. associate-/r* 9.3%

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

11. Simplified 9.3%

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

    1. div-inv 9.3%

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

13. Applied egg-rr 9.3%

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

14. Final simplification 9.3%

    \[\leadsto \frac{1}{r} \cdot \frac{\frac{0.25}{s}}{\pi} \]
15. Add Preprocessing

Alternative 17: 9.0% accurate, 33.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.25) / s) / single(pi)) / 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 99.2%

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

5. Taylor expanded in r around 0 9.8%

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

6. Taylor expanded in r around inf 9.8%

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

   1. associate-*r/ 9.8%

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

   2. *-commutative 9.8%

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

   3. *-commutative 9.8%

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

   4. associate-*l* 9.8%

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

   5. *-commutative 9.8%

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

   6. times-frac 9.8%

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

   7. mul-1-neg 9.8%

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

   8. distribute-neg-frac2 9.8%

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

8. Simplified 9.8%

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

9. Taylor expanded in r around 0 9.3%

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

    1. *-commutative 9.3%

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

    2. associate-/l/ 9.3%

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

    3. *-commutative 9.3%

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

    4. associate-/r* 9.3%

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

11. Simplified 9.3%

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

Alternative 18: 9.0% accurate, 33.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = (single(0.25) / (s * single(pi))) / 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 99.2%

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

5. Taylor expanded in r around 0 9.8%

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

6. Taylor expanded in r around inf 9.8%

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

   1. associate-*r/ 9.8%

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

   2. *-commutative 9.8%

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

   3. *-commutative 9.8%

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

   4. associate-*l* 9.8%

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

   5. *-commutative 9.8%

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

   6. times-frac 9.8%

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

   7. mul-1-neg 9.8%

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

   8. distribute-neg-frac2 9.8%

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

8. Simplified 9.8%

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

9. Taylor expanded in r around 0 9.3%

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

    1. *-commutative 9.3%

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

    2. associate-/l/ 9.3%

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

    3. *-commutative 9.3%

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

    4. associate-/r* 9.3%

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

11. Simplified 9.3%

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

12. Taylor expanded in s around 0 9.3%

    \[\leadsto \frac{\color{blue}{\frac{0.25}{s \cdot \pi}}}{r} \]
13. Add Preprocessing

Alternative 19: 9.0% accurate, 33.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Simplified 99.2%

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

5. Taylor expanded in r around 0 9.8%

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

6. Taylor expanded in s around inf 9.3%

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

Reproduce

herbie shell --seed 2024088 
(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))))