Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.6%

Time: 12.1s

Alternatives: 16

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

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

FPCore

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

C

float code(float s, float r) {
	return ((0.25f * expf((r / -s))) / (2.0f * ((r * s) * ((float) M_PI)))) + ((0.75f * expf(((r / -3.0f) / s))) / (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(2.0) * Float32(Float32(r * s) * Float32(pi)))) + Float32(Float32(Float32(0.75) * exp(Float32(Float32(r / Float32(-3.0)) / s))) / Float32(r * Float32(s * Float32(Float32(pi) * Float32(6.0))))))
end

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.25) * exp((r / -s))) / (single(2.0) * ((r * s) * single(pi)))) + ((single(0.75) * exp(((r / single(-3.0)) / s))) / (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. Taylor expanded in s around 0 99.7%

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

   1. associate-*r* 99.7%

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

5. Simplified 99.7%

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

   1. distribute-frac-neg 99.7%

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

   2. *-commutative 99.7%

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

7. Applied egg-rr 99.7%

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

   1. distribute-neg-frac2 99.7%

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

   2. distribute-rgt-neg-in 99.7%

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

   3. metadata-eval 99.7%

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

9. Simplified 99.7%

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

10. Taylor expanded in r around 0 99.7%

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

    1. *-commutative 99.7%

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

    2. metadata-eval 99.7%

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

    3. times-frac 99.7%

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

    4. *-rgt-identity 99.7%

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

    5. *-commutative 99.7%

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

    6. associate-/r* 99.7%

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

12. Simplified 99.7%

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

13. Final simplification 99.7%

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

Alternative 2: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

3. Taylor expanded in s around 0 99.7%

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

   1. associate-*r* 99.7%

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

5. Simplified 99.7%

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

   1. distribute-frac-neg 99.7%

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

   2. *-commutative 99.7%

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

7. Applied egg-rr 99.7%

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

   1. distribute-neg-frac2 99.7%

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

   2. distribute-rgt-neg-in 99.7%

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

   3. metadata-eval 99.7%

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

9. Simplified 99.7%

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

10. Final simplification 99.7%

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

Alternative 3: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

float code(float s, float r) {
	return ((0.125f / s) * (expf((r / -s)) / (r * ((float) M_PI)))) + ((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) / s) * Float32(exp(Float32(r / Float32(-s))) / Float32(r * Float32(pi)))) + Float32(Float32(Float32(0.75) * 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) * (exp((r / -s)) / (r * single(pi)))) + ((single(0.75) * exp((r / (s * -single(3.0))))) / (r * (single(pi) * (s * 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. Taylor expanded in s around 0 99.7%

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

   1. associate-*r* 99.7%

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

5. Simplified 99.7%

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

6. Taylor expanded in r around inf 99.7%

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

   1. associate-*r/ 99.7%

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

   2. *-commutative 99.7%

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

   3. associate-*l* 99.7%

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

   4. *-commutative 99.7%

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

   5. times-frac 99.7%

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

   6. mul-1-neg 99.7%

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

   7. distribute-neg-frac2 99.7%

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

8. Simplified 99.7%

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

9. Taylor expanded in s around 0 99.7%

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

    1. *-commutative 99.7%

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

    2. associate-*r* 99.7%

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

    3. *-commutative 99.7%

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

    4. associate-*l* 99.7%

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

11. Simplified 99.7%

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

12. Final simplification 99.7%

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

Alternative 4: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.75) * exp(Float32(r / Float32(s * Float32(-3.0))))) / Float32(r * Float32(s * Float32(Float32(pi) * Float32(6.0))))) + Float32(Float32(Float32(0.125) / s) * Float32(exp(Float32(r / Float32(-s))) / Float32(r * 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(0.125) / s) * (exp((r / -s)) / (r * 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 s around 0 99.7%

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

   1. associate-*r* 99.7%

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

5. Simplified 99.7%

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

6. Taylor expanded in r around inf 99.7%

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

   1. associate-*r/ 99.7%

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

   2. *-commutative 99.7%

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

   3. associate-*l* 99.7%

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

   4. *-commutative 99.7%

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

   5. times-frac 99.7%

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

   6. mul-1-neg 99.7%

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

   7. distribute-neg-frac2 99.7%

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

8. Simplified 99.7%

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

   1. distribute-frac-neg 99.7%

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

   2. *-commutative 99.7%

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

10. Applied egg-rr 99.7%

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

    1. distribute-neg-frac2 99.7%

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

    2. distribute-rgt-neg-in 99.7%

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

    3. metadata-eval 99.7%

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

12. Simplified 99.7%

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

13. Final simplification 99.7%

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

Alternative 5: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

MATLAB

function tmp = code(s, r)
	tmp = (single(0.125) / (s * single(pi))) * ((exp((r / -s)) / r) + (exp((r * (single(-0.3333333333333333) / s))) / 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 99.4%

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

5. Taylor expanded in r around inf 99.6%

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

   1. *-lft-identity 99.6%

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

   2. associate-*l/ 99.6%

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

   3. associate-*l* 99.6%

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

   4. metadata-eval 99.6%

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

   5. *-commutative 99.6%

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

   6. exp-prod 97.7%

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

   7. metadata-eval 97.7%

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

   8. associate-*r/ 97.7%

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

   9. metadata-eval 97.7%

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

7. Simplified 97.7%

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

   1. pow-exp 99.6%

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

9. Applied egg-rr 99.6%

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

Alternative 6: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

MATLAB

function tmp = code(s, r)
	tmp = (single(0.125) / (s * single(pi))) * ((exp((r / -s)) / r) + (exp((single(-0.3333333333333333) * (r / s))) / 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 99.4%

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

5. Taylor expanded in r around inf 99.6%

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

Alternative 7: 43.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.7%

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

3. Simplified 99.4%

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

5. Taylor expanded in r around 0 9.7%

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

6. Taylor expanded in r around inf 9.7%

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

   1. associate-*r/ 9.7%

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

   2. associate-*r* 9.7%

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

   3. times-frac 9.7%

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

   4. mul-1-neg 9.7%

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

   5. distribute-neg-frac2 9.7%

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

8. Simplified 9.7%

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

9. Taylor expanded in r around 0 9.2%

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

    1. associate-/r* 9.2%

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

    2. *-commutative 9.2%

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

    3. associate-/l/ 9.2%

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

    4. *-commutative 9.2%

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

    5. associate-*l* 9.2%

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

11. Simplified 9.2%

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

    1. log1p-expm1-u 45.4%

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

    2. *-commutative 45.4%

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

13. Applied egg-rr 45.4%

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

Alternative 8: 10.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \frac{0.125}{s} \cdot \left(\frac{\frac{-1}{\pi} - -0.5 \cdot \frac{r}{s \cdot \pi}}{s} + \frac{1}{r \cdot \pi}\right) + \frac{0.75 \cdot e^{\frac{r}{s \cdot \left(-3\right)}}}{r \cdot \left(s \cdot \left(\pi \cdot 6\right)\right)} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (*
   (/ 0.125 s)
   (+ (/ (- (/ -1.0 PI) (* -0.5 (/ r (* s PI)))) s) (/ 1.0 (* r PI))))
  (/ (* 0.75 (exp (/ r (* s (- 3.0))))) (* r (* s (* PI 6.0))))))

C

float code(float s, float r) {
	return ((0.125f / s) * ((((-1.0f / ((float) M_PI)) - (-0.5f * (r / (s * ((float) M_PI))))) / s) + (1.0f / (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(0.125) / s) * Float32(Float32(Float32(Float32(Float32(-1.0) / Float32(pi)) - Float32(Float32(-0.5) * Float32(r / Float32(s * Float32(pi))))) / s) + Float32(Float32(1.0) / 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(0.125) / s) * ((((single(-1.0) / single(pi)) - (single(-0.5) * (r / (s * single(pi))))) / s) + (single(1.0) / (r * single(pi))))) + ((single(0.75) * exp((r / (s * -single(3.0))))) / (r * (s * (single(pi) * single(6.0)))));
end

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in s around 0 99.7%

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

   1. associate-*r* 99.7%

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

5. Simplified 99.7%

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

6. Taylor expanded in r around inf 99.7%

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

   1. associate-*r/ 99.7%

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

   2. *-commutative 99.7%

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

   3. associate-*l* 99.7%

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

   4. *-commutative 99.7%

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

   5. times-frac 99.7%

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

   6. mul-1-neg 99.7%

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

   7. distribute-neg-frac2 99.7%

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

8. Simplified 99.7%

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

9. Taylor expanded in s around -inf 10.6%

   \[\leadsto \frac{0.125}{s} \cdot \color{blue}{\left(-1 \cdot \frac{-0.5 \cdot \frac{r}{s \cdot \pi} + \frac{1}{\pi}}{s} + \frac{1}{r \cdot \pi}\right)} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

10. Final simplification 10.6%

    \[\leadsto \frac{0.125}{s} \cdot \left(\frac{\frac{-1}{\pi} - -0.5 \cdot \frac{r}{s \cdot \pi}}{s} + \frac{1}{r \cdot \pi}\right) + \frac{0.75 \cdot e^{\frac{r}{s \cdot \left(-3\right)}}}{r \cdot \left(s \cdot \left(\pi \cdot 6\right)\right)} \]
11. Add Preprocessing

Alternative 9: 9.7% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

MATLAB

function tmp = code(s, r)
	tmp = (single(0.125) / (s * single(pi))) * ((exp((r / -s)) / r) + (((s / r) - single(0.3333333333333333)) / 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 99.4%

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

5. Taylor expanded in r around 0 10.0%

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

   1. associate-*r/ 10.0%

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

   2. associate-*l/ 10.0%

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

   3. associate-/r/ 10.0%

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

7. Simplified 10.0%

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

8. Taylor expanded in s around 0 10.1%

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

Alternative 10: 9.7% accurate, 1.9× speedup

Math

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

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 99.7%

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

3. Simplified 99.4%

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

5. Taylor expanded in r around 0 10.0%

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

   1. associate-*r/ 10.0%

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

   2. associate-*l/ 10.0%

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

   3. associate-/r/ 10.0%

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

7. Simplified 10.0%

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

8. Taylor expanded in s around inf 10.0%

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

   1. associate-*r/ 10.0%

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

   2. metadata-eval 10.0%

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

10. Simplified 10.0%

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

Alternative 11: 9.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Simplified 99.4%

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

5. Taylor expanded in r around 0 9.7%

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

6. Taylor expanded in r around inf 9.7%

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

   1. associate-*r/ 9.7%

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

   2. associate-*r* 9.7%

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

   3. times-frac 9.7%

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

   4. mul-1-neg 9.7%

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

   5. distribute-neg-frac2 9.7%

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

8. Simplified 9.7%

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

9. Final simplification 9.7%

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

Alternative 12: 9.6% accurate, 2.0× speedup

Math

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

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(Float32(r * 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 99.7%

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

3. Simplified 99.4%

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

5. Taylor expanded in r around 0 9.7%

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

6. Taylor expanded in r around inf 9.7%

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

   1. associate-*r/ 9.7%

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

   2. associate-*r* 9.7%

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

   3. times-frac 9.7%

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

   4. mul-1-neg 9.7%

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

   5. distribute-neg-frac2 9.7%

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

8. Simplified 9.7%

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

9. Taylor expanded in r around inf 9.7%

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

    1. mul-1-neg 9.7%

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

    2. associate-*r* 9.7%

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

    3. *-commutative 9.7%

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

    4. *-commutative 9.7%

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

11. Simplified 9.7%

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

12. Final simplification 9.7%

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

Alternative 13: 9.6% 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 99.7%

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

3. Simplified 99.4%

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

5. Taylor expanded in r around 0 9.7%

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

6. Taylor expanded in r around inf 9.7%

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

   1. mul-1-neg 9.7%

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

8. Simplified 9.7%

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

9. Final simplification 9.7%

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

Alternative 14: 9.1% accurate, 25.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Simplified 99.4%

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

5. Taylor expanded in r around 0 9.7%

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

6. Taylor expanded in r around inf 9.7%

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

   1. associate-*r/ 9.7%

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

   2. associate-*r* 9.7%

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

   3. times-frac 9.7%

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

   4. mul-1-neg 9.7%

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

   5. distribute-neg-frac2 9.7%

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

8. Simplified 9.7%

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

9. Taylor expanded in r around 0 9.2%

   \[\leadsto \frac{0.125}{r \cdot s} \cdot \color{blue}{\frac{2}{\pi}} \]
10. Add Preprocessing

Alternative 15: 9.1% accurate, 33.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(s, r)
	return Float32(Float32(0.25) / Float32(Float32(r * 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 99.4%

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

5. Taylor expanded in r around 0 9.7%

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

6. Taylor expanded in s around inf 9.2%

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

   1. pow1 9.2%

      \[\leadsto \frac{0.25}{\color{blue}{{\left(r \cdot \left(s \cdot \pi\right)\right)}^{1}}} \]

   2. *-commutative 9.2%

      \[\leadsto \frac{0.25}{{\left(r \cdot \color{blue}{\left(\pi \cdot s\right)}\right)}^{1}} \]

8. Applied egg-rr 9.2%

   \[\leadsto \frac{0.25}{\color{blue}{{\left(r \cdot \left(\pi \cdot s\right)\right)}^{1}}} \]
9. Step-by-step derivation

   1. unpow1 9.2%

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

   2. *-commutative 9.2%

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

   3. associate-*r* 9.2%

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

10. Simplified 9.2%

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

11. Final simplification 9.2%

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

Alternative 16: 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 99.4%

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

5. Taylor expanded in r around 0 9.7%

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

6. Taylor expanded in s around inf 9.2%

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

Reproduce

herbie shell --seed 2024105 
(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))))