Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.5%

Time: 1.5min

Alternatives: 18

Speedup: N/A×

Specification

\[\left(0 \leq s \land s \leq 256\right) \land \left(10^{-6} < r \land r < 1000000\right)\]

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.5% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/ (* 0.25 (exp (/ r (- s)))) (* r (* s (* 2.0 PI))))
  (/ (* 0.75 (exp (/ (* r -0.3333333333333333) s))) (* 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 * -0.3333333333333333f) / 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(r * Float32(s * Float32(Float32(2.0) * Float32(pi))))) + Float32(Float32(Float32(0.75) * exp(Float32(Float32(r * Float32(-0.3333333333333333)) / 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))) / (r * (s * (single(2.0) * single(pi))))) + ((single(0.75) * exp(((r * single(-0.3333333333333333)) / 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 r around 0 99.6%

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

   1. associate-*r/ 99.7%

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

5. Simplified 99.7%

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

6. Final simplification 99.7%

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

Alternative 2: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{r}{-s}}}{r} + 0.75 \cdot \frac{e^{\frac{r \cdot -0.3333333333333333}{s}}}{r \cdot \left(s \cdot \left(\pi \cdot 6\right)\right)} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (* (/ 0.125 (* s PI)) (/ (exp (/ r (- s))) r))
  (* 0.75 (/ (exp (/ (* r -0.3333333333333333) s)) (* r (* s (* PI 6.0)))))))

C

float code(float s, float r) {
	return ((0.125f / (s * ((float) M_PI))) * (expf((r / -s)) / r)) + (0.75f * (expf(((r * -0.3333333333333333f) / s)) / (r * (s * (((float) M_PI) * 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(Float32(r * Float32(-0.3333333333333333)) / s)) / Float32(r * Float32(s * Float32(Float32(pi) * 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 * single(-0.3333333333333333)) / 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. 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* 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in s around 0 99.6%

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

6. Taylor expanded in s around 0 99.6%

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

   1. *-commutative 99.6%

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

   2. associate-*r* 99.6%

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

8. Simplified 99.6%

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

9. Taylor expanded in r around 0 99.6%

   \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \frac{e^{-\frac{r}{s}}}{r} + 0.75 \cdot \frac{e^{\color{blue}{-0.3333333333333333 \cdot \frac{r}{s}}}}{r \cdot \left(s \cdot \left(\pi \cdot 6\right)\right)} \]
10. Step-by-step derivation

    1. *-commutative 99.6%

       \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \frac{e^{-\frac{r}{s}}}{r} + 0.75 \cdot \frac{e^{\color{blue}{\frac{r}{s} \cdot -0.3333333333333333}}}{r \cdot \left(s \cdot \left(\pi \cdot 6\right)\right)} \]

    2. associate-*l/ 99.7%

       \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \frac{e^{-\frac{r}{s}}}{r} + 0.75 \cdot \frac{e^{\color{blue}{\frac{r \cdot -0.3333333333333333}{s}}}}{r \cdot \left(s \cdot \left(\pi \cdot 6\right)\right)} \]

11. Simplified 99.7%

    \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \frac{e^{-\frac{r}{s}}}{r} + 0.75 \cdot \frac{e^{\color{blue}{\frac{r \cdot -0.3333333333333333}{s}}}}{r \cdot \left(s \cdot \left(\pi \cdot 6\right)\right)} \]

12. Final simplification 99.7%

    \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{r}{-s}}}{r} + 0.75 \cdot \frac{e^{\frac{r \cdot -0.3333333333333333}{s}}}{r \cdot \left(s \cdot \left(\pi \cdot 6\right)\right)} \]
13. Add Preprocessing

Alternative 3: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{r}{-s}}}{r} + 0.75 \cdot \frac{e^{\frac{r \cdot -0.3333333333333333}{s}}}{s \cdot \left(r \cdot \left(\pi \cdot 6\right)\right)} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (* (/ 0.125 (* s PI)) (/ (exp (/ r (- s))) r))
  (* 0.75 (/ (exp (/ (* r -0.3333333333333333) s)) (* s (* r (* PI 6.0)))))))

C

float code(float s, float r) {
	return ((0.125f / (s * ((float) M_PI))) * (expf((r / -s)) / r)) + (0.75f * (expf(((r * -0.3333333333333333f) / s)) / (s * (r * (((float) M_PI) * 6.0f)))));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.125) / Float32(s * Float32(pi))) * Float32(exp(Float32(r / Float32(-s))) / r)) + Float32(Float32(0.75) * Float32(exp(Float32(Float32(r * Float32(-0.3333333333333333)) / s)) / Float32(s * Float32(r * Float32(Float32(pi) * 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 * single(-0.3333333333333333)) / s)) / (s * (r * (single(pi) * single(6.0))))));
end

Derivation

1. Initial program 99.7%

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

   1. times-frac 99.6%

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

   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* 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in s around 0 99.6%

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

6. Taylor expanded in s around 0 99.6%

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

   1. *-commutative 99.6%

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

   2. associate-*r* 99.6%

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

8. Simplified 99.6%

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

9. Taylor expanded in r around 0 99.6%

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

    1. *-commutative 99.6%

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

    2. associate-*r* 99.6%

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

    3. associate-*r* 99.6%

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

    4. *-commutative 99.6%

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

    5. associate-*l* 99.7%

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

    6. *-commutative 99.7%

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

11. Simplified 99.7%

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

12. Taylor expanded in r around 0 99.6%

    \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \frac{e^{-\frac{r}{s}}}{r} + 0.75 \cdot \frac{e^{\color{blue}{-0.3333333333333333 \cdot \frac{r}{s}}}}{s \cdot \left(\left(6 \cdot \pi\right) \cdot r\right)} \]
13. Step-by-step derivation

    1. *-commutative 99.6%

       \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \frac{e^{-\frac{r}{s}}}{r} + 0.75 \cdot \frac{e^{\color{blue}{\frac{r}{s} \cdot -0.3333333333333333}}}{r \cdot \left(s \cdot \left(\pi \cdot 6\right)\right)} \]

    2. associate-*l/ 99.7%

       \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \frac{e^{-\frac{r}{s}}}{r} + 0.75 \cdot \frac{e^{\color{blue}{\frac{r \cdot -0.3333333333333333}{s}}}}{r \cdot \left(s \cdot \left(\pi \cdot 6\right)\right)} \]

14. Simplified 99.7%

    \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \frac{e^{-\frac{r}{s}}}{r} + 0.75 \cdot \frac{e^{\color{blue}{\frac{r \cdot -0.3333333333333333}{s}}}}{s \cdot \left(\left(6 \cdot \pi\right) \cdot r\right)} \]

15. Final simplification 99.7%

    \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{r}{-s}}}{r} + 0.75 \cdot \frac{e^{\frac{r \cdot -0.3333333333333333}{s}}}{s \cdot \left(r \cdot \left(\pi \cdot 6\right)\right)} \]
16. Add Preprocessing

Alternative 4: 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 99.3%

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

5. Taylor expanded in r around inf 99.5%

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

6. Final simplification 99.5%

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

Alternative 5: 99.5% accurate, 1.1× speedup

Math

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

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 99.3%

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

   1. add-sqr-sqrt 99.3%

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

   2. sqrt-unprod 98.8%

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

   3. pow-prod-down 98.8%

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

   4. prod-exp 99.0%

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

   5. metadata-eval 99.0%

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

6. Applied egg-rr 99.0%

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

   1. sqrt-pow1 99.6%

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

8. Applied egg-rr 99.6%

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

9. Taylor expanded in r around inf 99.5%

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

    1. associate-*r/ 99.6%

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

    2. *-commutative 99.6%

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

    3. associate-/l* 99.6%

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

11. Simplified 99.6%

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

12. Final simplification 99.6%

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

Alternative 6: 11.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

function code(s, r)
	return Float32(Float32(0.25) / log1p(expm1(Float32(r * Float32(s * 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 99.3%

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

5. Taylor expanded in r around 0 9.5%

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

6. Taylor expanded in s around inf 9.0%

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

   1. log1p-expm1-u 10.4%

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

8. Applied egg-rr 10.4%

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

9. Final simplification 10.4%

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

Alternative 7: 11.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

function code(s, r)
	return Float32(Float32(0.25) / log1p(expm1(Float32(s * 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 99.3%

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

5. Taylor expanded in r around 0 9.5%

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

6. Taylor expanded in s around inf 9.0%

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

   1. *-commutative 9.0%

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

   2. *-commutative 9.0%

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

   3. associate-*l* 9.0%

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

8. Simplified 9.0%

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

   1. add-sqr-sqrt 9.0%

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

   2. sqrt-unprod 8.7%

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

   3. sqr-neg 8.7%

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

   4. sqrt-unprod -0.0%

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

   5. add-sqr-sqrt 4.4%

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

   6. distribute-lft-neg-in 4.4%

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

   7. distribute-rgt-neg-in 4.4%

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

   8. log1p-expm1-u 6.7%

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

   9. *-commutative 6.7%

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

   10. distribute-lft-neg-in 6.7%

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

   11. distribute-lft-neg-in 6.7%

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

   12. add-sqr-sqrt -0.0%

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

   13. sqrt-unprod 10.2%

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

   14. sqr-neg 10.2%

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

   15. sqrt-unprod 10.4%

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

   16. add-sqr-sqrt 10.4%

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

   17. associate-*l* 10.4%

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

10. Applied egg-rr 10.4%

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

11. Final simplification 10.4%

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

Alternative 8: 9.9% accurate, 1.8× speedup

Math

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

FPCore

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

C

float code(float s, float r) {
	return (0.125f / (s * ((float) M_PI))) * (((1.0f - ((r + ((powf(r, 2.0f) / s) * -0.5f)) / s)) / r) + ((((r * -0.3333333333333333f) / s) + 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)) / s) * Float32(-0.5))) / s)) / r) + Float32(Float32(Float32(Float32(r * Float32(-0.3333333333333333)) / s) + 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)) / s) * single(-0.5))) / s)) / r) + ((((r * single(-0.3333333333333333)) / s) + single(1.0)) / r));
end

Derivation

1. Initial program 99.7%

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

3. Simplified 99.3%

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

5. Taylor expanded in r around 0 9.7%

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

   1. associate-*r/ 9.7%

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

7. Simplified 9.7%

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

8. Taylor expanded in s around -inf 9.9%

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

   1. mul-1-neg 9.9%

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

   2. *-commutative 9.9%

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

10. Simplified 9.9%

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

11. Final simplification 9.9%

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

Alternative 9: 9.8% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Simplified 99.3%

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

5. Taylor expanded in r around 0 9.7%

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

   1. associate-*r/ 9.7%

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

7. Simplified 9.7%

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

8. Final simplification 9.7%

   \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\frac{r \cdot -0.3333333333333333}{s} + 1}{r}\right) \]
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 99.3%

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

5. Taylor expanded in r around 0 9.7%

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

   1. associate-*r/ 9.7%

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

7. Simplified 9.7%

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

8. Taylor expanded in r around inf 9.7%

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

   1. associate-*r/ 9.7%

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

   2. metadata-eval 9.7%

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

10. Simplified 9.7%

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

11. Final simplification 9.7%

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

Alternative 11: 9.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Simplified 99.3%

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

5. Taylor expanded in r around 0 9.5%

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

6. Taylor expanded in r around inf 9.5%

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

   1. associate-*r/ 9.5%

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

   2. *-commutative 9.5%

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

   3. *-commutative 9.5%

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

   4. times-frac 9.5%

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

   5. *-commutative 9.5%

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

   6. mul-1-neg 9.5%

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

   7. distribute-neg-frac2 9.5%

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

8. Simplified 9.5%

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

   1. associate-/r* 9.5%

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

   2. div-inv 9.5%

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

10. Applied egg-rr 9.5%

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

11. Final simplification 9.5%

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

Alternative 12: 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 99.3%

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

5. Taylor expanded in r around 0 9.5%

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

6. Taylor expanded in r around inf 9.5%

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

   1. associate-*r/ 9.5%

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

   2. *-commutative 9.5%

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

   3. *-commutative 9.5%

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

   4. times-frac 9.5%

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

   5. *-commutative 9.5%

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

   6. mul-1-neg 9.5%

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

   7. distribute-neg-frac2 9.5%

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

8. Simplified 9.5%

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

9. Taylor expanded in s around 0 9.5%

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

    1. mul-1-neg 9.5%

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

11. Simplified 9.5%

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

12. Final simplification 9.5%

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

Alternative 13: 9.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Simplified 99.3%

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

5. Taylor expanded in r around 0 9.5%

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

6. Taylor expanded in r around -inf 9.5%

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

   1. associate-*r/ 9.5%

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

   2. *-commutative 9.5%

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

   3. *-commutative 9.5%

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

   4. associate-*l* 9.5%

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

   5. times-frac 9.5%

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

   6. sub-neg 9.5%

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

   7. metadata-eval 9.5%

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

   8. +-commutative 9.5%

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

   9. mul-1-neg 9.5%

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

   10. unsub-neg 9.5%

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

   11. mul-1-neg 9.5%

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

   12. distribute-neg-frac2 9.5%

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

8. Simplified 9.5%

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

9. Final simplification 9.5%

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

Alternative 14: 9.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Simplified 99.3%

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

5. Taylor expanded in r around 0 9.5%

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

6. Taylor expanded in r around inf 9.5%

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

   1. associate-*r/ 9.5%

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

   2. *-commutative 9.5%

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

   3. associate-*r/ 9.5%

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

   4. neg-mul-1 9.5%

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

   5. *-commutative 9.5%

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

   6. associate-*l* 9.5%

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

8. Simplified 9.5%

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

9. Final simplification 9.5%

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

Alternative 15: 9.1% accurate, 10.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = (single(0.125) / (s * single(pi))) * (((((r * single(-0.3333333333333333)) / s) + single(1.0)) / r) + ((single(1.0) - (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 99.3%

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

5. Taylor expanded in r around 0 9.7%

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

   1. associate-*r/ 9.7%

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

7. Simplified 9.7%

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

8. Taylor expanded in r around 0 9.2%

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

   1. mul-1-neg 9.2%

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

   2. unsub-neg 9.2%

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

10. Simplified 9.2%

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

11. Final simplification 9.2%

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

Alternative 16: 9.1% accurate, 25.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Simplified 99.3%

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

5. Taylor expanded in r around 0 9.5%

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

6. Taylor expanded in r around -inf 9.5%

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

   1. associate-*r/ 9.5%

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

   2. *-commutative 9.5%

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

   3. *-commutative 9.5%

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

   4. associate-*l* 9.5%

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

   5. times-frac 9.5%

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

   6. sub-neg 9.5%

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

   7. metadata-eval 9.5%

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

   8. +-commutative 9.5%

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

   9. mul-1-neg 9.5%

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

   10. unsub-neg 9.5%

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

   11. mul-1-neg 9.5%

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

   12. distribute-neg-frac2 9.5%

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

8. Simplified 9.5%

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

9. Taylor expanded in r around 0 9.0%

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

10. Final simplification 9.0%

    \[\leadsto \frac{-0.125}{\pi} \cdot \frac{-2}{r \cdot s} \]
11. Add Preprocessing

Alternative 17: 9.1% accurate, 33.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Simplified 99.3%

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

5. Taylor expanded in r around 0 9.5%

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

6. Taylor expanded in s around inf 9.0%

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

7. Final simplification 9.0%

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

Alternative 18: 9.1% accurate, 33.0× speedup

Math

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

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Simplified 99.3%

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

5. Taylor expanded in r around 0 9.5%

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

6. Taylor expanded in s around inf 9.0%

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

   1. associate-/r* 9.0%

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

8. Simplified 9.0%

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

9. Final simplification 9.0%

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

Reproduce

herbie shell --seed 2024115 
(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))))