Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.6%

Time: 16.3s

Alternatives: 16

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: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in s around 0 99.8%

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

   1. *-commutative 99.8%

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

5. Simplified 99.8%

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

6. Final simplification 99.8%

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

Alternative 2: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

float code(float s, float r) {
	return ((0.25f * expf((r / -s))) / (r * (s * (2.0f * ((float) M_PI))))) + ((0.75f * expf(((r / s) * -0.3333333333333333f))) / ((r * ((float) M_PI)) * (s * 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 / s) * Float32(-0.3333333333333333)))) / Float32(Float32(r * Float32(pi)) * Float32(s * Float32(6.0)))))
end

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.25) * exp((r / -s))) / (r * (s * (single(2.0) * single(pi))))) + ((single(0.75) * exp(((r / s) * single(-0.3333333333333333)))) / ((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 99.8%

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

   1. *-commutative 99.8%

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

5. Simplified 99.8%

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

6. Taylor expanded in r around 0 99.8%

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

   1. *-commutative 99.8%

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

   2. *-commutative 99.8%

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

   3. associate-*r* 99.7%

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

   4. associate-*l* 99.8%

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

   5. *-commutative 99.8%

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

8. Simplified 99.8%

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

9. Taylor expanded in r around 0 99.7%

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

    1. *-commutative 99.7%

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

11. Simplified 99.7%

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

12. Final simplification 99.7%

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

Alternative 3: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Simplified 99.6%

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

5. Taylor expanded in s around 0 99.7%

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

   1. associate-*r/ 99.7%

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

   2. times-frac 99.7%

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

   3. mul-1-neg 99.7%

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

   4. distribute-neg-frac2 99.7%

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

   5. *-commutative 99.7%

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

   6. associate-*l/ 99.7%

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

   7. associate-/l* 99.7%

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

7. Simplified 99.7%

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

8. Final simplification 99.7%

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

Alternative 4: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = (single(0.125) / (s * single(pi))) * ((exp((r / -s)) + exp(((r / s) * single(-0.3333333333333333)))) / 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 99.6%

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

5. Taylor expanded in r around inf 99.7%

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

   1. associate-*r/ 99.7%

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

   2. *-commutative 99.7%

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

   3. times-frac 99.6%

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

   4. mul-1-neg 99.6%

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

   5. distribute-neg-frac2 99.6%

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

   6. *-commutative 99.6%

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

   7. associate-*l/ 99.6%

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

   8. associate-/l* 99.6%

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

7. Simplified 99.6%

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

8. Taylor expanded in r around 0 99.6%

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

9. Final simplification 99.6%

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

Alternative 5: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Simplified 99.6%

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

5. Taylor expanded in r around inf 99.7%

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

6. Final simplification 99.7%

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

Alternative 6: 11.8% 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 99.6%

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

5. Taylor expanded in s around inf 8.7%

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

   1. log1p-expm1-u 9.9%

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

   2. *-commutative 9.9%

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

7. Applied egg-rr 9.9%

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

8. Final simplification 9.9%

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

Alternative 7: 44.2% 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 99.6%

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

5. Taylor expanded in s around inf 8.7%

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

   1. pow1 8.7%

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

   2. *-commutative 8.7%

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

7. Applied egg-rr 8.7%

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

   1. unpow1 8.7%

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

   2. *-commutative 8.7%

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

   3. *-commutative 8.7%

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

   4. associate-*l* 8.7%

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

9. Simplified 8.7%

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

    1. log1p-expm1-u 40.5%

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

11. Applied egg-rr 40.5%

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

12. Final simplification 40.5%

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

Alternative 8: 9.4% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Simplified 99.6%

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

5. Taylor expanded in s around 0 99.7%

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

   1. associate-*r/ 99.7%

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

   2. times-frac 99.7%

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

   3. mul-1-neg 99.7%

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

   4. distribute-neg-frac2 99.7%

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

   5. *-commutative 99.7%

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

   6. associate-*l/ 99.7%

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

   7. associate-/l* 99.7%

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

7. Simplified 99.7%

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

8. Taylor expanded in r around 0 9.4%

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

   1. mul-1-neg 9.4%

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

   2. unsub-neg 9.4%

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

10. Simplified 9.4%

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

11. Final simplification 9.4%

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

Alternative 9: 9.4% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = (single(0.125) / (s * single(pi))) * ((exp((r * (single(-0.3333333333333333) / s))) + (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 99.6%

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

5. Taylor expanded in r around inf 99.7%

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

   1. associate-*r/ 99.7%

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

   2. *-commutative 99.7%

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

   3. times-frac 99.6%

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

   4. mul-1-neg 99.6%

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

   5. distribute-neg-frac2 99.6%

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

   6. *-commutative 99.6%

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

   7. associate-*l/ 99.6%

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

   8. associate-/l* 99.6%

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

7. Simplified 99.6%

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

8. Taylor expanded in r around 0 9.4%

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

   1. mul-1-neg 9.4%

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

   2. unsub-neg 9.4%

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

10. Simplified 9.4%

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

11. Final simplification 9.4%

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

Alternative 10: 9.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = ((single(-0.16666666666666666) / single(pi)) / (s ^ single(2.0))) + (single(0.25) / (s * (r * single(pi))));
end

Derivation

1. Initial program 99.7%

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

3. Simplified 99.6%

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

5. Taylor expanded in r around inf 99.7%

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

   1. associate-*r/ 99.7%

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

   2. *-commutative 99.7%

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

   3. times-frac 99.6%

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

   4. mul-1-neg 99.6%

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

   5. distribute-neg-frac2 99.6%

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

   6. *-commutative 99.6%

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

   7. associate-*l/ 99.6%

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

   8. associate-/l* 99.6%

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

7. Simplified 99.6%

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

8. Taylor expanded in r around 0 99.6%

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

9. Taylor expanded in s around inf 9.3%

   \[\leadsto \color{blue}{0.25 \cdot \frac{1}{r \cdot \left(s \cdot \pi\right)} - 0.16666666666666666 \cdot \frac{1}{{s}^{2} \cdot \pi}} \]
10. Step-by-step derivation

    1. sub-neg 9.3%

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

    2. associate-*r/ 9.3%

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

    3. metadata-eval 9.3%

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

    4. *-commutative 9.3%

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

    5. *-commutative 9.3%

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

    6. *-commutative 9.3%

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

    7. associate-*l* 9.3%

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

    8. *-commutative 9.3%

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

    9. associate-*r/ 9.3%

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

    10. metadata-eval 9.3%

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

    11. *-commutative 9.3%

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

    12. distribute-neg-frac 9.3%

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

    13. metadata-eval 9.3%

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

    14. associate-/r* 9.3%

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

11. Simplified 9.3%

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

12. Final simplification 9.3%

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

Alternative 11: 9.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.25}{r}}{s \cdot \pi} + \frac{-0.16666666666666666}{\pi \cdot {s}^{2}} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.25) / r) / (s * single(pi))) + (single(-0.16666666666666666) / (single(pi) * (s ^ single(2.0))));
end

Derivation

1. Initial program 99.7%

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

3. Simplified 99.6%

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

5. Taylor expanded in s around 0 99.7%

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

   1. associate-*r/ 99.7%

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

   2. times-frac 99.7%

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

   3. mul-1-neg 99.7%

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

   4. distribute-neg-frac2 99.7%

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

   5. *-commutative 99.7%

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

   6. associate-*l/ 99.7%

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

   7. associate-/l* 99.7%

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

7. Simplified 99.7%

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

8. Taylor expanded in s around inf 9.3%

   \[\leadsto \color{blue}{0.25 \cdot \frac{1}{r \cdot \left(s \cdot \pi\right)} - 0.16666666666666666 \cdot \frac{1}{{s}^{2} \cdot \pi}} \]
9. Step-by-step derivation

   1. sub-neg 9.3%

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

   2. associate-*r/ 9.3%

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

   3. metadata-eval 9.3%

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

   4. associate-/r* 9.3%

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

   5. associate-*r/ 9.3%

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

   6. metadata-eval 9.3%

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

   7. distribute-neg-frac 9.3%

      \[\leadsto \frac{\frac{0.25}{r}}{s \cdot \pi} + \color{blue}{\frac{-0.16666666666666666}{{s}^{2} \cdot \pi}} \]

   8. metadata-eval 9.3%

      \[\leadsto \frac{\frac{0.25}{r}}{s \cdot \pi} + \frac{\color{blue}{-0.16666666666666666}}{{s}^{2} \cdot \pi} \]

   9. *-commutative 9.3%

      \[\leadsto \frac{\frac{0.25}{r}}{s \cdot \pi} + \frac{-0.16666666666666666}{\color{blue}{\pi \cdot {s}^{2}}} \]

10. Simplified 9.3%

    \[\leadsto \color{blue}{\frac{\frac{0.25}{r}}{s \cdot \pi} + \frac{-0.16666666666666666}{\pi \cdot {s}^{2}}} \]

11. Final simplification 9.3%

    \[\leadsto \frac{\frac{0.25}{r}}{s \cdot \pi} + \frac{-0.16666666666666666}{\pi \cdot {s}^{2}} \]
12. Add Preprocessing

Alternative 12: 9.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.25}{r}}{s \cdot \pi} + \frac{\frac{-0.16666666666666666}{\pi}}{{s}^{2}} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.25) / r) / (s * single(pi))) + ((single(-0.16666666666666666) / single(pi)) / (s ^ single(2.0)));
end

Derivation

1. Initial program 99.7%

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

3. Simplified 99.6%

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

5. Taylor expanded in r around 0 8.6%

   \[\leadsto \color{blue}{\left(0.06944444444444445 \cdot \frac{r}{{s}^{3} \cdot \pi} + 0.25 \cdot \frac{1}{r \cdot \left(s \cdot \pi\right)}\right) - 0.16666666666666666 \cdot \frac{1}{{s}^{2} \cdot \pi}} \]

6. Taylor expanded in s around 0 8.6%

   \[\leadsto \left(0.06944444444444445 \cdot \frac{r}{{s}^{3} \cdot \pi} + 0.25 \cdot \frac{1}{r \cdot \left(s \cdot \pi\right)}\right) - \color{blue}{\frac{0.16666666666666666}{{s}^{2} \cdot \pi}} \]
7. Step-by-step derivation

   1. associate--l+ 8.6%

      \[\leadsto \color{blue}{0.06944444444444445 \cdot \frac{r}{{s}^{3} \cdot \pi} + \left(0.25 \cdot \frac{1}{r \cdot \left(s \cdot \pi\right)} - \frac{0.16666666666666666}{{s}^{2} \cdot \pi}\right)} \]

   2. *-commutative 8.6%

      \[\leadsto 0.06944444444444445 \cdot \frac{r}{\color{blue}{\pi \cdot {s}^{3}}} + \left(0.25 \cdot \frac{1}{r \cdot \left(s \cdot \pi\right)} - \frac{0.16666666666666666}{{s}^{2} \cdot \pi}\right) \]

   3. un-div-inv 8.6%

      \[\leadsto 0.06944444444444445 \cdot \frac{r}{\pi \cdot {s}^{3}} + \left(\color{blue}{\frac{0.25}{r \cdot \left(s \cdot \pi\right)}} - \frac{0.16666666666666666}{{s}^{2} \cdot \pi}\right) \]

   4. *-commutative 8.6%

      \[\leadsto 0.06944444444444445 \cdot \frac{r}{\pi \cdot {s}^{3}} + \left(\frac{0.25}{r \cdot \color{blue}{\left(\pi \cdot s\right)}} - \frac{0.16666666666666666}{{s}^{2} \cdot \pi}\right) \]

   5. *-commutative 8.6%

      \[\leadsto 0.06944444444444445 \cdot \frac{r}{\pi \cdot {s}^{3}} + \left(\frac{0.25}{r \cdot \left(\pi \cdot s\right)} - \frac{0.16666666666666666}{\color{blue}{\pi \cdot {s}^{2}}}\right) \]

8. Applied egg-rr 8.6%

   \[\leadsto \color{blue}{0.06944444444444445 \cdot \frac{r}{\pi \cdot {s}^{3}} + \left(\frac{0.25}{r \cdot \left(\pi \cdot s\right)} - \frac{0.16666666666666666}{\pi \cdot {s}^{2}}\right)} \]

9. Taylor expanded in r around 0 9.3%

   \[\leadsto \color{blue}{0.25 \cdot \frac{1}{r \cdot \left(s \cdot \pi\right)} - 0.16666666666666666 \cdot \frac{1}{{s}^{2} \cdot \pi}} \]
10. Step-by-step derivation

    1. sub-neg 9.3%

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

    2. associate-*r/ 9.3%

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

    3. metadata-eval 9.3%

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

    4. associate-/r* 9.3%

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

    5. associate-*r/ 9.3%

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

    6. metadata-eval 9.3%

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

    7. *-commutative 9.3%

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

    8. distribute-neg-frac 9.3%

       \[\leadsto \frac{\frac{0.25}{r}}{s \cdot \pi} + \color{blue}{\frac{-0.16666666666666666}{\pi \cdot {s}^{2}}} \]

    9. metadata-eval 9.3%

       \[\leadsto \frac{\frac{0.25}{r}}{s \cdot \pi} + \frac{\color{blue}{-0.16666666666666666}}{\pi \cdot {s}^{2}} \]

    10. associate-/r* 9.3%

        \[\leadsto \frac{\frac{0.25}{r}}{s \cdot \pi} + \color{blue}{\frac{\frac{-0.16666666666666666}{\pi}}{{s}^{2}}} \]

11. Simplified 9.3%

    \[\leadsto \color{blue}{\frac{\frac{0.25}{r}}{s \cdot \pi} + \frac{\frac{-0.16666666666666666}{\pi}}{{s}^{2}}} \]

12. Final simplification 9.3%

    \[\leadsto \frac{\frac{0.25}{r}}{s \cdot \pi} + \frac{\frac{-0.16666666666666666}{\pi}}{{s}^{2}} \]
13. Add Preprocessing

Alternative 13: 9.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = (single(0.25) / (r * (s * single(pi)))) - (single(0.16666666666666666) / (single(pi) * (s ^ single(2.0))));
end

Derivation

1. Initial program 99.7%

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

3. Simplified 99.6%

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

5. Taylor expanded in s around inf 9.3%

   \[\leadsto \color{blue}{0.25 \cdot \frac{1}{r \cdot \left(s \cdot \pi\right)} - 0.16666666666666666 \cdot \frac{1}{{s}^{2} \cdot \pi}} \]
6. Step-by-step derivation

   1. associate-*r/ 9.3%

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

   2. metadata-eval 9.3%

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

   3. associate-*r/ 9.3%

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

   4. metadata-eval 9.3%

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

   5. *-commutative 9.3%

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

7. Simplified 9.3%

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

8. Final simplification 9.3%

   \[\leadsto \frac{0.25}{r \cdot \left(s \cdot \pi\right)} - \frac{0.16666666666666666}{\pi \cdot {s}^{2}} \]
9. Add Preprocessing

Alternative 14: 9.2% accurate, 25.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = (single(0.25) / r) * (single(1.0) / (s * single(pi)));
end

Derivation

1. Initial program 99.7%

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

3. Simplified 99.6%

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

5. Taylor expanded in s around 0 99.7%

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

   1. associate-*r/ 99.7%

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

   2. times-frac 99.7%

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

   3. mul-1-neg 99.7%

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

   4. distribute-neg-frac2 99.7%

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

   5. *-commutative 99.7%

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

   6. associate-*l/ 99.7%

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

   7. associate-/l* 99.7%

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

7. Simplified 99.7%

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

8. Taylor expanded in s around inf 8.7%

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

   1. associate-/r* 8.7%

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

10. Simplified 8.7%

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

    1. div-inv 8.7%

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

12. Applied egg-rr 8.7%

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

13. Final simplification 8.7%

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

Alternative 15: 9.2% 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 99.6%

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

5. Taylor expanded in s around inf 8.7%

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

6. Final simplification 8.7%

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

Alternative 16: 9.2% 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 99.6%

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

5. Taylor expanded in s around 0 99.7%

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

   1. associate-*r/ 99.7%

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

   2. times-frac 99.7%

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

   3. mul-1-neg 99.7%

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

   4. distribute-neg-frac2 99.7%

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

   5. *-commutative 99.7%

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

   6. associate-*l/ 99.7%

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

   7. associate-/l* 99.7%

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

7. Simplified 99.7%

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

8. Taylor expanded in s around inf 8.7%

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

   1. associate-/r* 8.7%

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

10. Simplified 8.7%

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

11. Final simplification 8.7%

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

Reproduce

herbie shell --seed 2024046 
(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))))