Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.5% → 99.5%

Time: 20.2s

Alternatives: 13

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: 99.5% 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

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

   1. times-frac 99.5%

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

   2. *-commutative 99.5%

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

   3. distribute-frac-neg 99.5%

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

   4. associate-/l* 99.5%

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

   5. *-commutative 99.5%

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

   6. *-commutative 99.5%

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

   7. associate-*l* 99.4%

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

3. Simplified 99.4%

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

   1. add-sqr-sqrt 99.4%

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

   2. pow2 99.4%

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

   3. associate-*r* 99.5%

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

   4. *-commutative 99.5%

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

   5. *-commutative 99.5%

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

6. Applied egg-rr 99.5%

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

7. Taylor expanded in s around 0 99.5%

   \[\leadsto \color{blue}{\frac{0.125}{s \cdot \pi}} \cdot \frac{e^{-\frac{r}{s}}}{r} + 0.75 \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r \cdot {\left(\sqrt{s \cdot \left(\pi \cdot 6\right)}\right)}^{2}} \]
8. Step-by-step derivation

   1. unpow2 99.5%

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

   2. add-sqr-sqrt 99.5%

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

   3. *-commutative 99.5%

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

9. Applied egg-rr 99.5%

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

10. Taylor expanded in r around 0 99.4%

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

    1. metadata-eval 99.4%

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

    2. times-frac 99.5%

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

    3. *-commutative 99.5%

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

    4. associate-*r/ 99.5%

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

    5. neg-mul-1 99.5%

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

    6. distribute-neg-frac2 99.5%

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

    7. distribute-rgt-neg-in 99.5%

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

    8. metadata-eval 99.5%

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

12. Simplified 99.5%

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

13. Final simplification 99.5%

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

Alternative 2: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

   1. times-frac 99.5%

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

   2. *-commutative 99.5%

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

   3. distribute-frac-neg 99.5%

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

   4. associate-/l* 99.5%

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

   5. *-commutative 99.5%

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

   6. *-commutative 99.5%

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

   7. associate-*l* 99.4%

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

3. Simplified 99.4%

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

5. Taylor expanded in s around 0 99.4%

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

   1. associate-*r/ 99.4%

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

   2. rec-exp 99.4%

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

   3. associate-*r/ 99.4%

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

   4. metadata-eval 99.4%

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

   5. *-commutative 99.4%

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

   6. associate-*l* 99.4%

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

   7. *-commutative 99.4%

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

7. Simplified 99.4%

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

8. Taylor expanded in r around 0 99.4%

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

   1. *-commutative 99.4%

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

10. Simplified 99.4%

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

11. Taylor expanded in s around 0 99.4%

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

    1. *-commutative 99.4%

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

    2. associate-*r* 99.5%

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

    3. *-commutative 99.5%

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

13. Simplified 99.5%

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

14. Final simplification 99.5%

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

Alternative 3: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

   1. times-frac 99.5%

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

   2. *-commutative 99.5%

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

   3. distribute-frac-neg 99.5%

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

   4. associate-/l* 99.5%

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

   5. *-commutative 99.5%

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

   6. *-commutative 99.5%

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

   7. associate-*l* 99.4%

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

3. Simplified 99.4%

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

5. Taylor expanded in s around 0 99.4%

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

   1. associate-*r/ 99.4%

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

   2. rec-exp 99.4%

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

   3. associate-*r/ 99.4%

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

   4. metadata-eval 99.4%

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

   5. *-commutative 99.4%

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

   6. associate-*l* 99.4%

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

   7. *-commutative 99.4%

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

7. Simplified 99.4%

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

8. Taylor expanded in r around 0 99.4%

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

   1. *-commutative 99.4%

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

10. Simplified 99.4%

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

11. Final simplification 99.4%

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

Alternative 4: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

3. Simplified 99.1%

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

5. Taylor expanded in r around inf 99.4%

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

6. Final simplification 99.4%

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

Alternative 5: 99.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

3. Simplified 99.1%

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

5. Taylor expanded in r around inf 99.4%

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

   1. +-commutative 99.4%

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

   2. associate-*r/ 99.4%

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

   3. neg-mul-1 99.4%

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

   4. distribute-neg-frac2 99.4%

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

7. Simplified 99.4%

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

8. Final simplification 99.4%

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

Alternative 6: 46.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 28:\\ \;\;\;\;0.75 \cdot \frac{e^{\frac{r}{s} \cdot -0.3333333333333333}}{r \cdot \left(\left(s \cdot \pi\right) \cdot 6\right)} + \frac{\frac{0.125}{\frac{r}{s} + 1}}{s \cdot \left(\pi \cdot r\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.25}{s \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\pi \cdot r\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (if (<= r 28.0)
   (+
    (* 0.75 (/ (exp (* (/ r s) -0.3333333333333333)) (* r (* (* s PI) 6.0))))
    (/ (/ 0.125 (+ (/ r s) 1.0)) (* s (* PI r))))
   (/ -0.25 (* s (log1p (expm1 (* PI r)))))))

C

float code(float s, float r) {
	float tmp;
	if (r <= 28.0f) {
		tmp = (0.75f * (expf(((r / s) * -0.3333333333333333f)) / (r * ((s * ((float) M_PI)) * 6.0f)))) + ((0.125f / ((r / s) + 1.0f)) / (s * (((float) M_PI) * r)));
	} else {
		tmp = -0.25f / (s * log1pf(expm1f((((float) M_PI) * r))));
	}
	return tmp;
}

Julia

function code(s, r)
	tmp = Float32(0.0)
	if (r <= Float32(28.0))
		tmp = Float32(Float32(Float32(0.75) * Float32(exp(Float32(Float32(r / s) * Float32(-0.3333333333333333))) / Float32(r * Float32(Float32(s * Float32(pi)) * Float32(6.0))))) + Float32(Float32(Float32(0.125) / Float32(Float32(r / s) + Float32(1.0))) / Float32(s * Float32(Float32(pi) * r))));
	else
		tmp = Float32(Float32(-0.25) / Float32(s * log1p(expm1(Float32(Float32(pi) * r)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if r < 28

   1. Initial program 99.2%

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

      1. times-frac 99.2%

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

      2. *-commutative 99.2%

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

      3. distribute-frac-neg 99.2%

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

      4. associate-/l* 99.2%

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

      5. *-commutative 99.2%

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

      6. *-commutative 99.2%

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

      7. associate-*l* 99.2%

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

   3. Simplified 99.2%

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

   5. Taylor expanded in s around 0 99.1%

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

      1. associate-*r/ 99.1%

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

      2. rec-exp 99.1%

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

      3. associate-*r/ 99.1%

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

      4. metadata-eval 99.1%

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

      5. *-commutative 99.1%

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

      6. associate-*l* 99.1%

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

      7. *-commutative 99.1%

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

   7. Simplified 99.1%

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

   8. Taylor expanded in r around 0 99.2%

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

      1. *-commutative 99.2%

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

   10. Simplified 99.2%

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

   11. Taylor expanded in r around 0 17.0%

       \[\leadsto \frac{\frac{0.125}{\color{blue}{1 + \frac{r}{s}}}}{s \cdot \left(r \cdot \pi\right)} + 0.75 \cdot \frac{e^{\frac{r}{s} \cdot -0.3333333333333333}}{r \cdot \left(6 \cdot \left(\pi \cdot s\right)\right)} \]

   if 28 < r

   1. Initial program 99.9%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in r around 0 5.9%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \color{blue}{\frac{2}{r}} \]
   6. Step-by-step derivation

      1. frac-2neg 5.9%

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

      2. frac-2neg 5.9%

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

      3. frac-times 5.9%

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

      4. metadata-eval 5.9%

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

      5. metadata-eval 5.9%

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

      6. metadata-eval 5.9%

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

      7. add-sqr-sqrt -0.0%

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

      8. sqrt-unprod 5.1%

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

      9. sqr-neg 5.1%

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

      10. sqrt-unprod 5.1%

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

      11. add-sqr-sqrt 5.1%

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

   7. Applied egg-rr 5.1%

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

      1. associate-/r* 5.1%

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

      2. neg-mul-1 5.1%

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

      3. associate-/r* 5.1%

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

      4. metadata-eval 5.1%

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

      5. associate-/r* 5.1%

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

      6. associate-*l* 5.1%

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

   9. Simplified 5.1%

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

       1. log1p-expm1-u 96.4%

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

   11. Applied egg-rr 96.4%

       \[\leadsto \frac{-0.25}{s \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\pi \cdot r\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 48.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 28:\\ \;\;\;\;0.75 \cdot \frac{e^{\frac{r}{s} \cdot -0.3333333333333333}}{r \cdot \left(\left(s \cdot \pi\right) \cdot 6\right)} + \frac{\frac{0.125}{\frac{r}{s} + 1}}{s \cdot \left(\pi \cdot r\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.25}{s \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\pi \cdot r\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 15.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ 0.75 \cdot \frac{e^{\frac{r}{s} \cdot -0.3333333333333333}}{r \cdot \left(\left(s \cdot \pi\right) \cdot 6\right)} + \frac{\frac{0.125}{\frac{r}{s} + 1}}{s \cdot \left(\pi \cdot r\right)} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (* 0.75 (/ (exp (* (/ r s) -0.3333333333333333)) (* r (* (* s PI) 6.0))))
  (/ (/ 0.125 (+ (/ r s) 1.0)) (* s (* PI r)))))

C

float code(float s, float r) {
	return (0.75f * (expf(((r / s) * -0.3333333333333333f)) / (r * ((s * ((float) M_PI)) * 6.0f)))) + ((0.125f / ((r / s) + 1.0f)) / (s * (((float) M_PI) * r)));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(0.75) * Float32(exp(Float32(Float32(r / s) * Float32(-0.3333333333333333))) / Float32(r * Float32(Float32(s * Float32(pi)) * Float32(6.0))))) + Float32(Float32(Float32(0.125) / Float32(Float32(r / s) + Float32(1.0))) / Float32(s * Float32(Float32(pi) * r))))
end

MATLAB

function tmp = code(s, r)
	tmp = (single(0.75) * (exp(((r / s) * single(-0.3333333333333333))) / (r * ((s * single(pi)) * single(6.0))))) + ((single(0.125) / ((r / s) + single(1.0))) / (s * (single(pi) * r)));
end

Derivation

1. Initial program 99.4%

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

   1. times-frac 99.5%

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

   2. *-commutative 99.5%

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

   3. distribute-frac-neg 99.5%

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

   4. associate-/l* 99.5%

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

   5. *-commutative 99.5%

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

   6. *-commutative 99.5%

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

   7. associate-*l* 99.4%

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

3. Simplified 99.4%

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

5. Taylor expanded in s around 0 99.4%

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

   1. associate-*r/ 99.4%

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

   2. rec-exp 99.4%

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

   3. associate-*r/ 99.4%

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

   4. metadata-eval 99.4%

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

   5. *-commutative 99.4%

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

   6. associate-*l* 99.4%

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

   7. *-commutative 99.4%

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

7. Simplified 99.4%

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

8. Taylor expanded in r around 0 99.4%

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

   1. *-commutative 99.4%

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

10. Simplified 99.4%

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

11. Taylor expanded in r around 0 16.4%

    \[\leadsto \frac{\frac{0.125}{\color{blue}{1 + \frac{r}{s}}}}{s \cdot \left(r \cdot \pi\right)} + 0.75 \cdot \frac{e^{\frac{r}{s} \cdot -0.3333333333333333}}{r \cdot \left(6 \cdot \left(\pi \cdot s\right)\right)} \]

12. Final simplification 16.4%

    \[\leadsto 0.75 \cdot \frac{e^{\frac{r}{s} \cdot -0.3333333333333333}}{r \cdot \left(\left(s \cdot \pi\right) \cdot 6\right)} + \frac{\frac{0.125}{\frac{r}{s} + 1}}{s \cdot \left(\pi \cdot r\right)} \]
13. Add Preprocessing

Alternative 8: 10.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \frac{0.125}{s \cdot \pi} \cdot \left(r \cdot \frac{0.5555555555555556}{{s}^{2}} + \left(\frac{2}{r} + \frac{-1.3333333333333333}{s}\right)\right) \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (*
  (/ 0.125 (* s PI))
  (+
   (* r (/ 0.5555555555555556 (pow s 2.0)))
   (+ (/ 2.0 r) (/ -1.3333333333333333 s)))))

C

float code(float s, float r) {
	return (0.125f / (s * ((float) M_PI))) * ((r * (0.5555555555555556f / powf(s, 2.0f))) + ((2.0f / r) + (-1.3333333333333333f / s)));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(0.125) / Float32(s * Float32(pi))) * Float32(Float32(r * Float32(Float32(0.5555555555555556) / (s ^ Float32(2.0)))) + Float32(Float32(Float32(2.0) / r) + Float32(Float32(-1.3333333333333333) / s))))
end

MATLAB

function tmp = code(s, r)
	tmp = (single(0.125) / (s * single(pi))) * ((r * (single(0.5555555555555556) / (s ^ single(2.0)))) + ((single(2.0) / r) + (single(-1.3333333333333333) / s)));
end

Derivation

1. Initial program 99.4%

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

3. Simplified 99.1%

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

5. Taylor expanded in s around inf 11.3%

   \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \color{blue}{\left(\left(0.05555555555555555 \cdot \frac{r}{{s}^{2}} + \left(0.5 \cdot \frac{r}{{s}^{2}} + 2 \cdot \frac{1}{r}\right)\right) - 1.3333333333333333 \cdot \frac{1}{s}\right)} \]
6. Step-by-step derivation

   1. associate-+r+ 11.3%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\color{blue}{\left(\left(0.05555555555555555 \cdot \frac{r}{{s}^{2}} + 0.5 \cdot \frac{r}{{s}^{2}}\right) + 2 \cdot \frac{1}{r}\right)} - 1.3333333333333333 \cdot \frac{1}{s}\right) \]

   2. distribute-rgt-out 11.3%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\left(\color{blue}{\frac{r}{{s}^{2}} \cdot \left(0.05555555555555555 + 0.5\right)} + 2 \cdot \frac{1}{r}\right) - 1.3333333333333333 \cdot \frac{1}{s}\right) \]

   3. metadata-eval 11.3%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\left(\frac{r}{{s}^{2}} \cdot \color{blue}{0.5555555555555556} + 2 \cdot \frac{1}{r}\right) - 1.3333333333333333 \cdot \frac{1}{s}\right) \]

   4. *-commutative 11.3%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\left(\color{blue}{0.5555555555555556 \cdot \frac{r}{{s}^{2}}} + 2 \cdot \frac{1}{r}\right) - 1.3333333333333333 \cdot \frac{1}{s}\right) \]

   5. associate--l+ 11.3%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \color{blue}{\left(0.5555555555555556 \cdot \frac{r}{{s}^{2}} + \left(2 \cdot \frac{1}{r} - 1.3333333333333333 \cdot \frac{1}{s}\right)\right)} \]

   6. *-commutative 11.3%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\color{blue}{\frac{r}{{s}^{2}} \cdot 0.5555555555555556} + \left(2 \cdot \frac{1}{r} - 1.3333333333333333 \cdot \frac{1}{s}\right)\right) \]

   7. associate-*l/ 11.3%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\color{blue}{\frac{r \cdot 0.5555555555555556}{{s}^{2}}} + \left(2 \cdot \frac{1}{r} - 1.3333333333333333 \cdot \frac{1}{s}\right)\right) \]

   8. associate-*r/ 11.3%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\color{blue}{r \cdot \frac{0.5555555555555556}{{s}^{2}}} + \left(2 \cdot \frac{1}{r} - 1.3333333333333333 \cdot \frac{1}{s}\right)\right) \]

   9. sub-neg 11.3%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(r \cdot \frac{0.5555555555555556}{{s}^{2}} + \color{blue}{\left(2 \cdot \frac{1}{r} + \left(-1.3333333333333333 \cdot \frac{1}{s}\right)\right)}\right) \]

   10. associate-*r/ 11.3%

       \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(r \cdot \frac{0.5555555555555556}{{s}^{2}} + \left(\color{blue}{\frac{2 \cdot 1}{r}} + \left(-1.3333333333333333 \cdot \frac{1}{s}\right)\right)\right) \]

   11. metadata-eval 11.3%

       \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(r \cdot \frac{0.5555555555555556}{{s}^{2}} + \left(\frac{\color{blue}{2}}{r} + \left(-1.3333333333333333 \cdot \frac{1}{s}\right)\right)\right) \]

   12. associate-*r/ 11.3%

       \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(r \cdot \frac{0.5555555555555556}{{s}^{2}} + \left(\frac{2}{r} + \left(-\color{blue}{\frac{1.3333333333333333 \cdot 1}{s}}\right)\right)\right) \]

   13. metadata-eval 11.3%

       \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(r \cdot \frac{0.5555555555555556}{{s}^{2}} + \left(\frac{2}{r} + \left(-\frac{\color{blue}{1.3333333333333333}}{s}\right)\right)\right) \]

   14. distribute-neg-frac 11.3%

       \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(r \cdot \frac{0.5555555555555556}{{s}^{2}} + \left(\frac{2}{r} + \color{blue}{\frac{-1.3333333333333333}{s}}\right)\right) \]

   15. metadata-eval 11.3%

       \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(r \cdot \frac{0.5555555555555556}{{s}^{2}} + \left(\frac{2}{r} + \frac{\color{blue}{-1.3333333333333333}}{s}\right)\right) \]

7. Simplified 11.3%

   \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \color{blue}{\left(r \cdot \frac{0.5555555555555556}{{s}^{2}} + \left(\frac{2}{r} + \frac{-1.3333333333333333}{s}\right)\right)} \]
8. Add Preprocessing

Alternative 9: 10.0% accurate, 12.2× speedup

Math

\[\begin{array}{l} \\ \frac{0.125}{s \cdot \pi} \cdot \left(\frac{2}{r} + \frac{\frac{r \cdot 0.5555555555555556}{s} - 1.3333333333333333}{s}\right) \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (*
  (/ 0.125 (* s PI))
  (+ (/ 2.0 r) (/ (- (/ (* r 0.5555555555555556) s) 1.3333333333333333) s))))

C

float code(float s, float r) {
	return (0.125f / (s * ((float) M_PI))) * ((2.0f / r) + ((((r * 0.5555555555555556f) / s) - 1.3333333333333333f) / s));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(0.125) / Float32(s * Float32(pi))) * Float32(Float32(Float32(2.0) / r) + Float32(Float32(Float32(Float32(r * Float32(0.5555555555555556)) / s) - Float32(1.3333333333333333)) / s)))
end

MATLAB

function tmp = code(s, r)
	tmp = (single(0.125) / (s * single(pi))) * ((single(2.0) / r) + ((((r * single(0.5555555555555556)) / s) - single(1.3333333333333333)) / s));
end

Derivation

1. Initial program 99.4%

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

3. Simplified 99.1%

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

5. Taylor expanded in s around -inf 11.3%

   \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \color{blue}{\left(-1 \cdot \frac{1.3333333333333333 + -1 \cdot \frac{0.05555555555555555 \cdot r + 0.5 \cdot r}{s}}{s} + 2 \cdot \frac{1}{r}\right)} \]
6. Step-by-step derivation

   1. +-commutative 11.3%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \color{blue}{\left(2 \cdot \frac{1}{r} + -1 \cdot \frac{1.3333333333333333 + -1 \cdot \frac{0.05555555555555555 \cdot r + 0.5 \cdot r}{s}}{s}\right)} \]

   2. mul-1-neg 11.3%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(2 \cdot \frac{1}{r} + \color{blue}{\left(-\frac{1.3333333333333333 + -1 \cdot \frac{0.05555555555555555 \cdot r + 0.5 \cdot r}{s}}{s}\right)}\right) \]

   3. unsub-neg 11.3%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \color{blue}{\left(2 \cdot \frac{1}{r} - \frac{1.3333333333333333 + -1 \cdot \frac{0.05555555555555555 \cdot r + 0.5 \cdot r}{s}}{s}\right)} \]

   4. associate-*r/ 11.3%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\color{blue}{\frac{2 \cdot 1}{r}} - \frac{1.3333333333333333 + -1 \cdot \frac{0.05555555555555555 \cdot r + 0.5 \cdot r}{s}}{s}\right) \]

   5. metadata-eval 11.3%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{\color{blue}{2}}{r} - \frac{1.3333333333333333 + -1 \cdot \frac{0.05555555555555555 \cdot r + 0.5 \cdot r}{s}}{s}\right) \]

   6. mul-1-neg 11.3%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{2}{r} - \frac{1.3333333333333333 + \color{blue}{\left(-\frac{0.05555555555555555 \cdot r + 0.5 \cdot r}{s}\right)}}{s}\right) \]

   7. unsub-neg 11.3%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{2}{r} - \frac{\color{blue}{1.3333333333333333 - \frac{0.05555555555555555 \cdot r + 0.5 \cdot r}{s}}}{s}\right) \]

   8. distribute-rgt-out 11.3%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{2}{r} - \frac{1.3333333333333333 - \frac{\color{blue}{r \cdot \left(0.05555555555555555 + 0.5\right)}}{s}}{s}\right) \]

   9. metadata-eval 11.3%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{2}{r} - \frac{1.3333333333333333 - \frac{r \cdot \color{blue}{0.5555555555555556}}{s}}{s}\right) \]

7. Simplified 11.3%

   \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \color{blue}{\left(\frac{2}{r} - \frac{1.3333333333333333 - \frac{r \cdot 0.5555555555555556}{s}}{s}\right)} \]

8. Final simplification 11.3%

   \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{2}{r} + \frac{\frac{r \cdot 0.5555555555555556}{s} - 1.3333333333333333}{s}\right) \]
9. Add Preprocessing

Alternative 10: 9.0% accurate, 33.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

3. Simplified 99.1%

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

5. Taylor expanded in r around 0 10.1%

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

6. Taylor expanded in s around 0 10.1%

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

   1. associate-/r* 10.1%

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

   2. metadata-eval 10.1%

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

   3. associate-*r/ 10.1%

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

   4. associate-*l/ 10.1%

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

   5. associate-/r* 10.1%

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

   6. times-frac 10.1%

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

   7. associate-/r* 10.1%

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

   8. associate-*l/ 10.1%

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

   9. metadata-eval 10.1%

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

8. Simplified 10.1%

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

Alternative 11: 9.0% accurate, 33.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

3. Simplified 99.1%

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

5. Taylor expanded in r around 0 10.1%

   \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \color{blue}{\frac{2}{r}} \]
6. Step-by-step derivation

   1. frac-times 10.1%

      \[\leadsto \color{blue}{\frac{0.125 \cdot 2}{\left(s \cdot \pi\right) \cdot r}} \]

   2. metadata-eval 10.1%

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

   3. *-commutative 10.1%

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

   4. associate-*r* 10.1%

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

7. Applied egg-rr 10.1%

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

8. Final simplification 10.1%

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

Alternative 12: 9.0% accurate, 33.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

3. Simplified 99.1%

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

5. Taylor expanded in r around 0 10.1%

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

6. Taylor expanded in s around 0 10.1%

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

7. Final simplification 10.1%

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

Alternative 13: 4.4% accurate, 33.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

3. Simplified 99.1%

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

5. Taylor expanded in r around 0 10.1%

   \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \color{blue}{\frac{2}{r}} \]
6. Step-by-step derivation

   1. frac-2neg 10.1%

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

   2. frac-2neg 10.1%

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

   3. frac-times 10.1%

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

   4. metadata-eval 10.1%

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

   5. metadata-eval 10.1%

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

   6. metadata-eval 10.1%

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

   7. add-sqr-sqrt -0.0%

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

   8. sqrt-unprod 4.4%

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

   9. sqr-neg 4.4%

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

   10. sqrt-unprod 4.4%

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

   11. add-sqr-sqrt 4.4%

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

7. Applied egg-rr 4.4%

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

   1. associate-/r* 4.4%

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

   2. neg-mul-1 4.4%

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

   3. associate-/r* 4.4%

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

   4. metadata-eval 4.4%

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

   5. associate-/r* 4.4%

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

   6. associate-*l* 4.4%

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

9. Simplified 4.4%

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

Reproduce

herbie shell --seed 2024110 
(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))))