Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.6%

Time: 14.7s

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

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

   1. pow1 99.4%

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

   2. associate-*l* 99.4%

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

   3. *-commutative 99.4%

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

4. Applied egg-rr 99.4%

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

   1. unpow1 99.4%

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

   2. *-commutative 99.4%

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

6. Simplified 99.4%

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

7. Taylor expanded in r around inf 99.4%

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

   1. neg-mul-1 99.4%

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

   2. rec-exp 99.4%

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

   3. associate-*r/ 99.4%

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

   4. metadata-eval 99.4%

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

9. Simplified 99.4%

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

10. Taylor expanded in r around inf 99.4%

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

    1. associate-/r* 99.4%

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

    2. associate-*r* 99.4%

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

12. Simplified 99.4%

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

13. Final simplification 99.4%

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

Alternative 2: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

   1. pow1 99.4%

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

   2. associate-*l* 99.4%

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

   3. *-commutative 99.4%

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

4. Applied egg-rr 99.4%

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

   1. unpow1 99.4%

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

   2. *-commutative 99.4%

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

6. Simplified 99.4%

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

7. Taylor expanded in r around inf 99.4%

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

   1. neg-mul-1 99.4%

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

   2. rec-exp 99.4%

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

   3. associate-*r/ 99.4%

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

   4. metadata-eval 99.4%

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

9. Simplified 99.4%

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

10. Taylor expanded in r around inf 99.4%

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

    1. associate-/r* 99.4%

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

    2. associate-*r* 99.4%

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

12. Simplified 99.4%

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

13. Taylor expanded in r around 0 99.4%

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

    1. associate-*r/ 99.4%

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

    2. associate-*l/ 99.4%

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

    3. *-commutative 99.4%

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

15. Simplified 99.4%

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

Alternative 3: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.125) * Float32(exp(Float32(Float32(-r) / s)) / Float32(r * Float32(pi)))) + Float32(Float32(0.125) * Float32((Float32(exp(1)) ^ Float32(r * Float32(Float32(-0.3333333333333333) / s))) / Float32(r * Float32(pi))))) / s)
end

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.125) * (exp((-r / s)) / (r * single(pi)))) + (single(0.125) * ((single(2.71828182845904523536) ^ (r * (single(-0.3333333333333333) / s))) / (r * single(pi))))) / s;
end

Derivation

1. Initial program 99.4%

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

   1. +-commutative 99.4%

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

   2. times-frac 99.4%

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

   3. fma-define 99.4%

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

   4. associate-*l* 99.3%

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

   5. associate-/r* 99.3%

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

   6. metadata-eval 99.3%

      \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{0.125}}{\pi \cdot s}, \frac{e^{\frac{-r}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   7. *-commutative 99.3%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{\color{blue}{s \cdot \pi}}, \frac{e^{\frac{-r}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   8. neg-mul-1 99.3%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\frac{\color{blue}{-1 \cdot r}}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   9. times-frac 99.3%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\color{blue}{\frac{-1}{3} \cdot \frac{r}{s}}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   10. metadata-eval 99.3%

       \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\color{blue}{-0.3333333333333333} \cdot \frac{r}{s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   11. times-frac 99.3%

       \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r}, \color{blue}{\frac{0.25}{\left(2 \cdot \pi\right) \cdot s} \cdot \frac{e^{\frac{-r}{s}}}{r}}\right) \]

3. Simplified 99.3%

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

5. Taylor expanded in s around 0 99.4%

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

   1. associate-*r/ 99.4%

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

   2. *-commutative 99.4%

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

   3. *-un-lft-identity 99.4%

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

   4. exp-prod 99.4%

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

   5. associate-/l* 99.4%

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

7. Applied egg-rr 99.4%

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

   1. exp-1-e 99.4%

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

9. Simplified 99.4%

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

10. Final simplification 99.4%

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

Alternative 4: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

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. Step-by-step derivation

   1. pow-exp 99.3%

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

   2. associate-*r/ 99.3%

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

6. Applied egg-rr 99.3%

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

   1. *-un-lft-identity 99.3%

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

   2. associate-/r* 99.4%

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

8. Applied egg-rr 99.4%

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

9. Final simplification 99.4%

   \[\leadsto \frac{\frac{0.125}{s}}{\pi} \cdot \left(\frac{e^{\frac{-r}{s}}}{r} + \frac{e^{\frac{r \cdot -0.3333333333333333}{s}}}{r}\right) \]
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(Float32(-r) / 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 99.4%

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

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. Step-by-step derivation

   1. add-sqr-sqrt 99.1%

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

   2. sqrt-unprod 98.9%

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

   3. pow-prod-down 98.9%

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

   4. prod-exp 99.2%

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

   5. metadata-eval 99.2%

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

6. Applied egg-rr 99.2%

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

7. Taylor expanded in s around 0 99.2%

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

   1. associate-*r/ 99.2%

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

   2. *-commutative 99.2%

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

   3. times-frac 99.1%

      \[\leadsto \color{blue}{\frac{0.125}{\pi} \cdot \frac{\frac{1}{r} \cdot \sqrt{e^{-0.6666666666666666 \cdot \frac{r}{s}}} + \frac{e^{-1 \cdot \frac{r}{s}}}{r}}{s}} \]

   4. +-commutative 99.1%

      \[\leadsto \frac{0.125}{\pi} \cdot \frac{\color{blue}{\frac{e^{-1 \cdot \frac{r}{s}}}{r} + \frac{1}{r} \cdot \sqrt{e^{-0.6666666666666666 \cdot \frac{r}{s}}}}}{s} \]

   5. mul-1-neg 99.1%

      \[\leadsto \frac{0.125}{\pi} \cdot \frac{\frac{e^{\color{blue}{-\frac{r}{s}}}}{r} + \frac{1}{r} \cdot \sqrt{e^{-0.6666666666666666 \cdot \frac{r}{s}}}}{s} \]

   6. distribute-neg-frac2 99.1%

      \[\leadsto \frac{0.125}{\pi} \cdot \frac{\frac{e^{\color{blue}{\frac{r}{-s}}}}{r} + \frac{1}{r} \cdot \sqrt{e^{-0.6666666666666666 \cdot \frac{r}{s}}}}{s} \]

   7. exp-prod 99.2%

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

   8. associate-*l/ 99.2%

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

   9. *-lft-identity 99.2%

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

9. Simplified 99.2%

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

10. Taylor expanded in r around inf 99.2%

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

    1. associate-*r/ 99.2%

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

    2. neg-mul-1 99.2%

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

    3. exp-prod 99.3%

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

    4. unpow1/2 99.3%

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

    5. exp-prod 99.2%

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

    6. *-commutative 99.2%

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

    7. exp-prod 99.4%

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

    8. associate-*l* 99.4%

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

    9. metadata-eval 99.4%

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

12. Simplified 99.4%

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

Alternative 6: 49.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 28:\\ \;\;\;\;\frac{0.75 \cdot e^{\frac{r}{s \cdot \left(-3\right)}}}{\left(\pi \cdot 6\right) \cdot \left(r \cdot s\right)} + \frac{\frac{0.125}{r}}{s \cdot \pi + r \cdot \left(\pi + 0.5 \cdot \frac{r \cdot \pi}{s}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.25}{\mathsf{log1p}\left(\mathsf{expm1}\left(r \cdot \pi\right)\right)}}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (if (<= r 28.0)
   (+
    (/ (* 0.75 (exp (/ r (* s (- 3.0))))) (* (* PI 6.0) (* r s)))
    (/ (/ 0.125 r) (+ (* s PI) (* r (+ PI (* 0.5 (/ (* r PI) s)))))))
   (/ (/ 0.25 (log1p (expm1 (* r PI)))) s)))

C

float code(float s, float r) {
	float tmp;
	if (r <= 28.0f) {
		tmp = ((0.75f * expf((r / (s * -3.0f)))) / ((((float) M_PI) * 6.0f) * (r * s))) + ((0.125f / r) / ((s * ((float) M_PI)) + (r * (((float) M_PI) + (0.5f * ((r * ((float) M_PI)) / s))))));
	} else {
		tmp = (0.25f / log1pf(expm1f((r * ((float) M_PI))))) / s;
	}
	return tmp;
}

Julia

function code(s, r)
	tmp = Float32(0.0)
	if (r <= Float32(28.0))
		tmp = Float32(Float32(Float32(Float32(0.75) * exp(Float32(r / Float32(s * Float32(-Float32(3.0)))))) / Float32(Float32(Float32(pi) * Float32(6.0)) * Float32(r * s))) + Float32(Float32(Float32(0.125) / r) / Float32(Float32(s * Float32(pi)) + Float32(r * Float32(Float32(pi) + Float32(Float32(0.5) * Float32(Float32(r * Float32(pi)) / s)))))));
	else
		tmp = Float32(Float32(Float32(0.25) / log1p(expm1(Float32(r * Float32(pi))))) / s);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if r < 28

   1. Initial program 99.1%

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

      1. pow1 99.1%

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

      2. associate-*l* 99.2%

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

      3. *-commutative 99.2%

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

   4. Applied egg-rr 99.2%

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

      1. unpow1 99.2%

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

      2. *-commutative 99.2%

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

   6. Simplified 99.2%

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

   7. Taylor expanded in r around inf 99.2%

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

      1. neg-mul-1 99.2%

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

      2. rec-exp 99.2%

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

      3. associate-*r/ 99.2%

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

      4. metadata-eval 99.2%

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

   9. Simplified 99.2%

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

   10. Taylor expanded in r around inf 99.2%

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

       1. associate-/r* 99.2%

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

       2. associate-*r* 99.2%

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

   12. Simplified 99.2%

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

   13. Taylor expanded in r around 0 20.7%

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

   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 99.9%

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

      1. add-sqr-sqrt 99.9%

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

      2. sqrt-unprod 99.9%

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

      3. pow-prod-down 99.9%

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

      4. prod-exp 99.9%

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

      5. metadata-eval 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in s around inf 5.8%

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

      1. associate-/r* 5.8%

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

      2. *-commutative 5.8%

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

      3. associate-/r* 5.8%

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

      4. associate-/r* 5.8%

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

   9. Simplified 5.8%

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

       1. log1p-expm1-u 96.8%

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

   11. Applied egg-rr 96.8%

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

4. Final simplification 46.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 28:\\ \;\;\;\;\frac{0.75 \cdot e^{\frac{r}{s \cdot \left(-3\right)}}}{\left(\pi \cdot 6\right) \cdot \left(r \cdot s\right)} + \frac{\frac{0.125}{r}}{s \cdot \pi + r \cdot \left(\pi + 0.5 \cdot \frac{r \cdot \pi}{s}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.25}{\mathsf{log1p}\left(\mathsf{expm1}\left(r \cdot \pi\right)\right)}}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 25.5% accurate, 1.7× speedup

Math

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

FPCore

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

C

float code(float s, float r) {
	return ((0.75f * expf((r / (s * -3.0f)))) / ((((float) M_PI) * 6.0f) * (r * s))) + ((0.125f / r) / ((s * ((float) M_PI)) + (r * (((float) M_PI) + (0.5f * ((r * ((float) M_PI)) / s))))));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.75) * exp(Float32(r / Float32(s * Float32(-Float32(3.0)))))) / Float32(Float32(Float32(pi) * Float32(6.0)) * Float32(r * s))) + Float32(Float32(Float32(0.125) / r) / Float32(Float32(s * Float32(pi)) + Float32(r * Float32(Float32(pi) + Float32(Float32(0.5) * Float32(Float32(r * Float32(pi)) / s)))))))
end

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.75) * exp((r / (s * -single(3.0))))) / ((single(pi) * single(6.0)) * (r * s))) + ((single(0.125) / r) / ((s * single(pi)) + (r * (single(pi) + (single(0.5) * ((r * single(pi)) / s))))));
end

Derivation

1. Initial program 99.4%

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

   1. pow1 99.4%

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

   2. associate-*l* 99.4%

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

   3. *-commutative 99.4%

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

4. Applied egg-rr 99.4%

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

   1. unpow1 99.4%

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

   2. *-commutative 99.4%

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

6. Simplified 99.4%

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

7. Taylor expanded in r around inf 99.4%

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

   1. neg-mul-1 99.4%

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

   2. rec-exp 99.4%

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

   3. associate-*r/ 99.4%

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

   4. metadata-eval 99.4%

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

9. Simplified 99.4%

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

10. Taylor expanded in r around inf 99.4%

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

    1. associate-/r* 99.4%

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

    2. associate-*r* 99.4%

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

12. Simplified 99.4%

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

13. Taylor expanded in r around 0 25.1%

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

14. Final simplification 25.1%

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

Alternative 8: 15.9% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

   1. pow1 99.4%

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

   2. associate-*l* 99.4%

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

   3. *-commutative 99.4%

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

4. Applied egg-rr 99.4%

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

   1. unpow1 99.4%

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

   2. *-commutative 99.4%

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

6. Simplified 99.4%

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

7. Taylor expanded in r around inf 99.4%

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

   1. neg-mul-1 99.4%

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

   2. rec-exp 99.4%

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

   3. associate-*r/ 99.4%

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

   4. metadata-eval 99.4%

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

9. Simplified 99.4%

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

10. Taylor expanded in r around inf 99.4%

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

    1. associate-/r* 99.4%

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

    2. associate-*r* 99.4%

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

12. Simplified 99.4%

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

13. Taylor expanded in s around inf 16.6%

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

14. Final simplification 16.6%

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

Alternative 9: 12.4% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

   1. pow1 99.4%

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

   2. associate-*l* 99.4%

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

   3. *-commutative 99.4%

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

4. Applied egg-rr 99.4%

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

   1. unpow1 99.4%

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

   2. *-commutative 99.4%

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

6. Simplified 99.4%

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

7. Taylor expanded in r around inf 99.4%

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

   1. neg-mul-1 99.4%

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

   2. rec-exp 99.4%

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

   3. associate-*r/ 99.4%

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

   4. metadata-eval 99.4%

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

9. Simplified 99.4%

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

10. Taylor expanded in r around inf 99.4%

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

    1. associate-/r* 99.4%

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

    2. associate-*r* 99.4%

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

12. Simplified 99.4%

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

13. Taylor expanded in r around 0 13.7%

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

    1. +-commutative 13.7%

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

    2. distribute-rgt-out 13.7%

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

15. Simplified 13.7%

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

16. Final simplification 13.7%

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

Alternative 10: 10.1% accurate, 11.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\frac{r}{s \cdot \pi} \cdot 0.06944444444444445 - \frac{0.16666666666666666}{\pi}}{s} + \frac{0.25}{r \cdot \pi}}{s} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (/
  (+
   (/ (- (* (/ r (* s PI)) 0.06944444444444445) (/ 0.16666666666666666 PI)) s)
   (/ 0.25 (* r PI)))
  s))

C

float code(float s, float r) {
	return (((((r / (s * ((float) M_PI))) * 0.06944444444444445f) - (0.16666666666666666f / ((float) M_PI))) / s) + (0.25f / (r * ((float) M_PI)))) / s;
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(Float32(Float32(r / Float32(s * Float32(pi))) * Float32(0.06944444444444445)) - Float32(Float32(0.16666666666666666) / Float32(pi))) / s) + Float32(Float32(0.25) / Float32(r * Float32(pi)))) / s)
end

MATLAB

function tmp = code(s, r)
	tmp = (((((r / (s * single(pi))) * single(0.06944444444444445)) - (single(0.16666666666666666) / single(pi))) / s) + (single(0.25) / (r * single(pi)))) / s;
end

Derivation

1. Initial program 99.4%

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

   1. +-commutative 99.4%

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

   2. times-frac 99.4%

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

   3. fma-define 99.4%

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

   4. associate-*l* 99.3%

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

   5. associate-/r* 99.3%

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

   6. metadata-eval 99.3%

      \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{0.125}}{\pi \cdot s}, \frac{e^{\frac{-r}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   7. *-commutative 99.3%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{\color{blue}{s \cdot \pi}}, \frac{e^{\frac{-r}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   8. neg-mul-1 99.3%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\frac{\color{blue}{-1 \cdot r}}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   9. times-frac 99.3%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\color{blue}{\frac{-1}{3} \cdot \frac{r}{s}}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   10. metadata-eval 99.3%

       \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\color{blue}{-0.3333333333333333} \cdot \frac{r}{s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   11. times-frac 99.3%

       \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r}, \color{blue}{\frac{0.25}{\left(2 \cdot \pi\right) \cdot s} \cdot \frac{e^{\frac{-r}{s}}}{r}}\right) \]

3. Simplified 99.3%

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

5. Taylor expanded in s around 0 99.4%

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

6. Taylor expanded in s around -inf 11.3%

   \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{\left(0.006944444444444444 \cdot \frac{r}{s \cdot \pi} + 0.0625 \cdot \frac{r}{s \cdot \pi}\right) - 0.16666666666666666 \cdot \frac{1}{\pi}}{s} - 0.25 \cdot \frac{1}{r \cdot \pi}}{s}} \]
7. Step-by-step derivation

   1. mul-1-neg 11.3%

      \[\leadsto \color{blue}{-\frac{-1 \cdot \frac{\left(0.006944444444444444 \cdot \frac{r}{s \cdot \pi} + 0.0625 \cdot \frac{r}{s \cdot \pi}\right) - 0.16666666666666666 \cdot \frac{1}{\pi}}{s} - 0.25 \cdot \frac{1}{r \cdot \pi}}{s}} \]

8. Simplified 11.3%

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

9. Final simplification 11.3%

   \[\leadsto \frac{\frac{\frac{r}{s \cdot \pi} \cdot 0.06944444444444445 - \frac{0.16666666666666666}{\pi}}{s} + \frac{0.25}{r \cdot \pi}}{s} \]
10. Add Preprocessing

Alternative 11: 9.1% accurate, 17.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

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 10.3%

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

   1. associate-*r/ 10.3%

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

   2. metadata-eval 10.3%

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

   3. associate-*r/ 10.3%

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

   4. metadata-eval 10.3%

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

7. Simplified 10.3%

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

Alternative 12: 9.1% accurate, 17.8× speedup

Math

\[\begin{array}{l} \\ \frac{0.125}{\pi} \cdot \frac{\frac{2}{r} - \frac{1.3333333333333333}{s}}{s} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = (single(0.125) / single(pi)) * (((single(2.0) / r) - (single(1.3333333333333333) / s)) / 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. Step-by-step derivation

   1. add-sqr-sqrt 99.1%

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

   2. sqrt-unprod 98.9%

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

   3. pow-prod-down 98.9%

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

   4. prod-exp 99.2%

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

   5. metadata-eval 99.2%

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

6. Applied egg-rr 99.2%

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

7. Taylor expanded in s around 0 99.2%

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

   1. associate-*r/ 99.2%

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

   2. *-commutative 99.2%

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

   3. times-frac 99.1%

      \[\leadsto \color{blue}{\frac{0.125}{\pi} \cdot \frac{\frac{1}{r} \cdot \sqrt{e^{-0.6666666666666666 \cdot \frac{r}{s}}} + \frac{e^{-1 \cdot \frac{r}{s}}}{r}}{s}} \]

   4. +-commutative 99.1%

      \[\leadsto \frac{0.125}{\pi} \cdot \frac{\color{blue}{\frac{e^{-1 \cdot \frac{r}{s}}}{r} + \frac{1}{r} \cdot \sqrt{e^{-0.6666666666666666 \cdot \frac{r}{s}}}}}{s} \]

   5. mul-1-neg 99.1%

      \[\leadsto \frac{0.125}{\pi} \cdot \frac{\frac{e^{\color{blue}{-\frac{r}{s}}}}{r} + \frac{1}{r} \cdot \sqrt{e^{-0.6666666666666666 \cdot \frac{r}{s}}}}{s} \]

   6. distribute-neg-frac2 99.1%

      \[\leadsto \frac{0.125}{\pi} \cdot \frac{\frac{e^{\color{blue}{\frac{r}{-s}}}}{r} + \frac{1}{r} \cdot \sqrt{e^{-0.6666666666666666 \cdot \frac{r}{s}}}}{s} \]

   7. exp-prod 99.2%

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

   8. associate-*l/ 99.2%

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

   9. *-lft-identity 99.2%

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

9. Simplified 99.2%

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

10. Taylor expanded in s around inf 10.2%

    \[\leadsto \frac{0.125}{\pi} \cdot \color{blue}{\frac{2 \cdot \frac{1}{r} - 1.3333333333333333 \cdot \frac{1}{s}}{s}} \]
11. Step-by-step derivation

    1. associate-*r/ 10.2%

       \[\leadsto \frac{0.125}{\pi} \cdot \frac{\color{blue}{\frac{2 \cdot 1}{r}} - 1.3333333333333333 \cdot \frac{1}{s}}{s} \]

    2. metadata-eval 10.2%

       \[\leadsto \frac{0.125}{\pi} \cdot \frac{\frac{\color{blue}{2}}{r} - 1.3333333333333333 \cdot \frac{1}{s}}{s} \]

    3. associate-*r/ 10.2%

       \[\leadsto \frac{0.125}{\pi} \cdot \frac{\frac{2}{r} - \color{blue}{\frac{1.3333333333333333 \cdot 1}{s}}}{s} \]

    4. metadata-eval 10.2%

       \[\leadsto \frac{0.125}{\pi} \cdot \frac{\frac{2}{r} - \frac{\color{blue}{1.3333333333333333}}{s}}{s} \]

12. Simplified 10.2%

    \[\leadsto \frac{0.125}{\pi} \cdot \color{blue}{\frac{\frac{2}{r} - \frac{1.3333333333333333}{s}}{s}} \]
13. Add Preprocessing

Alternative 13: 9.0% accurate, 25.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = single(1.0) / ((r * single(pi)) / (single(0.25) / 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. Step-by-step derivation

   1. add-sqr-sqrt 99.1%

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

   2. sqrt-unprod 98.9%

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

   3. pow-prod-down 98.9%

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

   4. prod-exp 99.2%

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

   5. metadata-eval 99.2%

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

6. Applied egg-rr 99.2%

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

7. Taylor expanded in s around inf 10.0%

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

   1. associate-/r* 10.0%

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

   2. *-commutative 10.0%

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

   3. associate-/r* 10.0%

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

   4. associate-/r* 10.0%

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

9. Simplified 10.0%

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

    1. clear-num 10.0%

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

    2. inv-pow 10.0%

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

11. Applied egg-rr 10.0%

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

    1. associate-/r/ 10.0%

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

    2. unpow-prod-down 10.0%

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

    3. inv-pow 10.0%

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

    4. clear-num 10.0%

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

    5. inv-pow 10.0%

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

    6. div-inv 10.0%

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

    7. clear-num 10.0%

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

13. Applied egg-rr 10.0%

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

Alternative 14: 9.0% accurate, 25.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = single(1.0) / ((r * single(pi)) * (s / single(0.25)));
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. Step-by-step derivation

   1. add-sqr-sqrt 99.1%

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

   2. sqrt-unprod 98.9%

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

   3. pow-prod-down 98.9%

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

   4. prod-exp 99.2%

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

   5. metadata-eval 99.2%

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

6. Applied egg-rr 99.2%

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

7. Taylor expanded in s around inf 10.0%

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

   1. associate-/r* 10.0%

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

   2. *-commutative 10.0%

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

   3. associate-/r* 10.0%

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

   4. associate-/r* 10.0%

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

9. Simplified 10.0%

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

    1. clear-num 10.0%

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

    2. inv-pow 10.0%

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

11. Applied egg-rr 10.0%

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

    1. unpow-1 10.0%

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

    2. associate-/r/ 10.0%

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

13. Simplified 10.0%

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

14. Final simplification 10.0%

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

Alternative 15: 9.0% accurate, 33.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = single(0.25) / (s * (r * single(pi)));
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. Step-by-step derivation

   1. add-sqr-sqrt 99.1%

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

   2. sqrt-unprod 98.9%

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

   3. pow-prod-down 98.9%

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

   4. prod-exp 99.2%

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

   5. metadata-eval 99.2%

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

6. Applied egg-rr 99.2%

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

7. Taylor expanded in s around inf 10.0%

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

   1. associate-/r* 10.0%

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

   2. *-commutative 10.0%

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

   3. associate-/r* 10.0%

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

   4. associate-/r* 10.0%

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

9. Simplified 10.0%

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

    1. clear-num 10.0%

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

    2. inv-pow 10.0%

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

11. Applied egg-rr 10.0%

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

    1. associate-/r/ 10.0%

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

    2. unpow-prod-down 10.0%

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

    3. inv-pow 10.0%

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

    4. clear-num 10.0%

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

    5. inv-pow 10.0%

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

    6. div-inv 10.0%

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

    7. associate-/l/ 10.0%

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

13. Applied egg-rr 10.0%

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

14. Final simplification 10.0%

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

Alternative 16: 9.0% accurate, 33.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

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 10.0%

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

Reproduce

herbie shell --seed 2024151 
(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))))