Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.5%

Time: 9.5s

Alternatives: 21

Speedup: N/A×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

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. Add Preprocessing

3. Taylor expanded in r around inf 99.2%

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

   1. associate-*r/ 99.2%

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

   2. *-commutative 99.2%

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

   3. associate-*l* 99.2%

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

   4. *-commutative 99.2%

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

   5. times-frac 99.3%

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

   6. mul-1-neg 99.3%

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

   7. distribute-neg-frac2 99.3%

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

5. Simplified 99.3%

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

6. Final simplification 99.3%

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

Alternative 2: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

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. Add Preprocessing

3. Taylor expanded in r around inf 99.2%

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

   1. associate-*r/ 99.2%

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

   2. *-commutative 99.2%

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

   3. associate-*l* 99.2%

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

   4. *-commutative 99.2%

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

   5. times-frac 99.3%

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

   6. mul-1-neg 99.3%

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

   7. distribute-neg-frac2 99.3%

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

5. Simplified 99.3%

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

   1. neg-mul-1 99.3%

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

   2. times-frac 99.2%

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

   3. metadata-eval 99.2%

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

   4. rem-log-exp 99.2%

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

   5. pow-exp 99.0%

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

   6. log-pow 99.0%

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

   7. rem-log-exp 99.2%

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

7. Applied egg-rr 99.2%

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

8. Final simplification 99.2%

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

Alternative 3: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.2%

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

3. Simplified 99.0%

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

5. Taylor expanded in s around 0 99.2%

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

6. Final simplification 99.2%

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

Alternative 4: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Derivation

1. Initial program 99.2%

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

3. Simplified 99.0%

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

   1. add-exp-log 99.0%

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

   2. log-pow 99.0%

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

   3. rem-log-exp 99.2%

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

6. Applied egg-rr 99.2%

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

7. Final simplification 99.2%

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

Alternative 5: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.2%

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

3. Simplified 99.0%

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

5. Taylor expanded in r around inf 99.1%

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

   1. mul-1-neg 99.1%

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

   2. exp-neg 99.2%

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

7. Applied egg-rr 99.2%

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

   1. rec-exp 99.1%

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

   2. distribute-neg-frac2 99.1%

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

9. Simplified 99.1%

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

10. Final simplification 99.1%

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

Alternative 6: 91.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;s \leq 0.05999999865889549:\\ \;\;\;\;0.125 \cdot \frac{e^{\frac{r}{-s}}}{r \cdot \left(s \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.75 \cdot e^{\frac{r}{s} \cdot -0.3333333333333333}}{r \cdot \left(s \cdot \left(\pi \cdot 6\right)\right)} + \frac{0.125}{s} \cdot \left(\frac{1}{r \cdot \pi} + \frac{\frac{-1}{\pi} - \frac{r}{s \cdot \pi} \cdot -0.5}{s}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (if (<= s 0.05999999865889549)
   (* 0.125 (/ (exp (/ r (- s))) (* r (* s PI))))
   (+
    (/ (* 0.75 (exp (* (/ r s) -0.3333333333333333))) (* r (* s (* PI 6.0))))
    (*
     (/ 0.125 s)
     (+ (/ 1.0 (* r PI)) (/ (- (/ -1.0 PI) (* (/ r (* s PI)) -0.5)) s))))))

C

float code(float s, float r) {
	float tmp;
	if (s <= 0.05999999865889549f) {
		tmp = 0.125f * (expf((r / -s)) / (r * (s * ((float) M_PI))));
	} else {
		tmp = ((0.75f * expf(((r / s) * -0.3333333333333333f))) / (r * (s * (((float) M_PI) * 6.0f)))) + ((0.125f / s) * ((1.0f / (r * ((float) M_PI))) + (((-1.0f / ((float) M_PI)) - ((r / (s * ((float) M_PI))) * -0.5f)) / s)));
	}
	return tmp;
}

Julia

function code(s, r)
	tmp = Float32(0.0)
	if (s <= Float32(0.05999999865889549))
		tmp = Float32(Float32(0.125) * Float32(exp(Float32(r / Float32(-s))) / Float32(r * Float32(s * Float32(pi)))));
	else
		tmp = Float32(Float32(Float32(Float32(0.75) * exp(Float32(Float32(r / s) * Float32(-0.3333333333333333)))) / Float32(r * Float32(s * Float32(Float32(pi) * Float32(6.0))))) + Float32(Float32(Float32(0.125) / s) * Float32(Float32(Float32(1.0) / Float32(r * Float32(pi))) + Float32(Float32(Float32(Float32(-1.0) / Float32(pi)) - Float32(Float32(r / Float32(s * Float32(pi))) * Float32(-0.5))) / s))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(s, r)
	tmp = single(0.0);
	if (s <= single(0.05999999865889549))
		tmp = single(0.125) * (exp((r / -s)) / (r * (s * single(pi))));
	else
		tmp = ((single(0.75) * exp(((r / s) * single(-0.3333333333333333)))) / (r * (s * (single(pi) * single(6.0))))) + ((single(0.125) / s) * ((single(1.0) / (r * single(pi))) + (((single(-1.0) / single(pi)) - ((r / (s * single(pi))) * single(-0.5))) / s)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if s < 0.0599999987

   1. Initial program 99.5%

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

   3. Taylor expanded in r around inf 99.5%

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

      1. associate-*r/ 99.5%

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

      2. *-commutative 99.5%

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

      3. associate-*l* 99.5%

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

      4. *-commutative 99.5%

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

      5. times-frac 99.5%

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

      6. mul-1-neg 99.5%

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

      7. distribute-neg-frac2 99.5%

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

   5. Simplified 99.5%

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

      1. neg-mul-1 99.5%

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

      2. times-frac 99.5%

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

      3. metadata-eval 99.5%

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

      4. rem-log-exp 99.5%

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

      5. pow-exp 99.4%

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

      6. log-pow 99.4%

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

      7. rem-log-exp 99.5%

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

   7. Applied egg-rr 99.5%

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

      1. add-exp-log 99.4%

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

      2. log-div 99.4%

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

      3. *-commutative 99.4%

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

      4. log-prod 99.4%

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

      5. add-log-exp 99.7%

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

      6. associate-*l* 99.7%

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

      7. *-commutative 99.7%

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

   9. Applied egg-rr 99.7%

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

   10. Taylor expanded in s around 0 95.3%

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

       1. mul-1-neg 95.3%

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

       2. distribute-frac-neg2 95.3%

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

   12. Simplified 95.3%

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

   if 0.0599999987 < s

   1. Initial program 96.9%

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

   3. Taylor expanded in r around inf 96.9%

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

      1. associate-*r/ 96.9%

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

      2. *-commutative 96.9%

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

      3. associate-*l* 97.0%

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

      4. *-commutative 97.0%

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

      5. times-frac 97.1%

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

      6. mul-1-neg 97.1%

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

      7. distribute-neg-frac2 97.1%

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

   5. Simplified 97.1%

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

      1. neg-mul-1 97.1%

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

      2. times-frac 96.9%

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

      3. metadata-eval 96.9%

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

      4. rem-log-exp 96.9%

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

      5. pow-exp 95.4%

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

      6. log-pow 95.8%

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

      7. rem-log-exp 96.9%

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

   7. Applied egg-rr 96.9%

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

   8. Taylor expanded in s around -inf 65.0%

      \[\leadsto \frac{0.125}{s} \cdot \color{blue}{\left(-1 \cdot \frac{-0.5 \cdot \frac{r}{s \cdot \pi} + \frac{1}{\pi}}{s} + \frac{1}{r \cdot \pi}\right)} + \frac{0.75 \cdot e^{\frac{r}{s} \cdot -0.3333333333333333}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \leq 0.05999999865889549:\\ \;\;\;\;0.125 \cdot \frac{e^{\frac{r}{-s}}}{r \cdot \left(s \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.75 \cdot e^{\frac{r}{s} \cdot -0.3333333333333333}}{r \cdot \left(s \cdot \left(\pi \cdot 6\right)\right)} + \frac{0.125}{s} \cdot \left(\frac{1}{r \cdot \pi} + \frac{\frac{-1}{\pi} - \frac{r}{s \cdot \pi} \cdot -0.5}{s}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 91.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{r}{-s}}\\ \mathbf{if}\;s \leq 0.05999999865889549:\\ \;\;\;\;0.125 \cdot \frac{t\_0}{r \cdot \left(s \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.125}{s} \cdot \frac{t\_0}{r \cdot \pi} + \frac{\frac{0.006944444444444444 \cdot \frac{r}{s \cdot \pi} + 0.041666666666666664 \cdot \frac{-1}{\pi}}{s} + 0.125 \cdot \frac{1}{r \cdot \pi}}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (let* ((t_0 (exp (/ r (- s)))))
   (if (<= s 0.05999999865889549)
     (* 0.125 (/ t_0 (* r (* s PI))))
     (+
      (* (/ 0.125 s) (/ t_0 (* r PI)))
      (/
       (+
        (/
         (+
          (* 0.006944444444444444 (/ r (* s PI)))
          (* 0.041666666666666664 (/ -1.0 PI)))
         s)
        (* 0.125 (/ 1.0 (* r PI))))
       s)))))

C

float code(float s, float r) {
	float t_0 = expf((r / -s));
	float tmp;
	if (s <= 0.05999999865889549f) {
		tmp = 0.125f * (t_0 / (r * (s * ((float) M_PI))));
	} else {
		tmp = ((0.125f / s) * (t_0 / (r * ((float) M_PI)))) + (((((0.006944444444444444f * (r / (s * ((float) M_PI)))) + (0.041666666666666664f * (-1.0f / ((float) M_PI)))) / s) + (0.125f * (1.0f / (r * ((float) M_PI))))) / s);
	}
	return tmp;
}

Julia

function code(s, r)
	t_0 = exp(Float32(r / Float32(-s)))
	tmp = Float32(0.0)
	if (s <= Float32(0.05999999865889549))
		tmp = Float32(Float32(0.125) * Float32(t_0 / Float32(r * Float32(s * Float32(pi)))));
	else
		tmp = Float32(Float32(Float32(Float32(0.125) / s) * Float32(t_0 / Float32(r * Float32(pi)))) + Float32(Float32(Float32(Float32(Float32(Float32(0.006944444444444444) * Float32(r / Float32(s * Float32(pi)))) + Float32(Float32(0.041666666666666664) * Float32(Float32(-1.0) / Float32(pi)))) / s) + Float32(Float32(0.125) * Float32(Float32(1.0) / Float32(r * Float32(pi))))) / s));
	end
	return tmp
end

MATLAB

function tmp_2 = code(s, r)
	t_0 = exp((r / -s));
	tmp = single(0.0);
	if (s <= single(0.05999999865889549))
		tmp = single(0.125) * (t_0 / (r * (s * single(pi))));
	else
		tmp = ((single(0.125) / s) * (t_0 / (r * single(pi)))) + (((((single(0.006944444444444444) * (r / (s * single(pi)))) + (single(0.041666666666666664) * (single(-1.0) / single(pi)))) / s) + (single(0.125) * (single(1.0) / (r * single(pi))))) / s);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if s < 0.0599999987

   1. Initial program 99.5%

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

   3. Taylor expanded in r around inf 99.5%

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

      1. associate-*r/ 99.5%

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

      2. *-commutative 99.5%

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

      3. associate-*l* 99.5%

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

      4. *-commutative 99.5%

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

      5. times-frac 99.5%

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

      6. mul-1-neg 99.5%

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

      7. distribute-neg-frac2 99.5%

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

   5. Simplified 99.5%

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

      1. neg-mul-1 99.5%

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

      2. times-frac 99.5%

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

      3. metadata-eval 99.5%

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

      4. rem-log-exp 99.5%

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

      5. pow-exp 99.4%

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

      6. log-pow 99.4%

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

      7. rem-log-exp 99.5%

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

   7. Applied egg-rr 99.5%

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

      1. add-exp-log 99.4%

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

      2. log-div 99.4%

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

      3. *-commutative 99.4%

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

      4. log-prod 99.4%

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

      5. add-log-exp 99.7%

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

      6. associate-*l* 99.7%

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

      7. *-commutative 99.7%

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

   9. Applied egg-rr 99.7%

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

   10. Taylor expanded in s around 0 95.3%

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

       1. mul-1-neg 95.3%

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

       2. distribute-frac-neg2 95.3%

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

   12. Simplified 95.3%

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

   if 0.0599999987 < s

   1. Initial program 96.9%

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

   3. Taylor expanded in r around inf 96.9%

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

      1. associate-*r/ 96.9%

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

      2. *-commutative 96.9%

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

      3. associate-*l* 97.0%

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

      4. *-commutative 97.0%

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

      5. times-frac 97.1%

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

      6. mul-1-neg 97.1%

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

      7. distribute-neg-frac2 97.1%

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

   5. Simplified 97.1%

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

   6. Taylor expanded in s around -inf 67.3%

      \[\leadsto \frac{0.125}{s} \cdot \frac{e^{\frac{r}{-s}}}{r \cdot \pi} + \color{blue}{-1 \cdot \frac{-1 \cdot \frac{0.006944444444444444 \cdot \frac{r}{s \cdot \pi} - 0.041666666666666664 \cdot \frac{1}{\pi}}{s} - 0.125 \cdot \frac{1}{r \cdot \pi}}{s}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \leq 0.05999999865889549:\\ \;\;\;\;0.125 \cdot \frac{e^{\frac{r}{-s}}}{r \cdot \left(s \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.125}{s} \cdot \frac{e^{\frac{r}{-s}}}{r \cdot \pi} + \frac{\frac{0.006944444444444444 \cdot \frac{r}{s \cdot \pi} + 0.041666666666666664 \cdot \frac{-1}{\pi}}{s} + 0.125 \cdot \frac{1}{r \cdot \pi}}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 91.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := r \cdot \left(s \cdot \pi\right)\\ \mathbf{if}\;s \leq 0.05999999865889549:\\ \;\;\;\;0.125 \cdot \frac{e^{\frac{r}{-s}}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;0.125 \cdot \frac{2 - \frac{r \cdot 1.3333333333333333 - \frac{{r}^{2} \cdot 0.5555555555555556}{s}}{s}}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (let* ((t_0 (* r (* s PI))))
   (if (<= s 0.05999999865889549)
     (* 0.125 (/ (exp (/ r (- s))) t_0))
     (*
      0.125
      (/
       (-
        2.0
        (/
         (- (* r 1.3333333333333333) (/ (* (pow r 2.0) 0.5555555555555556) s))
         s))
       t_0)))))

C

float code(float s, float r) {
	float t_0 = r * (s * ((float) M_PI));
	float tmp;
	if (s <= 0.05999999865889549f) {
		tmp = 0.125f * (expf((r / -s)) / t_0);
	} else {
		tmp = 0.125f * ((2.0f - (((r * 1.3333333333333333f) - ((powf(r, 2.0f) * 0.5555555555555556f) / s)) / s)) / t_0);
	}
	return tmp;
}

Julia

function code(s, r)
	t_0 = Float32(r * Float32(s * Float32(pi)))
	tmp = Float32(0.0)
	if (s <= Float32(0.05999999865889549))
		tmp = Float32(Float32(0.125) * Float32(exp(Float32(r / Float32(-s))) / t_0));
	else
		tmp = Float32(Float32(0.125) * Float32(Float32(Float32(2.0) - Float32(Float32(Float32(r * Float32(1.3333333333333333)) - Float32(Float32((r ^ Float32(2.0)) * Float32(0.5555555555555556)) / s)) / s)) / t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(s, r)
	t_0 = r * (s * single(pi));
	tmp = single(0.0);
	if (s <= single(0.05999999865889549))
		tmp = single(0.125) * (exp((r / -s)) / t_0);
	else
		tmp = single(0.125) * ((single(2.0) - (((r * single(1.3333333333333333)) - (((r ^ single(2.0)) * single(0.5555555555555556)) / s)) / s)) / t_0);
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if s < 0.0599999987

   1. Initial program 99.5%

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

   3. Taylor expanded in r around inf 99.5%

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

      1. associate-*r/ 99.5%

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

      2. *-commutative 99.5%

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

      3. associate-*l* 99.5%

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

      4. *-commutative 99.5%

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

      5. times-frac 99.5%

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

      6. mul-1-neg 99.5%

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

      7. distribute-neg-frac2 99.5%

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

   5. Simplified 99.5%

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

      1. neg-mul-1 99.5%

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

      2. times-frac 99.5%

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

      3. metadata-eval 99.5%

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

      4. rem-log-exp 99.5%

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

      5. pow-exp 99.4%

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

      6. log-pow 99.4%

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

      7. rem-log-exp 99.5%

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

   7. Applied egg-rr 99.5%

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

      1. add-exp-log 99.4%

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

      2. log-div 99.4%

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

      3. *-commutative 99.4%

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

      4. log-prod 99.4%

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

      5. add-log-exp 99.7%

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

      6. associate-*l* 99.7%

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

      7. *-commutative 99.7%

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

   9. Applied egg-rr 99.7%

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

   10. Taylor expanded in s around 0 95.3%

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

       1. mul-1-neg 95.3%

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

       2. distribute-frac-neg2 95.3%

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

   12. Simplified 95.3%

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

   if 0.0599999987 < s

   1. Initial program 96.9%

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

   3. Simplified 95.5%

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

   5. Taylor expanded in r around inf 96.1%

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

      1. associate-*r/ 95.5%

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

      2. *-commutative 95.5%

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

   7. Applied egg-rr 95.5%

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

   8. Taylor expanded in s around -inf 64.9%

      \[\leadsto 0.125 \cdot \frac{\color{blue}{2 + -1 \cdot \frac{r + \left(-1 \cdot \frac{0.05555555555555555 \cdot {r}^{2} + 0.5 \cdot {r}^{2}}{s} + 0.3333333333333333 \cdot r\right)}{s}}}{r \cdot \left(s \cdot \pi\right)} \]
   9. Step-by-step derivation

      1. mul-1-neg 64.9%

         \[\leadsto 0.125 \cdot \frac{2 + \color{blue}{\left(-\frac{r + \left(-1 \cdot \frac{0.05555555555555555 \cdot {r}^{2} + 0.5 \cdot {r}^{2}}{s} + 0.3333333333333333 \cdot r\right)}{s}\right)}}{r \cdot \left(s \cdot \pi\right)} \]

      2. unsub-neg 64.9%

         \[\leadsto 0.125 \cdot \frac{\color{blue}{2 - \frac{r + \left(-1 \cdot \frac{0.05555555555555555 \cdot {r}^{2} + 0.5 \cdot {r}^{2}}{s} + 0.3333333333333333 \cdot r\right)}{s}}}{r \cdot \left(s \cdot \pi\right)} \]

      3. +-commutative 64.9%

         \[\leadsto 0.125 \cdot \frac{2 - \frac{r + \color{blue}{\left(0.3333333333333333 \cdot r + -1 \cdot \frac{0.05555555555555555 \cdot {r}^{2} + 0.5 \cdot {r}^{2}}{s}\right)}}{s}}{r \cdot \left(s \cdot \pi\right)} \]

      4. associate-+r+ 64.9%

         \[\leadsto 0.125 \cdot \frac{2 - \frac{\color{blue}{\left(r + 0.3333333333333333 \cdot r\right) + -1 \cdot \frac{0.05555555555555555 \cdot {r}^{2} + 0.5 \cdot {r}^{2}}{s}}}{s}}{r \cdot \left(s \cdot \pi\right)} \]

      5. distribute-rgt1-in 64.9%

         \[\leadsto 0.125 \cdot \frac{2 - \frac{\color{blue}{\left(0.3333333333333333 + 1\right) \cdot r} + -1 \cdot \frac{0.05555555555555555 \cdot {r}^{2} + 0.5 \cdot {r}^{2}}{s}}{s}}{r \cdot \left(s \cdot \pi\right)} \]

      6. metadata-eval 64.9%

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

      7. mul-1-neg 64.9%

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

      8. distribute-neg-frac2 64.9%

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

      9. distribute-rgt-out 64.9%

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

      10. metadata-eval 64.9%

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

   10. Simplified 64.9%

       \[\leadsto 0.125 \cdot \frac{\color{blue}{2 - \frac{1.3333333333333333 \cdot r + \frac{{r}^{2} \cdot 0.5555555555555556}{-s}}{s}}}{r \cdot \left(s \cdot \pi\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \leq 0.05999999865889549:\\ \;\;\;\;0.125 \cdot \frac{e^{\frac{r}{-s}}}{r \cdot \left(s \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;0.125 \cdot \frac{2 - \frac{r \cdot 1.3333333333333333 - \frac{{r}^{2} \cdot 0.5555555555555556}{s}}{s}}{r \cdot \left(s \cdot \pi\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 91.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;s \leq 0.05999999865889549:\\ \;\;\;\;0.125 \cdot \frac{e^{\frac{r}{-s}}}{r \cdot \left(s \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.125 \cdot \frac{0.05555555555555555 \cdot \frac{r}{\pi} + \frac{r}{\pi} \cdot 0.5}{s} + 0.16666666666666666 \cdot \frac{-1}{\pi}}{s} + \frac{1}{r \cdot \pi} \cdot 0.25}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (if (<= s 0.05999999865889549)
   (* 0.125 (/ (exp (/ r (- s))) (* r (* s PI))))
   (/
    (+
     (/
      (+
       (* 0.125 (/ (+ (* 0.05555555555555555 (/ r PI)) (* (/ r PI) 0.5)) s))
       (* 0.16666666666666666 (/ -1.0 PI)))
      s)
     (* (/ 1.0 (* r PI)) 0.25))
    s)))

C

float code(float s, float r) {
	float tmp;
	if (s <= 0.05999999865889549f) {
		tmp = 0.125f * (expf((r / -s)) / (r * (s * ((float) M_PI))));
	} else {
		tmp = ((((0.125f * (((0.05555555555555555f * (r / ((float) M_PI))) + ((r / ((float) M_PI)) * 0.5f)) / s)) + (0.16666666666666666f * (-1.0f / ((float) M_PI)))) / s) + ((1.0f / (r * ((float) M_PI))) * 0.25f)) / s;
	}
	return tmp;
}

Julia

function code(s, r)
	tmp = Float32(0.0)
	if (s <= Float32(0.05999999865889549))
		tmp = Float32(Float32(0.125) * Float32(exp(Float32(r / Float32(-s))) / Float32(r * Float32(s * Float32(pi)))));
	else
		tmp = Float32(Float32(Float32(Float32(Float32(Float32(0.125) * Float32(Float32(Float32(Float32(0.05555555555555555) * Float32(r / Float32(pi))) + Float32(Float32(r / Float32(pi)) * Float32(0.5))) / s)) + Float32(Float32(0.16666666666666666) * Float32(Float32(-1.0) / Float32(pi)))) / s) + Float32(Float32(Float32(1.0) / Float32(r * Float32(pi))) * Float32(0.25))) / s);
	end
	return tmp
end

MATLAB

function tmp_2 = code(s, r)
	tmp = single(0.0);
	if (s <= single(0.05999999865889549))
		tmp = single(0.125) * (exp((r / -s)) / (r * (s * single(pi))));
	else
		tmp = ((((single(0.125) * (((single(0.05555555555555555) * (r / single(pi))) + ((r / single(pi)) * single(0.5))) / s)) + (single(0.16666666666666666) * (single(-1.0) / single(pi)))) / s) + ((single(1.0) / (r * single(pi))) * single(0.25))) / s;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if s < 0.0599999987

   1. Initial program 99.5%

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

   3. Taylor expanded in r around inf 99.5%

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

      1. associate-*r/ 99.5%

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

      2. *-commutative 99.5%

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

      3. associate-*l* 99.5%

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

      4. *-commutative 99.5%

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

      5. times-frac 99.5%

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

      6. mul-1-neg 99.5%

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

      7. distribute-neg-frac2 99.5%

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

   5. Simplified 99.5%

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

      1. neg-mul-1 99.5%

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

      2. times-frac 99.5%

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

      3. metadata-eval 99.5%

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

      4. rem-log-exp 99.5%

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

      5. pow-exp 99.4%

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

      6. log-pow 99.4%

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

      7. rem-log-exp 99.5%

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

   7. Applied egg-rr 99.5%

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

      1. add-exp-log 99.4%

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

      2. log-div 99.4%

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

      3. *-commutative 99.4%

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

      4. log-prod 99.4%

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

      5. add-log-exp 99.7%

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

      6. associate-*l* 99.7%

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

      7. *-commutative 99.7%

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

   9. Applied egg-rr 99.7%

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

   10. Taylor expanded in s around 0 95.3%

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

       1. mul-1-neg 95.3%

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

       2. distribute-frac-neg2 95.3%

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

   12. Simplified 95.3%

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

   if 0.0599999987 < s

   1. Initial program 96.9%

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

   3. Simplified 95.5%

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

   5. Taylor expanded in s around -inf 64.5%

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{0.125 \cdot \frac{0.05555555555555555 \cdot \frac{r}{\pi} + 0.5 \cdot \frac{r}{\pi}}{s} - 0.16666666666666666 \cdot \frac{1}{\pi}}{s} - 0.25 \cdot \frac{1}{r \cdot \pi}}{s}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \leq 0.05999999865889549:\\ \;\;\;\;0.125 \cdot \frac{e^{\frac{r}{-s}}}{r \cdot \left(s \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.125 \cdot \frac{0.05555555555555555 \cdot \frac{r}{\pi} + \frac{r}{\pi} \cdot 0.5}{s} + 0.16666666666666666 \cdot \frac{-1}{\pi}}{s} + \frac{1}{r \cdot \pi} \cdot 0.25}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 9.9% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.125 \cdot \frac{0.05555555555555555 \cdot \frac{r}{\pi} + \frac{r}{\pi} \cdot 0.5}{s} + 0.16666666666666666 \cdot \frac{-1}{\pi}}{s} + \frac{1}{r \cdot \pi} \cdot 0.25}{s} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (/
  (+
   (/
    (+
     (* 0.125 (/ (+ (* 0.05555555555555555 (/ r PI)) (* (/ r PI) 0.5)) s))
     (* 0.16666666666666666 (/ -1.0 PI)))
    s)
   (* (/ 1.0 (* r PI)) 0.25))
  s))

C

float code(float s, float r) {
	return ((((0.125f * (((0.05555555555555555f * (r / ((float) M_PI))) + ((r / ((float) M_PI)) * 0.5f)) / s)) + (0.16666666666666666f * (-1.0f / ((float) M_PI)))) / s) + ((1.0f / (r * ((float) M_PI))) * 0.25f)) / s;
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(Float32(Float32(0.125) * Float32(Float32(Float32(Float32(0.05555555555555555) * Float32(r / Float32(pi))) + Float32(Float32(r / Float32(pi)) * Float32(0.5))) / s)) + Float32(Float32(0.16666666666666666) * Float32(Float32(-1.0) / Float32(pi)))) / s) + Float32(Float32(Float32(1.0) / Float32(r * Float32(pi))) * Float32(0.25))) / s)
end

MATLAB

function tmp = code(s, r)
	tmp = ((((single(0.125) * (((single(0.05555555555555555) * (r / single(pi))) + ((r / single(pi)) * single(0.5))) / s)) + (single(0.16666666666666666) * (single(-1.0) / single(pi)))) / s) + ((single(1.0) / (r * single(pi))) * single(0.25))) / s;
end

Derivation

1. Initial program 99.2%

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

3. Simplified 99.0%

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

5. Taylor expanded in s around -inf 11.9%

   \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{0.125 \cdot \frac{0.05555555555555555 \cdot \frac{r}{\pi} + 0.5 \cdot \frac{r}{\pi}}{s} - 0.16666666666666666 \cdot \frac{1}{\pi}}{s} - 0.25 \cdot \frac{1}{r \cdot \pi}}{s}} \]

6. Final simplification 11.9%

   \[\leadsto \frac{\frac{0.125 \cdot \frac{0.05555555555555555 \cdot \frac{r}{\pi} + \frac{r}{\pi} \cdot 0.5}{s} + 0.16666666666666666 \cdot \frac{-1}{\pi}}{s} + \frac{1}{r \cdot \pi} \cdot 0.25}{s} \]
7. Add Preprocessing

Alternative 11: 9.7% accurate, 8.0× speedup

Math

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

FPCore

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

C

float code(float s, float r) {
	return (0.125f / (s * ((float) M_PI))) * ((((-1.0f - ((r / s) * -0.5f)) / s) + (1.0f / r)) + ((((r / s) * -0.3333333333333333f) + 1.0f) / r));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(0.125) / Float32(s * Float32(pi))) * Float32(Float32(Float32(Float32(Float32(-1.0) - Float32(Float32(r / s) * Float32(-0.5))) / s) + Float32(Float32(1.0) / r)) + Float32(Float32(Float32(Float32(r / s) * Float32(-0.3333333333333333)) + Float32(1.0)) / r)))
end

MATLAB

function tmp = code(s, r)
	tmp = (single(0.125) / (s * single(pi))) * ((((single(-1.0) - ((r / s) * single(-0.5))) / s) + (single(1.0) / r)) + ((((r / s) * single(-0.3333333333333333)) + single(1.0)) / r));
end

Derivation

1. Initial program 99.2%

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

3. Simplified 99.0%

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

5. Taylor expanded in r around 0 11.5%

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

6. Taylor expanded in s around -inf 11.6%

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

7. Final simplification 11.6%

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

Alternative 12: 8.9% accurate, 10.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.2%

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

3. Simplified 99.0%

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

5. Taylor expanded in r around 0 11.5%

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

6. Taylor expanded in r around 0 10.9%

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

   1. neg-mul-1 10.9%

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

   2. unsub-neg 10.9%

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

8. Simplified 10.9%

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

9. Final simplification 10.9%

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

Alternative 13: 8.9% accurate, 11.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.2%

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

3. Simplified 99.0%

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

5. Taylor expanded in r around 0 11.5%

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

6. Taylor expanded in r around 0 10.9%

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

   1. neg-mul-1 10.9%

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

   2. unsub-neg 10.9%

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

8. Simplified 10.9%

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

9. Taylor expanded in s around 0 10.9%

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

Alternative 14: 8.9% accurate, 15.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.2%

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

3. Simplified 99.0%

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

5. Taylor expanded in r around inf 99.1%

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

   1. associate-*r/ 99.1%

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

   2. *-commutative 99.1%

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

7. Applied egg-rr 99.1%

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

8. Taylor expanded in r around 0 10.9%

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

9. Final simplification 10.9%

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

Alternative 15: 8.9% 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 99.2%

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

3. Simplified 99.0%

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

5. Taylor expanded in s around inf 10.8%

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

   1. sub-neg 10.8%

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

   2. associate-*r/ 10.8%

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

   3. metadata-eval 10.8%

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

   4. associate-*r/ 10.8%

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

   5. metadata-eval 10.8%

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

   6. distribute-neg-frac 10.8%

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

   7. metadata-eval 10.8%

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

7. Simplified 10.8%

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

Alternative 16: 8.9% accurate, 17.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.2%

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

3. Simplified 99.0%

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

5. Taylor expanded in s around inf 10.8%

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

   1. sub-neg 10.8%

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

   2. associate-*r/ 10.8%

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

   3. metadata-eval 10.8%

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

   4. associate-*r/ 10.8%

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

   5. metadata-eval 10.8%

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

   6. distribute-neg-frac 10.8%

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

   7. metadata-eval 10.8%

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

7. Simplified 10.8%

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

8. Taylor expanded in r around inf 10.7%

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

   1. associate-*r/ 10.8%

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

   2. metadata-eval 10.8%

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

   3. associate-/r* 10.7%

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

   4. associate-/l/ 10.7%

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

   5. associate-*r/ 10.7%

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

   6. metadata-eval 10.7%

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

   7. unpow2 10.7%

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

   8. associate-*l* 10.7%

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

   9. associate-/l/ 10.7%

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

   10. metadata-eval 10.7%

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

   11. associate-*r/ 10.7%

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

   12. div-sub 10.8%

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

10. Simplified 10.8%

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

Alternative 17: 8.9% accurate, 33.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.2%

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

3. Simplified 99.0%

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

5. Taylor expanded in s around inf 10.8%

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

   1. sub-neg 10.8%

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

   2. associate-*r/ 10.8%

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

   3. metadata-eval 10.8%

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

   4. associate-*r/ 10.8%

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

   5. metadata-eval 10.8%

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

   6. distribute-neg-frac 10.8%

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

   7. metadata-eval 10.8%

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

7. Simplified 10.8%

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

8. Taylor expanded in r around 0 10.5%

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

   1. *-commutative 10.5%

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

   2. associate-/r* 10.5%

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

10. Simplified 10.5%

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

Alternative 18: 8.9% accurate, 33.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.2%

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

3. Simplified 99.0%

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

5. Taylor expanded in s around inf 10.8%

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

   1. sub-neg 10.8%

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

   2. associate-*r/ 10.8%

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

   3. metadata-eval 10.8%

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

   4. associate-*r/ 10.8%

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

   5. metadata-eval 10.8%

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

   6. distribute-neg-frac 10.8%

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

   7. metadata-eval 10.8%

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

7. Simplified 10.8%

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

8. Taylor expanded in r around 0 10.5%

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

Alternative 19: 8.9% 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 99.2%

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

3. Simplified 99.0%

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

5. Taylor expanded in s around inf 10.8%

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

   1. sub-neg 10.8%

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

   2. associate-*r/ 10.8%

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

   3. metadata-eval 10.8%

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

   4. associate-*r/ 10.8%

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

   5. metadata-eval 10.8%

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

   6. distribute-neg-frac 10.8%

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

   7. metadata-eval 10.8%

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

7. Simplified 10.8%

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

8. Taylor expanded in r around 0 10.5%

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

   1. *-commutative 10.5%

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

   2. *-commutative 10.5%

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

   3. associate-*l* 10.5%

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

   4. *-commutative 10.5%

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

10. Simplified 10.5%

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

11. Final simplification 10.5%

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

Alternative 20: 8.9% 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 99.2%

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

3. Simplified 99.0%

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

5. Taylor expanded in s around inf 10.5%

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

Alternative 21: 7.1% accurate, 33.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

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. Add Preprocessing

3. Taylor expanded in r around inf 99.2%

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

   1. associate-*r/ 99.2%

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

   2. *-commutative 99.2%

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

   3. associate-*l* 99.2%

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

   4. *-commutative 99.2%

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

   5. times-frac 99.3%

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

   6. mul-1-neg 99.3%

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

   7. distribute-neg-frac2 99.3%

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

5. Simplified 99.3%

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

   1. neg-mul-1 99.3%

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

   2. times-frac 99.2%

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

   3. metadata-eval 99.2%

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

   4. rem-log-exp 99.2%

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

   5. pow-exp 99.0%

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

   6. log-pow 99.0%

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

   7. rem-log-exp 99.2%

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

7. Applied egg-rr 99.2%

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

   1. add-exp-log 99.1%

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

   2. log-div 98.9%

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

   3. *-commutative 98.9%

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

   4. log-prod 99.0%

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

   5. add-log-exp 99.2%

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

   6. associate-*l* 99.2%

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

   7. *-commutative 99.2%

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

9. Applied egg-rr 99.2%

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

10. Taylor expanded in r around 0 7.5%

    \[\leadsto \color{blue}{\frac{0.125}{r \cdot \left(s \cdot \pi\right)}} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024170 
(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))))