Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.1%

Time: 17.1s

Alternatives: 12

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: 99.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ 0.125 \cdot \frac{e^{\frac{r}{-s}} + \sqrt{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s}\right)}}}{\pi \cdot \left(r \cdot s\right)} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (*
  0.125
  (/
   (+ (exp (/ r (- s))) (sqrt (pow (exp -0.6666666666666666) (/ r s))))
   (* PI (* r s)))))

C

float code(float s, float r) {
	return 0.125f * ((expf((r / -s)) + sqrtf(powf(expf(-0.6666666666666666f), (r / s)))) / (((float) M_PI) * (r * s)));
}

Julia

function code(s, r)
	return Float32(Float32(0.125) * Float32(Float32(exp(Float32(r / Float32(-s))) + sqrt((exp(Float32(-0.6666666666666666)) ^ Float32(r / s)))) / Float32(Float32(pi) * Float32(r * s))))
end

MATLAB

function tmp = code(s, r)
	tmp = single(0.125) * ((exp((r / -s)) + sqrt((exp(single(-0.6666666666666666)) ^ (r / s)))) / (single(pi) * (r * s)));
end

Derivation

1. Initial program 99.7%

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

3. Simplified 99.6%

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

   1. add-sqr-sqrt 99.6%

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

   2. sqrt-unprod 99.6%

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

   3. pow-prod-down 99.6%

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

   4. prod-exp 99.7%

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

   5. metadata-eval 99.7%

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

6. Applied egg-rr 99.7%

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

7. Taylor expanded in r around inf 99.6%

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

   1. mul-1-neg 99.6%

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

   2. distribute-neg-frac2 99.6%

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

   3. exp-prod 99.7%

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

   4. *-commutative 99.7%

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

   5. *-commutative 99.7%

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

   6. associate-*l* 99.7%

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

9. Simplified 99.7%

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

10. Final simplification 99.7%

    \[\leadsto 0.125 \cdot \frac{e^{\frac{r}{-s}} + \sqrt{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s}\right)}}}{\pi \cdot \left(r \cdot s\right)} \]
11. Add Preprocessing

Alternative 2: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

   1. times-frac 99.7%

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

   2. *-commutative 99.7%

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

   3. distribute-frac-neg 99.7%

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

   4. associate-/l* 99.7%

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

   5. *-commutative 99.7%

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

   6. *-commutative 99.7%

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

   7. associate-*l* 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in s around 0 99.7%

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

   1. *-commutative 99.7%

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

   2. *-commutative 99.7%

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

   3. associate-*r* 99.7%

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

7. Simplified 99.7%

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

8. Taylor expanded in s around 0 99.7%

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

9. Final simplification 99.7%

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

Alternative 3: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Simplified 99.6%

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

   1. pow-to-exp 99.6%

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

   2. associate-*r/ 99.6%

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

   3. *-commutative 99.6%

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

   4. rem-log-exp 99.7%

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

6. Applied egg-rr 99.7%

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

   1. associate-/r* 99.7%

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

   2. div-inv 99.7%

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

8. Applied egg-rr 99.7%

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

9. Final simplification 99.7%

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

Alternative 4: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Simplified 99.6%

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

5. Taylor expanded in r around inf 99.7%

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

6. Final simplification 99.7%

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

Alternative 5: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Simplified 99.6%

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

   1. pow-to-exp 99.6%

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

   2. associate-*r/ 99.6%

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

   3. *-commutative 99.6%

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

   4. rem-log-exp 99.7%

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

6. Applied egg-rr 99.7%

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

7. Final simplification 99.7%

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

Alternative 6: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Simplified 99.6%

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

   1. add-sqr-sqrt 99.6%

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

   2. sqrt-unprod 99.6%

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

   3. pow-prod-down 99.6%

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

   4. prod-exp 99.7%

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

   5. metadata-eval 99.7%

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

6. Applied egg-rr 99.7%

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

7. Taylor expanded in r around inf 99.6%

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

   1. mul-1-neg 99.6%

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

   2. distribute-neg-frac2 99.6%

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

   3. exp-prod 99.7%

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

   4. *-commutative 99.7%

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

   5. *-commutative 99.7%

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

   6. associate-*l* 99.7%

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

9. Simplified 99.7%

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

    1. +-commutative 99.7%

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

    2. add-sqr-sqrt 99.7%

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

    3. fma-define 99.7%

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

    4. pow1/2 99.7%

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

    5. sqrt-pow1 99.8%

       \[\leadsto 0.125 \cdot \frac{\mathsf{fma}\left(\color{blue}{{\left({\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s}\right)}\right)}^{\left(\frac{0.5}{2}\right)}}, \sqrt{\sqrt{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s}\right)}}}, e^{\frac{r}{-s}}\right)}{\pi \cdot \left(s \cdot r\right)} \]

    6. metadata-eval 99.8%

       \[\leadsto 0.125 \cdot \frac{\mathsf{fma}\left({\left({\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s}\right)}\right)}^{\color{blue}{0.25}}, \sqrt{\sqrt{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s}\right)}}}, e^{\frac{r}{-s}}\right)}{\pi \cdot \left(s \cdot r\right)} \]

    7. pow1/2 99.8%

       \[\leadsto 0.125 \cdot \frac{\mathsf{fma}\left({\left({\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s}\right)}\right)}^{0.25}, \sqrt{\color{blue}{{\left({\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s}\right)}\right)}^{0.5}}}, e^{\frac{r}{-s}}\right)}{\pi \cdot \left(s \cdot r\right)} \]

    8. sqrt-pow1 99.8%

       \[\leadsto 0.125 \cdot \frac{\mathsf{fma}\left({\left({\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s}\right)}\right)}^{0.25}, \color{blue}{{\left({\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s}\right)}\right)}^{\left(\frac{0.5}{2}\right)}}, e^{\frac{r}{-s}}\right)}{\pi \cdot \left(s \cdot r\right)} \]

    9. metadata-eval 99.8%

       \[\leadsto 0.125 \cdot \frac{\mathsf{fma}\left({\left({\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s}\right)}\right)}^{0.25}, {\left({\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s}\right)}\right)}^{\color{blue}{0.25}}, e^{\frac{r}{-s}}\right)}{\pi \cdot \left(s \cdot r\right)} \]

11. Applied egg-rr 99.8%

    \[\leadsto 0.125 \cdot \frac{\color{blue}{\mathsf{fma}\left({\left({\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s}\right)}\right)}^{0.25}, {\left({\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s}\right)}\right)}^{0.25}, e^{\frac{r}{-s}}\right)}}{\pi \cdot \left(s \cdot r\right)} \]
12. Step-by-step derivation

    1. fma-undefine 99.8%

       \[\leadsto 0.125 \cdot \frac{\color{blue}{{\left({\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s}\right)}\right)}^{0.25} \cdot {\left({\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s}\right)}\right)}^{0.25} + e^{\frac{r}{-s}}}}{\pi \cdot \left(s \cdot r\right)} \]

    2. pow-sqr 99.7%

       \[\leadsto 0.125 \cdot \frac{\color{blue}{{\left({\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s}\right)}\right)}^{\left(2 \cdot 0.25\right)}} + e^{\frac{r}{-s}}}{\pi \cdot \left(s \cdot r\right)} \]

    3. exp-prod 99.7%

       \[\leadsto 0.125 \cdot \frac{{\color{blue}{\left(e^{-0.6666666666666666 \cdot \frac{r}{s}}\right)}}^{\left(2 \cdot 0.25\right)} + e^{\frac{r}{-s}}}{\pi \cdot \left(s \cdot r\right)} \]

    4. metadata-eval 99.7%

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

    5. exp-prod 99.6%

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

    6. *-commutative 99.6%

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

    7. *-commutative 99.6%

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

    8. +-commutative 99.6%

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

    9. *-commutative 99.6%

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

    10. associate-*r* 99.6%

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

    11. metadata-eval 99.6%

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

    12. *-commutative 99.6%

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

13. Simplified 99.6%

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

14. Final simplification 99.6%

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

Alternative 7: 11.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

function code(s, r)
	return Float32(Float32(0.25) / log1p(expm1(Float32(r * Float32(s * Float32(pi))))))
end

Derivation

1. Initial program 99.7%

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

3. Simplified 99.6%

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

5. Taylor expanded in r around 0 10.4%

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

6. Taylor expanded in s around inf 9.9%

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

   1. *-commutative 9.9%

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

   2. *-commutative 9.9%

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

   3. associate-*l* 9.9%

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

8. Simplified 9.9%

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

   1. log1p-expm1-u 13.7%

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

   2. associate-*r* 13.7%

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

   3. *-commutative 13.7%

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

   4. *-commutative 13.7%

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

   5. *-commutative 13.7%

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

10. Applied egg-rr 13.7%

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

11. Final simplification 13.7%

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

Alternative 8: 9.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Simplified 99.6%

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

   1. add-sqr-sqrt 99.6%

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

   2. sqrt-unprod 99.6%

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

   3. pow-prod-down 99.6%

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

   4. prod-exp 99.7%

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

   5. metadata-eval 99.7%

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

6. Applied egg-rr 99.7%

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

7. Taylor expanded in r around inf 99.6%

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

   1. mul-1-neg 99.6%

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

   2. distribute-neg-frac2 99.6%

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

   3. exp-prod 99.7%

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

   4. *-commutative 99.7%

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

   5. *-commutative 99.7%

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

   6. associate-*l* 99.7%

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

9. Simplified 99.7%

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

10. Taylor expanded in r around 0 11.2%

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

    1. associate-*r/ 11.2%

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

    2. metadata-eval 11.2%

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

12. Simplified 11.2%

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

13. Final simplification 11.2%

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

Alternative 9: 9.6% accurate, 8.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.125 \cdot \frac{0.5 \cdot \left(\frac{r}{\pi} + \frac{r}{\pi} \cdot 0.1111111111111111\right)}{s} - \frac{0.16666666666666666}{\pi}}{s} + \frac{0.25}{r \cdot \pi}}{s} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (/
  (+
   (/
    (-
     (* 0.125 (/ (* 0.5 (+ (/ r PI) (* (/ r PI) 0.1111111111111111))) s))
     (/ 0.16666666666666666 PI))
    s)
   (/ 0.25 (* r PI)))
  s))

C

float code(float s, float r) {
	return ((((0.125f * ((0.5f * ((r / ((float) M_PI)) + ((r / ((float) M_PI)) * 0.1111111111111111f))) / s)) - (0.16666666666666666f / ((float) M_PI))) / s) + (0.25f / (r * ((float) M_PI)))) / s;
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(Float32(Float32(0.125) * Float32(Float32(Float32(0.5) * Float32(Float32(r / Float32(pi)) + Float32(Float32(r / Float32(pi)) * Float32(0.1111111111111111)))) / s)) - Float32(Float32(0.16666666666666666) / Float32(pi))) / s) + Float32(Float32(0.25) / Float32(r * Float32(pi)))) / s)
end

MATLAB

function tmp = code(s, r)
	tmp = ((((single(0.125) * ((single(0.5) * ((r / single(pi)) + ((r / single(pi)) * single(0.1111111111111111)))) / s)) - (single(0.16666666666666666) / single(pi))) / s) + (single(0.25) / (r * single(pi)))) / s;
end

Derivation

1. Initial program 99.7%

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

3. Simplified 99.6%

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

   1. add-sqr-sqrt 99.6%

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

   2. sqrt-unprod 99.6%

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

   3. pow-prod-down 99.6%

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

   4. prod-exp 99.7%

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

   5. metadata-eval 99.7%

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

6. Applied egg-rr 99.7%

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

7. Taylor expanded in s around -inf 11.2%

   \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{0.125 \cdot \frac{0.5 \cdot \frac{r}{\pi} + 0.5 \cdot \frac{0.2222222222222222 \cdot {r}^{2} - 0.1111111111111111 \cdot {r}^{2}}{r \cdot \pi}}{s} - 0.16666666666666666 \cdot \frac{1}{\pi}}{s} - 0.25 \cdot \frac{1}{r \cdot \pi}}{s}} \]

9. Simplified 11.2%

   \[\leadsto \color{blue}{-\frac{\left(-\frac{0.125 \cdot \frac{0.5 \cdot \left(\frac{r}{\pi} + \frac{{r}^{2} \cdot 0.1111111111111111}{\pi \cdot r}\right)}{s} - \frac{0.16666666666666666}{\pi}}{s}\right) - \frac{0.25}{\pi \cdot r}}{s}} \]

10. Taylor expanded in r around 0 11.2%

    \[\leadsto -\frac{\left(-\frac{0.125 \cdot \frac{0.5 \cdot \left(\frac{r}{\pi} + \color{blue}{0.1111111111111111 \cdot \frac{r}{\pi}}\right)}{s} - \frac{0.16666666666666666}{\pi}}{s}\right) - \frac{0.25}{\pi \cdot r}}{s} \]

11. Final simplification 11.2%

    \[\leadsto \frac{\frac{0.125 \cdot \frac{0.5 \cdot \left(\frac{r}{\pi} + \frac{r}{\pi} \cdot 0.1111111111111111\right)}{s} - \frac{0.16666666666666666}{\pi}}{s} + \frac{0.25}{r \cdot \pi}}{s} \]
12. Add Preprocessing

Alternative 10: 8.7% accurate, 17.8× speedup

Math

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

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(Float32(0.25) / 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 99.7%

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

3. Simplified 99.6%

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

   1. add-sqr-sqrt 99.6%

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

   2. sqrt-unprod 99.6%

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

   3. pow-prod-down 99.6%

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

   4. prod-exp 99.7%

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

   5. metadata-eval 99.7%

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

6. Applied egg-rr 99.7%

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

7. Taylor expanded in s around inf 10.2%

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

   1. sub-neg 10.2%

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

   2. associate-*r/ 10.2%

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

   3. metadata-eval 10.2%

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

   4. associate-/r* 10.2%

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

   5. associate-*r/ 10.2%

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

   6. metadata-eval 10.2%

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

   7. *-commutative 10.2%

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

   8. distribute-neg-frac 10.2%

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

   9. metadata-eval 10.2%

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

   10. *-commutative 10.2%

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

9. Simplified 10.2%

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

10. Final simplification 10.2%

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

Alternative 11: 8.8% accurate, 33.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Simplified 99.6%

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

5. Taylor expanded in r around 0 10.4%

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

6. Taylor expanded in s around inf 9.9%

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

7. Final simplification 9.9%

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

Alternative 12: 8.8% accurate, 33.0× speedup

Math

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

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Simplified 99.6%

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

5. Taylor expanded in r around 0 10.4%

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

6. Taylor expanded in s around inf 9.9%

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

   1. add-log-exp 6.3%

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

   2. *-commutative 6.3%

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

8. Applied egg-rr 6.3%

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

   1. rem-log-exp 9.9%

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

   2. *-commutative 9.9%

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

   3. associate-*r* 9.9%

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

   4. associate-/r* 9.9%

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

10. Applied egg-rr 9.9%

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

11. Final simplification 9.9%

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

Reproduce

herbie shell --seed 2024071 
(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))))