Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.5% → 99.5%

Time: 12.6s

Alternatives: 13

Speedup: N/A×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.125}{s}}{\pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s} \cdot 0.5\right)}}{r}\right) \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (*
  (/ (/ 0.125 s) PI)
  (+
   (/ (exp (/ r (- s))) r)
   (/ (pow (exp -0.6666666666666666) (* (/ r s) 0.5)) r))))

C

float code(float s, float r) {
	return ((0.125f / s) / ((float) M_PI)) * ((expf((r / -s)) / r) + (powf(expf(-0.6666666666666666f), ((r / s) * 0.5f)) / r));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.125) / s) / Float32(pi)) * Float32(Float32(exp(Float32(r / Float32(-s))) / r) + Float32((exp(Float32(-0.6666666666666666)) ^ Float32(Float32(r / s) * Float32(0.5))) / r)))
end

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.125) / s) / single(pi)) * ((exp((r / -s)) / r) + ((exp(single(-0.6666666666666666)) ^ ((r / s) * single(0.5))) / r));
end

Derivation

1. Initial program 99.6%

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

3. Simplified 99.3%

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

   1. add-sqr-sqrt 99.3%

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

   2. sqrt-unprod 98.9%

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

   3. pow-prod-down 98.9%

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

   4. prod-exp 99.2%

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

   5. metadata-eval 99.2%

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

5. Applied egg-rr 99.2%

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

   1. pow1/2 99.2%

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

   2. pow-pow 99.7%

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

7. Applied egg-rr 99.7%

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

   1. metadata-eval 99.7%

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

   2. associate-/r* 99.7%

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

   3. *-commutative 99.7%

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

   4. associate-*l* 99.7%

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

   5. associate-/l/ 99.7%

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

   6. div-inv 99.7%

      \[\leadsto \color{blue}{\left(\frac{0.25}{s} \cdot \frac{1}{2 \cdot \pi}\right)} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s} \cdot 0.5\right)}}{r}\right) \]

   7. *-commutative 99.7%

      \[\leadsto \left(\frac{0.25}{s} \cdot \frac{1}{\color{blue}{\pi \cdot 2}}\right) \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s} \cdot 0.5\right)}}{r}\right) \]

9. Applied egg-rr 99.7%

   \[\leadsto \color{blue}{\left(\frac{0.25}{s} \cdot \frac{1}{\pi \cdot 2}\right)} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s} \cdot 0.5\right)}}{r}\right) \]
10. Step-by-step derivation

    1. *-commutative 99.7%

       \[\leadsto \left(\frac{0.25}{s} \cdot \frac{1}{\color{blue}{2 \cdot \pi}}\right) \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s} \cdot 0.5\right)}}{r}\right) \]

    2. associate-/r* 99.7%

       \[\leadsto \left(\frac{0.25}{s} \cdot \color{blue}{\frac{\frac{1}{2}}{\pi}}\right) \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s} \cdot 0.5\right)}}{r}\right) \]

    3. metadata-eval 99.7%

       \[\leadsto \left(\frac{0.25}{s} \cdot \frac{\color{blue}{0.5}}{\pi}\right) \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s} \cdot 0.5\right)}}{r}\right) \]

11. Simplified 99.7%

    \[\leadsto \color{blue}{\left(\frac{0.25}{s} \cdot \frac{0.5}{\pi}\right)} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s} \cdot 0.5\right)}}{r}\right) \]

12. Taylor expanded in s around 0 99.7%

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

    1. associate-/r* 99.7%

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

14. Simplified 99.7%

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

15. Final simplification 99.7%

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

Alternative 2: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s} \cdot 0.5\right)}}{r}\right) \cdot \frac{0.125}{s \cdot \pi} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (*
  (+
   (/ (exp (/ r (- s))) r)
   (/ (pow (exp -0.6666666666666666) (* (/ r s) 0.5)) r))
  (/ 0.125 (* s PI))))

C

float code(float s, float r) {
	return ((expf((r / -s)) / r) + (powf(expf(-0.6666666666666666f), ((r / s) * 0.5f)) / r)) * (0.125f / (s * ((float) M_PI)));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(exp(Float32(r / Float32(-s))) / r) + Float32((exp(Float32(-0.6666666666666666)) ^ Float32(Float32(r / s) * Float32(0.5))) / r)) * Float32(Float32(0.125) / Float32(s * Float32(pi))))
end

MATLAB

function tmp = code(s, r)
	tmp = ((exp((r / -s)) / r) + ((exp(single(-0.6666666666666666)) ^ ((r / s) * single(0.5))) / r)) * (single(0.125) / (s * single(pi)));
end

Derivation

1. Initial program 99.6%

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

3. Simplified 99.3%

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

   1. add-sqr-sqrt 99.3%

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

   2. sqrt-unprod 98.9%

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

   3. pow-prod-down 98.9%

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

   4. prod-exp 99.2%

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

   5. metadata-eval 99.2%

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

5. Applied egg-rr 99.2%

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

   1. pow1/2 99.2%

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

   2. pow-pow 99.7%

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

7. Applied egg-rr 99.7%

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

8. Final simplification 99.7%

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

Alternative 3: 99.5% accurate, 1.3× speedup

Math

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

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(r / Float32(s / Float32(-0.3333333333333333)))) / r) + Float32(exp(Float32(Float32(-r) / 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 99.6%

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

3. Simplified 99.3%

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

   1. add-sqr-sqrt 99.3%

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

   2. sqrt-unprod 98.9%

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

   3. pow-prod-down 98.9%

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

   4. prod-exp 99.2%

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

   5. metadata-eval 99.2%

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

5. Applied egg-rr 99.2%

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

   1. pow1/2 99.2%

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

   2. pow-pow 99.7%

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

7. Applied egg-rr 99.7%

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

   1. metadata-eval 99.7%

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

   2. associate-/r* 99.7%

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

   3. *-commutative 99.7%

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

   4. associate-*l* 99.7%

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

   5. associate-/l/ 99.7%

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

   6. div-inv 99.7%

      \[\leadsto \color{blue}{\left(\frac{0.25}{s} \cdot \frac{1}{2 \cdot \pi}\right)} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s} \cdot 0.5\right)}}{r}\right) \]

   7. *-commutative 99.7%

      \[\leadsto \left(\frac{0.25}{s} \cdot \frac{1}{\color{blue}{\pi \cdot 2}}\right) \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s} \cdot 0.5\right)}}{r}\right) \]

9. Applied egg-rr 99.7%

   \[\leadsto \color{blue}{\left(\frac{0.25}{s} \cdot \frac{1}{\pi \cdot 2}\right)} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s} \cdot 0.5\right)}}{r}\right) \]
10. Step-by-step derivation

    1. *-commutative 99.7%

       \[\leadsto \left(\frac{0.25}{s} \cdot \frac{1}{\color{blue}{2 \cdot \pi}}\right) \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s} \cdot 0.5\right)}}{r}\right) \]

    2. associate-/r* 99.7%

       \[\leadsto \left(\frac{0.25}{s} \cdot \color{blue}{\frac{\frac{1}{2}}{\pi}}\right) \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s} \cdot 0.5\right)}}{r}\right) \]

    3. metadata-eval 99.7%

       \[\leadsto \left(\frac{0.25}{s} \cdot \frac{\color{blue}{0.5}}{\pi}\right) \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s} \cdot 0.5\right)}}{r}\right) \]

11. Simplified 99.7%

    \[\leadsto \color{blue}{\left(\frac{0.25}{s} \cdot \frac{0.5}{\pi}\right)} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s} \cdot 0.5\right)}}{r}\right) \]

12. Taylor expanded in s around 0 99.5%

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

    1. +-commutative 99.5%

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

    2. *-commutative 99.5%

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

    3. associate-/r/ 99.6%

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

    4. neg-mul-1 99.6%

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

    5. distribute-frac-neg 99.6%

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

14. Simplified 99.6%

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

15. Final simplification 99.6%

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

Alternative 4: 99.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Simplified 99.3%

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

   1. add-sqr-sqrt 99.3%

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

   2. sqrt-unprod 98.9%

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

   3. pow-prod-down 98.9%

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

   4. prod-exp 99.2%

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

   5. metadata-eval 99.2%

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

5. Applied egg-rr 99.2%

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

   1. pow1/2 99.2%

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

   2. pow-pow 99.7%

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

7. Applied egg-rr 99.7%

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

   1. metadata-eval 99.7%

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

   2. associate-/r* 99.7%

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

   3. *-commutative 99.7%

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

   4. associate-*l* 99.7%

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

   5. associate-/l/ 99.7%

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

   6. div-inv 99.7%

      \[\leadsto \color{blue}{\left(\frac{0.25}{s} \cdot \frac{1}{2 \cdot \pi}\right)} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s} \cdot 0.5\right)}}{r}\right) \]

   7. *-commutative 99.7%

      \[\leadsto \left(\frac{0.25}{s} \cdot \frac{1}{\color{blue}{\pi \cdot 2}}\right) \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s} \cdot 0.5\right)}}{r}\right) \]

9. Applied egg-rr 99.7%

   \[\leadsto \color{blue}{\left(\frac{0.25}{s} \cdot \frac{1}{\pi \cdot 2}\right)} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s} \cdot 0.5\right)}}{r}\right) \]
10. Step-by-step derivation

    1. *-commutative 99.7%

       \[\leadsto \left(\frac{0.25}{s} \cdot \frac{1}{\color{blue}{2 \cdot \pi}}\right) \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s} \cdot 0.5\right)}}{r}\right) \]

    2. associate-/r* 99.7%

       \[\leadsto \left(\frac{0.25}{s} \cdot \color{blue}{\frac{\frac{1}{2}}{\pi}}\right) \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s} \cdot 0.5\right)}}{r}\right) \]

    3. metadata-eval 99.7%

       \[\leadsto \left(\frac{0.25}{s} \cdot \frac{\color{blue}{0.5}}{\pi}\right) \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s} \cdot 0.5\right)}}{r}\right) \]

11. Simplified 99.7%

    \[\leadsto \color{blue}{\left(\frac{0.25}{s} \cdot \frac{0.5}{\pi}\right)} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s} \cdot 0.5\right)}}{r}\right) \]

12. Taylor expanded in r around inf 99.5%

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

    1. associate-*r/ 99.5%

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

    2. neg-mul-1 99.5%

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

    3. distribute-frac-neg 99.5%

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

    4. *-commutative 99.5%

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

    5. associate-/r/ 99.5%

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

14. Simplified 99.5%

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

15. Final simplification 99.5%

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

Alternative 5: 11.5% accurate, 1.4× 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 99.6%

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

3. Simplified 99.3%

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

4. Taylor expanded in r around 0 9.9%

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

5. Taylor expanded in s around inf 9.4%

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

   1. log1p-expm1-u 13.5%

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

7. Applied egg-rr 13.5%

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

8. Final simplification 13.5%

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

Alternative 6: 9.7% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Simplified 99.3%

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

   1. add-cbrt-cube 41.5%

      \[\leadsto \frac{0.125}{\color{blue}{\sqrt[3]{\left(\left(s \cdot \pi\right) \cdot \left(s \cdot \pi\right)\right) \cdot \left(s \cdot \pi\right)}}} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{{\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}}{r}\right) \]

   2. pow1/3 41.4%

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

   3. pow3 41.4%

      \[\leadsto \frac{0.125}{{\color{blue}{\left({\left(s \cdot \pi\right)}^{3}\right)}}^{0.3333333333333333}} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{{\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}}{r}\right) \]

5. Applied egg-rr 41.4%

   \[\leadsto \frac{0.125}{\color{blue}{{\left({\left(s \cdot \pi\right)}^{3}\right)}^{0.3333333333333333}}} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{{\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}}{r}\right) \]
6. Step-by-step derivation

   1. unpow1/3 41.5%

      \[\leadsto \frac{0.125}{\color{blue}{\sqrt[3]{{\left(s \cdot \pi\right)}^{3}}}} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{{\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}}{r}\right) \]

   2. rem-cbrt-cube 99.3%

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

   3. clear-num 99.3%

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

   4. associate-/r/ 99.3%

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

   5. associate-/r* 99.3%

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

7. Applied egg-rr 99.3%

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

8. Taylor expanded in r around 0 10.2%

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

9. Final simplification 10.2%

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

Alternative 7: 9.7% accurate, 1.9× speedup

Math

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

FPCore

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

C

float code(float s, float r) {
	return (0.125f / (s * ((float) M_PI))) * ((expf((r / -s)) / r) + ((1.0f + ((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(Float32(Float32(1.0) + 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) + ((single(1.0) + ((r / s) * single(-0.3333333333333333))) / r));
end

Derivation

1. Initial program 99.6%

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

3. Simplified 99.3%

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

4. Taylor expanded in r around 0 10.2%

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

5. Final simplification 10.2%

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

Alternative 8: 9.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Simplified 99.3%

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

4. Taylor expanded in r around 0 9.9%

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

5. Taylor expanded in s around 0 9.9%

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

   1. associate-*r/ 9.9%

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

   2. mul-1-neg 9.9%

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

7. Simplified 9.9%

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

   1. times-frac 10.0%

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

   2. distribute-neg-frac 10.0%

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

9. Applied egg-rr 10.0%

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

10. Final simplification 10.0%

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

Alternative 9: 9.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Simplified 99.3%

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

4. Taylor expanded in r around 0 9.9%

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

5. Taylor expanded in s around 0 9.9%

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

   1. associate-*r/ 9.9%

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

   2. mul-1-neg 9.9%

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

7. Simplified 9.9%

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

8. Taylor expanded in r around -inf 9.9%

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

   1. associate-*r/ 9.9%

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

   2. *-commutative 9.9%

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

   3. times-frac 9.9%

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

   4. sub-neg 9.9%

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

   5. metadata-eval 9.9%

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

   6. +-commutative 9.9%

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

   7. neg-mul-1 9.9%

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

   8. rec-exp 9.9%

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

   9. associate-*r/ 9.9%

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

   10. metadata-eval 9.9%

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

10. Simplified 9.9%

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

11. Final simplification 9.9%

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

Alternative 10: 9.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Simplified 99.3%

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

4. Taylor expanded in r around 0 9.9%

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

5. Taylor expanded in r around inf 9.9%

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

   1. associate-*r/ 9.9%

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

   2. mul-1-neg 9.9%

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

7. Simplified 9.9%

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

8. Final simplification 9.9%

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

Alternative 11: 9.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Simplified 99.3%

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

4. Taylor expanded in r around 0 9.9%

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

5. Taylor expanded in s around 0 9.9%

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

   1. associate-*r/ 9.9%

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

   2. mul-1-neg 9.9%

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

7. Simplified 9.9%

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

8. Taylor expanded in r around inf 9.9%

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

   1. associate-*r/ 9.9%

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

   2. associate-/l* 9.9%

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

   3. distribute-frac-neg 9.9%

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

10. Simplified 9.9%

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

11. Final simplification 9.9%

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

Alternative 12: 8.9% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Simplified 99.3%

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

4. Taylor expanded in r around 0 9.9%

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

5. Taylor expanded in s around inf 9.4%

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

   1. associate-/r* 9.4%

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

   2. div-inv 9.4%

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

7. Applied egg-rr 9.4%

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

   1. associate-*l/ 9.5%

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

9. Applied egg-rr 9.5%

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

10. Final simplification 9.5%

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

Alternative 13: 8.9% accurate, 4.0× 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 99.6%

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

3. Simplified 99.3%

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

4. Taylor expanded in r around 0 9.9%

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

5. Taylor expanded in s around inf 9.4%

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

6. Final simplification 9.4%

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

Reproduce

herbie shell --seed 2023333 
(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))))