Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.5% → 99.1%

Time: 14.3s

Alternatives: 18

Speedup: 1.0×

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: 99.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \frac{0.125}{s} \cdot \frac{\sqrt[3]{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s}\right)}} \cdot e^{\frac{r \cdot -0.1111111111111111}{s}} + e^{\frac{r}{-s}}}{r \cdot \pi} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (*
  (/ 0.125 s)
  (/
   (+
    (*
     (cbrt (pow (exp -0.6666666666666666) (/ r s)))
     (exp (/ (* r -0.1111111111111111) s)))
    (exp (/ r (- s))))
   (* r PI))))

C

float code(float s, float r) {
	return (0.125f / s) * (((cbrtf(powf(expf(-0.6666666666666666f), (r / s))) * expf(((r * -0.1111111111111111f) / s))) + expf((r / -s))) / (r * ((float) M_PI)));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(0.125) / s) * Float32(Float32(Float32(cbrt((exp(Float32(-0.6666666666666666)) ^ Float32(r / s))) * exp(Float32(Float32(r * Float32(-0.1111111111111111)) / s))) + exp(Float32(r / Float32(-s)))) / Float32(r * 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. Add Preprocessing
5. 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 99.3%

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

   3. pow-prod-down 99.3%

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

   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 99.5%

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

   1. associate-*r* 99.5%

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

   2. associate-*r/ 99.5%

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

   3. *-commutative 99.5%

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

   4. associate-*l* 99.5%

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

   5. *-commutative 99.5%

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

   6. times-frac 99.5%

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

   7. +-commutative 99.5%

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

   8. exp-prod 99.7%

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

   9. neg-mul-1 99.7%

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

   10. distribute-frac-neg 99.7%

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

   11. *-commutative 99.7%

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

9. Simplified 99.7%

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

    1. add-cbrt-cube 99.2%

       \[\leadsto \frac{0.125}{s} \cdot \frac{\color{blue}{\sqrt[3]{\left(\sqrt{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s}\right)}} \cdot \sqrt{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s}\right)}}\right) \cdot \sqrt{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s}\right)}}}} + e^{\frac{-r}{s}}}{r \cdot \pi} \]

    2. add-sqr-sqrt 99.2%

       \[\leadsto \frac{0.125}{s} \cdot \frac{\sqrt[3]{\color{blue}{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s}\right)}} \cdot \sqrt{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s}\right)}}} + e^{\frac{-r}{s}}}{r \cdot \pi} \]

    3. cbrt-prod 99.6%

       \[\leadsto \frac{0.125}{s} \cdot \frac{\color{blue}{\sqrt[3]{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s}\right)}} \cdot \sqrt[3]{\sqrt{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s}\right)}}}} + e^{\frac{-r}{s}}}{r \cdot \pi} \]

11. Applied egg-rr 99.6%

    \[\leadsto \frac{0.125}{s} \cdot \frac{\color{blue}{\sqrt[3]{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s}\right)}} \cdot \sqrt[3]{\sqrt{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s}\right)}}}} + e^{\frac{-r}{s}}}{r \cdot \pi} \]
12. Step-by-step derivation

    1. unpow1/2 99.6%

       \[\leadsto \frac{0.125}{s} \cdot \frac{\sqrt[3]{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s}\right)}} \cdot \sqrt[3]{\color{blue}{{\left({\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s}\right)}\right)}^{0.5}}} + e^{\frac{-r}{s}}}{r \cdot \pi} \]

    2. exp-prod 99.7%

       \[\leadsto \frac{0.125}{s} \cdot \frac{\sqrt[3]{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s}\right)}} \cdot \sqrt[3]{{\color{blue}{\left(e^{-0.6666666666666666 \cdot \frac{r}{s}}\right)}}^{0.5}} + e^{\frac{-r}{s}}}{r \cdot \pi} \]

    3. *-commutative 99.7%

       \[\leadsto \frac{0.125}{s} \cdot \frac{\sqrt[3]{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s}\right)}} \cdot \sqrt[3]{{\left(e^{\color{blue}{\frac{r}{s} \cdot -0.6666666666666666}}\right)}^{0.5}} + e^{\frac{-r}{s}}}{r \cdot \pi} \]

    4. exp-prod 99.7%

       \[\leadsto \frac{0.125}{s} \cdot \frac{\sqrt[3]{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s}\right)}} \cdot \sqrt[3]{\color{blue}{e^{\left(\frac{r}{s} \cdot -0.6666666666666666\right) \cdot 0.5}}} + e^{\frac{-r}{s}}}{r \cdot \pi} \]

    5. associate-*l* 99.7%

       \[\leadsto \frac{0.125}{s} \cdot \frac{\sqrt[3]{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s}\right)}} \cdot \sqrt[3]{e^{\color{blue}{\frac{r}{s} \cdot \left(-0.6666666666666666 \cdot 0.5\right)}}} + e^{\frac{-r}{s}}}{r \cdot \pi} \]

    6. metadata-eval 99.7%

       \[\leadsto \frac{0.125}{s} \cdot \frac{\sqrt[3]{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s}\right)}} \cdot \sqrt[3]{e^{\frac{r}{s} \cdot \color{blue}{-0.3333333333333333}}} + e^{\frac{-r}{s}}}{r \cdot \pi} \]

13. Simplified 99.7%

    \[\leadsto \frac{0.125}{s} \cdot \frac{\color{blue}{\sqrt[3]{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s}\right)}} \cdot \sqrt[3]{e^{\frac{r}{s} \cdot -0.3333333333333333}}} + e^{\frac{-r}{s}}}{r \cdot \pi} \]

14. Taylor expanded in r around inf 99.7%

    \[\leadsto \frac{0.125}{s} \cdot \frac{\sqrt[3]{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s}\right)}} \cdot \color{blue}{\sqrt[3]{e^{-0.3333333333333333 \cdot \frac{r}{s}}}} + e^{\frac{-r}{s}}}{r \cdot \pi} \]
15. Step-by-step derivation

    1. exp-prod 99.5%

       \[\leadsto \frac{0.125}{s} \cdot \frac{\sqrt[3]{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s}\right)}} \cdot \sqrt[3]{\color{blue}{{\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}}} + e^{\frac{-r}{s}}}{r \cdot \pi} \]

    2. unpow1/3 99.5%

       \[\leadsto \frac{0.125}{s} \cdot \frac{\sqrt[3]{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s}\right)}} \cdot \color{blue}{{\left({\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}\right)}^{0.3333333333333333}} + e^{\frac{-r}{s}}}{r \cdot \pi} \]

    3. exp-prod 99.6%

       \[\leadsto \frac{0.125}{s} \cdot \frac{\sqrt[3]{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s}\right)}} \cdot {\color{blue}{\left(e^{-0.3333333333333333 \cdot \frac{r}{s}}\right)}}^{0.3333333333333333} + e^{\frac{-r}{s}}}{r \cdot \pi} \]

    4. associate-*r/ 99.6%

       \[\leadsto \frac{0.125}{s} \cdot \frac{\sqrt[3]{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s}\right)}} \cdot {\left(e^{\color{blue}{\frac{-0.3333333333333333 \cdot r}{s}}}\right)}^{0.3333333333333333} + e^{\frac{-r}{s}}}{r \cdot \pi} \]

    5. *-commutative 99.6%

       \[\leadsto \frac{0.125}{s} \cdot \frac{\sqrt[3]{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s}\right)}} \cdot {\left(e^{\frac{\color{blue}{r \cdot -0.3333333333333333}}{s}}\right)}^{0.3333333333333333} + e^{\frac{-r}{s}}}{r \cdot \pi} \]

    6. exp-prod 99.6%

       \[\leadsto \frac{0.125}{s} \cdot \frac{\sqrt[3]{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s}\right)}} \cdot \color{blue}{e^{\frac{r \cdot -0.3333333333333333}{s} \cdot 0.3333333333333333}} + e^{\frac{-r}{s}}}{r \cdot \pi} \]

    7. *-commutative 99.6%

       \[\leadsto \frac{0.125}{s} \cdot \frac{\sqrt[3]{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s}\right)}} \cdot e^{\color{blue}{0.3333333333333333 \cdot \frac{r \cdot -0.3333333333333333}{s}}} + e^{\frac{-r}{s}}}{r \cdot \pi} \]

    8. associate-*r/ 99.6%

       \[\leadsto \frac{0.125}{s} \cdot \frac{\sqrt[3]{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s}\right)}} \cdot e^{\color{blue}{\frac{0.3333333333333333 \cdot \left(r \cdot -0.3333333333333333\right)}{s}}} + e^{\frac{-r}{s}}}{r \cdot \pi} \]

    9. *-commutative 99.6%

       \[\leadsto \frac{0.125}{s} \cdot \frac{\sqrt[3]{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s}\right)}} \cdot e^{\frac{0.3333333333333333 \cdot \color{blue}{\left(-0.3333333333333333 \cdot r\right)}}{s}} + e^{\frac{-r}{s}}}{r \cdot \pi} \]

    10. associate-*r* 99.6%

        \[\leadsto \frac{0.125}{s} \cdot \frac{\sqrt[3]{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s}\right)}} \cdot e^{\frac{\color{blue}{\left(0.3333333333333333 \cdot -0.3333333333333333\right) \cdot r}}{s}} + e^{\frac{-r}{s}}}{r \cdot \pi} \]

    11. metadata-eval 99.7%

        \[\leadsto \frac{0.125}{s} \cdot \frac{\sqrt[3]{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s}\right)}} \cdot e^{\frac{\color{blue}{-0.1111111111111111} \cdot r}{s}} + e^{\frac{-r}{s}}}{r \cdot \pi} \]

16. Simplified 99.7%

    \[\leadsto \frac{0.125}{s} \cdot \frac{\sqrt[3]{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s}\right)}} \cdot \color{blue}{e^{\frac{-0.1111111111111111 \cdot r}{s}}} + e^{\frac{-r}{s}}}{r \cdot \pi} \]

17. Final simplification 99.7%

    \[\leadsto \frac{0.125}{s} \cdot \frac{\sqrt[3]{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{r}{s}\right)}} \cdot e^{\frac{r \cdot -0.1111111111111111}{s}} + e^{\frac{r}{-s}}}{r \cdot \pi} \]
18. Add Preprocessing

Alternative 2: 99.1% accurate, 0.6× speedup

Math

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

FPCore

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

C

float code(float s, float r) {
	return (0.125f / (s * ((float) M_PI))) * ((expf((r / -s)) / r) + (sqrtf(powf(expf(-0.6666666666666666f), (r / 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(sqrt((exp(Float32(-0.6666666666666666)) ^ Float32(r / s))) / r)))
end

MATLAB

function tmp = code(s, r)
	tmp = (single(0.125) / (s * single(pi))) * ((exp((r / -s)) / r) + (sqrt((exp(single(-0.6666666666666666)) ^ (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. Add Preprocessing
5. 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 99.3%

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

   3. pow-prod-down 99.3%

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

   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. Add Preprocessing

Alternative 3: 99.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \frac{0.125}{s} \cdot \frac{e^{\frac{r}{-s}} + {\left(e^{-0.6666666666666666}\right)}^{\left(\frac{\frac{r}{s}}{2}\right)}}{r \cdot \pi} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (*
  (/ 0.125 s)
  (/
   (+ (exp (/ r (- s))) (pow (exp -0.6666666666666666) (/ (/ r s) 2.0)))
   (* r PI))))

C

float code(float s, float r) {
	return (0.125f / s) * ((expf((r / -s)) + powf(expf(-0.6666666666666666f), ((r / s) / 2.0f))) / (r * ((float) M_PI)));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(0.125) / s) * Float32(Float32(exp(Float32(r / Float32(-s))) + (exp(Float32(-0.6666666666666666)) ^ Float32(Float32(r / s) / Float32(2.0)))) / Float32(r * Float32(pi))))
end

MATLAB

function tmp = code(s, r)
	tmp = (single(0.125) / s) * ((exp((r / -s)) + (exp(single(-0.6666666666666666)) ^ ((r / s) / single(2.0)))) / (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. Add Preprocessing
5. 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 99.3%

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

   3. pow-prod-down 99.3%

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

   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 99.5%

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

   1. associate-*r* 99.5%

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

   2. associate-*r/ 99.5%

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

   3. *-commutative 99.5%

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

   4. associate-*l* 99.5%

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

   5. *-commutative 99.5%

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

   6. times-frac 99.5%

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

   7. +-commutative 99.5%

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

   8. exp-prod 99.7%

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

   9. neg-mul-1 99.7%

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

   10. distribute-frac-neg 99.7%

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

   11. *-commutative 99.7%

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

9. Simplified 99.7%

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

    1. sqrt-pow1 99.7%

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

11. Applied egg-rr 99.7%

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

12. Final simplification 99.7%

    \[\leadsto \frac{0.125}{s} \cdot \frac{e^{\frac{r}{-s}} + {\left(e^{-0.6666666666666666}\right)}^{\left(\frac{\frac{r}{s}}{2}\right)}}{r \cdot \pi} \]
13. Add Preprocessing

Alternative 4: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in r around inf 99.6%

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

   1. neg-mul-1 99.6%

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

   2. rec-exp 99.6%

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

   3. associate-*r/ 99.6%

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

   4. metadata-eval 99.6%

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

5. Simplified 99.6%

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

6. Final simplification 99.6%

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

Alternative 5: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = single(0.125) * ((exp((r / -s)) + (single(2.71828182845904523536) ^ ((r / s) * single(-0.3333333333333333)))) / (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. Add Preprocessing

5. Taylor expanded in r around inf 99.5%

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

   1. *-un-lft-identity 99.5%

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

   2. exp-prod 99.6%

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

   3. *-commutative 99.6%

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

7. Applied egg-rr 99.6%

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

   1. exp-1-e 99.6%

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

9. Simplified 99.6%

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

10. Final simplification 99.6%

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

Alternative 6: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = single(0.125) * ((exp((r / -s)) + exp(((r * single(-0.3333333333333333)) / 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. Add Preprocessing

5. Taylor expanded in r around inf 99.5%

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

   1. associate-*r/ 99.5%

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

7. Applied egg-rr 99.5%

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

8. Final simplification 99.5%

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

Alternative 7: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

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. Add Preprocessing

5. Taylor expanded in r around inf 99.5%

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

   1. mul-1-neg 99.5%

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

   2. distribute-frac-neg2 99.5%

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

   3. expm1-log1p-u 99.5%

      \[\leadsto 0.125 \cdot \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{\frac{r}{-s}}\right)\right)} + e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r \cdot \left(s \cdot \pi\right)} \]

   4. expm1-undefine 99.4%

      \[\leadsto 0.125 \cdot \frac{\color{blue}{\left(e^{\mathsf{log1p}\left(e^{\frac{r}{-s}}\right)} - 1\right)} + e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r \cdot \left(s \cdot \pi\right)} \]

7. Applied egg-rr 99.4%

   \[\leadsto 0.125 \cdot \frac{\color{blue}{\left(e^{\mathsf{log1p}\left(e^{\frac{r}{-s}}\right)} - 1\right)} + e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r \cdot \left(s \cdot \pi\right)} \]
8. Step-by-step derivation

   1. log1p-undefine 99.5%

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

   2. rem-exp-log 99.5%

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

   3. associate-+r- 99.4%

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

   4. expm1-undefine 99.4%

      \[\leadsto 0.125 \cdot \frac{\left(1 + \color{blue}{\mathsf{expm1}\left(\frac{r}{-s}\right)}\right) + e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r \cdot \left(s \cdot \pi\right)} \]

   5. rem-exp-log 99.4%

      \[\leadsto 0.125 \cdot \frac{\color{blue}{e^{\log \left(1 + \mathsf{expm1}\left(\frac{r}{-s}\right)\right)}} + e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r \cdot \left(s \cdot \pi\right)} \]

   6. log1p-define 99.4%

      \[\leadsto 0.125 \cdot \frac{e^{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{r}{-s}\right)\right)}} + e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r \cdot \left(s \cdot \pi\right)} \]

   7. log1p-expm1 99.5%

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

   8. distribute-frac-neg2 99.5%

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

   9. distribute-frac-neg 99.5%

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

9. Simplified 99.5%

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

10. Final simplification 99.5%

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

Alternative 8: 43.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

function code(s, r)
	return Float32(Float32(Float32(0.25) / log1p(expm1(Float32(r * Float32(pi))))) / s)
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. Add Preprocessing

5. Taylor expanded in s around inf 8.4%

   \[\leadsto \color{blue}{\frac{\left(0.125 \cdot \frac{0.05555555555555555 \cdot \frac{r}{\pi} + 0.5 \cdot \frac{r}{\pi}}{{s}^{2}} + 0.25 \cdot \frac{1}{r \cdot \pi}\right) - \frac{0.16666666666666666}{s \cdot \pi}}{s}} \]
6. Step-by-step derivation

7. Simplified 8.4%

   \[\leadsto \color{blue}{\frac{\frac{0.25}{r \cdot \pi} + \frac{\mathsf{fma}\left(0.125, \frac{r}{\pi} \cdot \frac{0.5555555555555556}{s}, \frac{-0.16666666666666666}{\pi}\right)}{s}}{s}} \]

8. Taylor expanded in r around 0 8.0%

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

   1. log1p-expm1-u 40.1%

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

   2. *-commutative 40.1%

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

10. Applied egg-rr 40.1%

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

11. Final simplification 40.1%

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

Alternative 9: 15.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.125}{\frac{r}{s} + 1}}{r \cdot \left(s \cdot \pi\right)} + 0.75 \cdot \frac{e^{\frac{r}{s \cdot -3}}}{r \cdot \left(\left(s \cdot \pi\right) \cdot 6\right)} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/ (/ 0.125 (+ (/ r s) 1.0)) (* r (* s PI)))
  (* 0.75 (/ (exp (/ r (* s -3.0))) (* r (* (* s PI) 6.0))))))

C

float code(float s, float r) {
	return ((0.125f / ((r / s) + 1.0f)) / (r * (s * ((float) M_PI)))) + (0.75f * (expf((r / (s * -3.0f))) / (r * ((s * ((float) M_PI)) * 6.0f))));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.125) / Float32(Float32(r / s) + Float32(1.0))) / Float32(r * Float32(s * Float32(pi)))) + Float32(Float32(0.75) * Float32(exp(Float32(r / Float32(s * Float32(-3.0)))) / Float32(r * Float32(Float32(s * Float32(pi)) * Float32(6.0))))))
end

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.125) / ((r / s) + single(1.0))) / (r * (s * single(pi)))) + (single(0.75) * (exp((r / (s * single(-3.0)))) / (r * ((s * single(pi)) * single(6.0)))));
end

Derivation

1. Initial program 99.6%

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

   1. times-frac 99.6%

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

   2. *-commutative 99.6%

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

   3. distribute-frac-neg 99.6%

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

   4. associate-/l* 99.6%

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

   5. *-commutative 99.6%

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

   6. *-commutative 99.6%

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

   7. associate-*l* 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in s around 0 99.6%

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

   1. associate-*r/ 99.6%

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

   2. rec-exp 99.6%

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

   3. associate-*r/ 99.6%

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

   4. metadata-eval 99.6%

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

7. Simplified 99.6%

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

   1. distribute-frac-neg 99.6%

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

9. Applied egg-rr 99.6%

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

    1. distribute-neg-frac2 99.6%

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

    2. distribute-rgt-neg-in 99.6%

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

    3. metadata-eval 99.6%

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

11. Simplified 99.6%

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

12. Taylor expanded in r around 0 12.3%

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

13. Final simplification 12.3%

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

Alternative 10: 9.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{s}{\frac{0.25}{r \cdot \pi} + \frac{\mathsf{fma}\left(0.06944444444444445, \frac{r}{s \cdot \pi}, \frac{-0.16666666666666666}{\pi}\right)}{s}}} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (/
  1.0
  (/
   s
   (+
    (/ 0.25 (* r PI))
    (/
     (fma 0.06944444444444445 (/ r (* s PI)) (/ -0.16666666666666666 PI))
     s)))))

C

float code(float s, float r) {
	return 1.0f / (s / ((0.25f / (r * ((float) M_PI))) + (fmaf(0.06944444444444445f, (r / (s * ((float) M_PI))), (-0.16666666666666666f / ((float) M_PI))) / s)));
}

Julia

function code(s, r)
	return Float32(Float32(1.0) / Float32(s / Float32(Float32(Float32(0.25) / Float32(r * Float32(pi))) + Float32(fma(Float32(0.06944444444444445), Float32(r / Float32(s * Float32(pi))), Float32(Float32(-0.16666666666666666) / Float32(pi))) / s))))
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. Add Preprocessing

5. Taylor expanded in s around inf 8.4%

   \[\leadsto \color{blue}{\frac{\left(0.125 \cdot \frac{0.05555555555555555 \cdot \frac{r}{\pi} + 0.5 \cdot \frac{r}{\pi}}{{s}^{2}} + 0.25 \cdot \frac{1}{r \cdot \pi}\right) - \frac{0.16666666666666666}{s \cdot \pi}}{s}} \]
6. Step-by-step derivation

7. Simplified 8.4%

   \[\leadsto \color{blue}{\frac{\frac{0.25}{r \cdot \pi} + \frac{\mathsf{fma}\left(0.125, \frac{r}{\pi} \cdot \frac{0.5555555555555556}{s}, \frac{-0.16666666666666666}{\pi}\right)}{s}}{s}} \]
8. Step-by-step derivation

   1. clear-num 8.4%

      \[\leadsto \color{blue}{\frac{1}{\frac{s}{\frac{0.25}{r \cdot \pi} + \frac{\mathsf{fma}\left(0.125, \frac{r}{\pi} \cdot \frac{0.5555555555555556}{s}, \frac{-0.16666666666666666}{\pi}\right)}{s}}}} \]

   2. inv-pow 8.4%

      \[\leadsto \color{blue}{{\left(\frac{s}{\frac{0.25}{r \cdot \pi} + \frac{\mathsf{fma}\left(0.125, \frac{r}{\pi} \cdot \frac{0.5555555555555556}{s}, \frac{-0.16666666666666666}{\pi}\right)}{s}}\right)}^{-1}} \]

   3. associate-/r* 8.4%

      \[\leadsto {\left(\frac{s}{\color{blue}{\frac{\frac{0.25}{r}}{\pi}} + \frac{\mathsf{fma}\left(0.125, \frac{r}{\pi} \cdot \frac{0.5555555555555556}{s}, \frac{-0.16666666666666666}{\pi}\right)}{s}}\right)}^{-1} \]

   4. frac-times 8.4%

      \[\leadsto {\left(\frac{s}{\frac{\frac{0.25}{r}}{\pi} + \frac{\mathsf{fma}\left(0.125, \color{blue}{\frac{r \cdot 0.5555555555555556}{\pi \cdot s}}, \frac{-0.16666666666666666}{\pi}\right)}{s}}\right)}^{-1} \]

   5. *-commutative 8.4%

      \[\leadsto {\left(\frac{s}{\frac{\frac{0.25}{r}}{\pi} + \frac{\mathsf{fma}\left(0.125, \frac{r \cdot 0.5555555555555556}{\color{blue}{s \cdot \pi}}, \frac{-0.16666666666666666}{\pi}\right)}{s}}\right)}^{-1} \]

9. Applied egg-rr 8.4%

   \[\leadsto \color{blue}{{\left(\frac{s}{\frac{\frac{0.25}{r}}{\pi} + \frac{\mathsf{fma}\left(0.125, \frac{r \cdot 0.5555555555555556}{s \cdot \pi}, \frac{-0.16666666666666666}{\pi}\right)}{s}}\right)}^{-1}} \]
10. Step-by-step derivation

    1. unpow-1 8.4%

       \[\leadsto \color{blue}{\frac{1}{\frac{s}{\frac{\frac{0.25}{r}}{\pi} + \frac{\mathsf{fma}\left(0.125, \frac{r \cdot 0.5555555555555556}{s \cdot \pi}, \frac{-0.16666666666666666}{\pi}\right)}{s}}}} \]

    2. associate-/r* 8.4%

       \[\leadsto \frac{1}{\frac{s}{\color{blue}{\frac{0.25}{r \cdot \pi}} + \frac{\mathsf{fma}\left(0.125, \frac{r \cdot 0.5555555555555556}{s \cdot \pi}, \frac{-0.16666666666666666}{\pi}\right)}{s}}} \]

11. Simplified 8.4%

    \[\leadsto \color{blue}{\frac{1}{\frac{s}{\frac{0.25}{r \cdot \pi} + \frac{\mathsf{fma}\left(0.06944444444444445, \frac{r}{s \cdot \pi}, \frac{-0.16666666666666666}{\pi}\right)}{s}}}} \]
12. Add Preprocessing

Alternative 11: 9.9% accurate, 8.6× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.25}{r \cdot \pi} + \frac{r \cdot \left(0.06944444444444445 \cdot \frac{1}{s \cdot \pi} + 0.16666666666666666 \cdot \frac{-1}{r \cdot \pi}\right)}{s}}{s} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (/
  (+
   (/ 0.25 (* r PI))
   (/
    (*
     r
     (+
      (* 0.06944444444444445 (/ 1.0 (* s PI)))
      (* 0.16666666666666666 (/ -1.0 (* r PI)))))
    s))
  s))

C

float code(float s, float r) {
	return ((0.25f / (r * ((float) M_PI))) + ((r * ((0.06944444444444445f * (1.0f / (s * ((float) M_PI)))) + (0.16666666666666666f * (-1.0f / (r * ((float) M_PI)))))) / s)) / s;
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.25) / Float32(r * Float32(pi))) + Float32(Float32(r * Float32(Float32(Float32(0.06944444444444445) * Float32(Float32(1.0) / Float32(s * Float32(pi)))) + Float32(Float32(0.16666666666666666) * Float32(Float32(-1.0) / Float32(r * Float32(pi)))))) / s)) / s)
end

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.25) / (r * single(pi))) + ((r * ((single(0.06944444444444445) * (single(1.0) / (s * single(pi)))) + (single(0.16666666666666666) * (single(-1.0) / (r * single(pi)))))) / s)) / s;
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. Add Preprocessing

5. Taylor expanded in s around inf 8.4%

   \[\leadsto \color{blue}{\frac{\left(0.125 \cdot \frac{0.05555555555555555 \cdot \frac{r}{\pi} + 0.5 \cdot \frac{r}{\pi}}{{s}^{2}} + 0.25 \cdot \frac{1}{r \cdot \pi}\right) - \frac{0.16666666666666666}{s \cdot \pi}}{s}} \]
6. Step-by-step derivation

7. Simplified 8.4%

   \[\leadsto \color{blue}{\frac{\frac{0.25}{r \cdot \pi} + \frac{\mathsf{fma}\left(0.125, \frac{r}{\pi} \cdot \frac{0.5555555555555556}{s}, \frac{-0.16666666666666666}{\pi}\right)}{s}}{s}} \]

8. Taylor expanded in r around inf 8.4%

   \[\leadsto \frac{\frac{0.25}{r \cdot \pi} + \frac{\color{blue}{r \cdot \left(0.06944444444444445 \cdot \frac{1}{s \cdot \pi} - 0.16666666666666666 \cdot \frac{1}{r \cdot \pi}\right)}}{s}}{s} \]

9. Final simplification 8.4%

   \[\leadsto \frac{\frac{0.25}{r \cdot \pi} + \frac{r \cdot \left(0.06944444444444445 \cdot \frac{1}{s \cdot \pi} + 0.16666666666666666 \cdot \frac{-1}{r \cdot \pi}\right)}{s}}{s} \]
10. Add Preprocessing

Alternative 12: 9.9% accurate, 10.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

5. Taylor expanded in s around inf 8.4%

   \[\leadsto \color{blue}{\frac{\left(0.125 \cdot \frac{0.05555555555555555 \cdot \frac{r}{\pi} + 0.5 \cdot \frac{r}{\pi}}{{s}^{2}} + 0.25 \cdot \frac{1}{r \cdot \pi}\right) - \frac{0.16666666666666666}{s \cdot \pi}}{s}} \]
6. Step-by-step derivation

7. Simplified 8.4%

   \[\leadsto \color{blue}{\frac{\frac{0.25}{r \cdot \pi} + \frac{\mathsf{fma}\left(0.125, \frac{r}{\pi} \cdot \frac{0.5555555555555556}{s}, \frac{-0.16666666666666666}{\pi}\right)}{s}}{s}} \]

8. Taylor expanded in r around inf 8.4%

   \[\leadsto \frac{\frac{0.25}{r \cdot \pi} + \frac{\color{blue}{r \cdot \left(0.06944444444444445 \cdot \frac{1}{s \cdot \pi} - 0.16666666666666666 \cdot \frac{1}{r \cdot \pi}\right)}}{s}}{s} \]
9. Step-by-step derivation

   1. associate-*r/ 8.4%

      \[\leadsto \frac{\frac{0.25}{r \cdot \pi} + \frac{r \cdot \left(\color{blue}{\frac{0.06944444444444445 \cdot 1}{s \cdot \pi}} - 0.16666666666666666 \cdot \frac{1}{r \cdot \pi}\right)}{s}}{s} \]

   2. metadata-eval 8.4%

      \[\leadsto \frac{\frac{0.25}{r \cdot \pi} + \frac{r \cdot \left(\frac{\color{blue}{0.06944444444444445}}{s \cdot \pi} - 0.16666666666666666 \cdot \frac{1}{r \cdot \pi}\right)}{s}}{s} \]

   3. associate-*r/ 8.4%

      \[\leadsto \frac{\frac{0.25}{r \cdot \pi} + \frac{r \cdot \left(\frac{0.06944444444444445}{s \cdot \pi} - \color{blue}{\frac{0.16666666666666666 \cdot 1}{r \cdot \pi}}\right)}{s}}{s} \]

   4. metadata-eval 8.4%

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

10. Simplified 8.4%

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

Alternative 13: 9.9% accurate, 11.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

5. Taylor expanded in s around inf 8.4%

   \[\leadsto \color{blue}{\frac{\left(0.125 \cdot \frac{0.05555555555555555 \cdot \frac{r}{\pi} + 0.5 \cdot \frac{r}{\pi}}{{s}^{2}} + 0.25 \cdot \frac{1}{r \cdot \pi}\right) - \frac{0.16666666666666666}{s \cdot \pi}}{s}} \]
6. Step-by-step derivation

7. Simplified 8.4%

   \[\leadsto \color{blue}{\frac{\frac{0.25}{r \cdot \pi} + \frac{\mathsf{fma}\left(0.125, \frac{r}{\pi} \cdot \frac{0.5555555555555556}{s}, \frac{-0.16666666666666666}{\pi}\right)}{s}}{s}} \]

8. Taylor expanded in s around inf 8.4%

   \[\leadsto \frac{\frac{0.25}{r \cdot \pi} + \color{blue}{\frac{0.06944444444444445 \cdot \frac{r}{s \cdot \pi} - 0.16666666666666666 \cdot \frac{1}{\pi}}{s}}}{s} \]
9. Step-by-step derivation

   1. associate-*r/ 8.4%

      \[\leadsto \frac{\frac{0.25}{r \cdot \pi} + \frac{\color{blue}{\frac{0.06944444444444445 \cdot r}{s \cdot \pi}} - 0.16666666666666666 \cdot \frac{1}{\pi}}{s}}{s} \]

   2. associate-*r/ 8.4%

      \[\leadsto \frac{\frac{0.25}{r \cdot \pi} + \frac{\frac{0.06944444444444445 \cdot r}{s \cdot \pi} - \color{blue}{\frac{0.16666666666666666 \cdot 1}{\pi}}}{s}}{s} \]

   3. metadata-eval 8.4%

      \[\leadsto \frac{\frac{0.25}{r \cdot \pi} + \frac{\frac{0.06944444444444445 \cdot r}{s \cdot \pi} - \frac{\color{blue}{0.16666666666666666}}{\pi}}{s}}{s} \]

10. Simplified 8.4%

    \[\leadsto \frac{\frac{0.25}{r \cdot \pi} + \color{blue}{\frac{\frac{0.06944444444444445 \cdot r}{s \cdot \pi} - \frac{0.16666666666666666}{\pi}}{s}}}{s} \]

11. Final simplification 8.4%

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

Alternative 14: 9.9% accurate, 11.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

5. Taylor expanded in s around inf 8.4%

   \[\leadsto \color{blue}{\frac{\left(0.125 \cdot \frac{0.05555555555555555 \cdot \frac{r}{\pi} + 0.5 \cdot \frac{r}{\pi}}{{s}^{2}} + 0.25 \cdot \frac{1}{r \cdot \pi}\right) - \frac{0.16666666666666666}{s \cdot \pi}}{s}} \]
6. Step-by-step derivation

7. Simplified 8.4%

   \[\leadsto \color{blue}{\frac{\frac{0.25}{r \cdot \pi} + \frac{\mathsf{fma}\left(0.125, \frac{r}{\pi} \cdot \frac{0.5555555555555556}{s}, \frac{-0.16666666666666666}{\pi}\right)}{s}}{s}} \]

8. Taylor expanded in r around inf 8.4%

   \[\leadsto \frac{\frac{0.25}{r \cdot \pi} + \frac{\color{blue}{r \cdot \left(0.06944444444444445 \cdot \frac{1}{s \cdot \pi} - 0.16666666666666666 \cdot \frac{1}{r \cdot \pi}\right)}}{s}}{s} \]
9. Step-by-step derivation

   1. associate-*r/ 8.4%

      \[\leadsto \frac{\frac{0.25}{r \cdot \pi} + \frac{r \cdot \left(\color{blue}{\frac{0.06944444444444445 \cdot 1}{s \cdot \pi}} - 0.16666666666666666 \cdot \frac{1}{r \cdot \pi}\right)}{s}}{s} \]

   2. metadata-eval 8.4%

      \[\leadsto \frac{\frac{0.25}{r \cdot \pi} + \frac{r \cdot \left(\frac{\color{blue}{0.06944444444444445}}{s \cdot \pi} - 0.16666666666666666 \cdot \frac{1}{r \cdot \pi}\right)}{s}}{s} \]

   3. associate-*r/ 8.4%

      \[\leadsto \frac{\frac{0.25}{r \cdot \pi} + \frac{r \cdot \left(\frac{0.06944444444444445}{s \cdot \pi} - \color{blue}{\frac{0.16666666666666666 \cdot 1}{r \cdot \pi}}\right)}{s}}{s} \]

   4. metadata-eval 8.4%

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

10. Simplified 8.4%

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

11. Taylor expanded in s around inf 8.4%

    \[\leadsto \frac{\frac{0.25}{r \cdot \pi} + \frac{r \cdot \color{blue}{\left(0.06944444444444445 \cdot \frac{1}{s \cdot \pi} - 0.16666666666666666 \cdot \frac{1}{r \cdot \pi}\right)}}{s}}{s} \]
12. Step-by-step derivation

    1. associate-*r/ 8.4%

       \[\leadsto \frac{\frac{0.25}{r \cdot \pi} + \frac{r \cdot \left(\color{blue}{\frac{0.06944444444444445 \cdot 1}{s \cdot \pi}} - 0.16666666666666666 \cdot \frac{1}{r \cdot \pi}\right)}{s}}{s} \]

    2. metadata-eval 8.4%

       \[\leadsto \frac{\frac{0.25}{r \cdot \pi} + \frac{r \cdot \left(\frac{\color{blue}{0.06944444444444445}}{s \cdot \pi} - 0.16666666666666666 \cdot \frac{1}{r \cdot \pi}\right)}{s}}{s} \]

    3. associate-/r* 8.4%

       \[\leadsto \frac{\frac{0.25}{r \cdot \pi} + \frac{r \cdot \left(\color{blue}{\frac{\frac{0.06944444444444445}{s}}{\pi}} - 0.16666666666666666 \cdot \frac{1}{r \cdot \pi}\right)}{s}}{s} \]

    4. associate-*r/ 8.4%

       \[\leadsto \frac{\frac{0.25}{r \cdot \pi} + \frac{r \cdot \left(\frac{\frac{0.06944444444444445}{s}}{\pi} - \color{blue}{\frac{0.16666666666666666 \cdot 1}{r \cdot \pi}}\right)}{s}}{s} \]

    5. metadata-eval 8.4%

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

    6. associate-/r* 8.4%

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

    7. div-sub 8.4%

       \[\leadsto \frac{\frac{0.25}{r \cdot \pi} + \frac{r \cdot \color{blue}{\frac{\frac{0.06944444444444445}{s} - \frac{0.16666666666666666}{r}}{\pi}}}{s}}{s} \]

13. Simplified 8.4%

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

Alternative 15: 9.0% accurate, 25.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = (single(1.0) / s) * ((single(0.25) / 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. Add Preprocessing
5. 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 99.3%

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

   3. pow-prod-down 99.3%

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

   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 8.0%

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

   1. associate-/r* 8.0%

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

9. Simplified 8.0%

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

    1. *-un-lft-identity 8.0%

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

    2. times-frac 8.0%

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

11. Applied egg-rr 8.0%

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

Alternative 16: 9.0% accurate, 25.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = (single(0.125) / s) * (single(2.0) / (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. Add Preprocessing
5. 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 99.3%

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

   3. pow-prod-down 99.3%

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

   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 99.5%

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

   1. associate-*r* 99.5%

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

   2. associate-*r/ 99.5%

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

   3. *-commutative 99.5%

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

   4. associate-*l* 99.5%

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

   5. *-commutative 99.5%

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

   6. times-frac 99.5%

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

   7. +-commutative 99.5%

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

   8. exp-prod 99.7%

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

   9. neg-mul-1 99.7%

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

   10. distribute-frac-neg 99.7%

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

   11. *-commutative 99.7%

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

9. Simplified 99.7%

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

10. Taylor expanded in r around 0 8.0%

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

Alternative 17: 9.0% accurate, 33.0× speedup

Math

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

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(Float32(0.25) / 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. Add Preprocessing
5. 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 99.3%

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

   3. pow-prod-down 99.3%

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

   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 8.0%

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

   1. associate-/r* 8.0%

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

9. Simplified 8.0%

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

Alternative 18: 9.0% accurate, 33.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

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. Add Preprocessing

5. Taylor expanded in s around inf 8.0%

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

Reproduce

herbie shell --seed 2024150 
(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))))