Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.5%

Time: 16.8s

Alternatives: 15

Speedup: N/A×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.3%

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

3. Simplified 99.0%

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

   1. pow-to-exp 99.1%

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

   2. rem-log-exp 99.3%

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

   3. metadata-eval 99.3%

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

   4. times-frac 99.3%

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

   5. neg-mul-1 99.3%

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

   6. distribute-frac-neg 99.3%

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

   7. *-commutative 99.3%

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

   8. exp-neg 99.1%

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

   9. add-sqr-sqrt 99.0%

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

   10. sqrt-unprod 99.1%

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

   11. sqr-neg 99.1%

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

   12. sqrt-unprod -0.0%

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

   13. add-sqr-sqrt 9.9%

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

   14. associate-/r* 9.9%

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

   15. exp-cbrt 9.9%

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

   16. frac-2neg 9.9%

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

   17. remove-double-neg 9.9%

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

   18. add-sqr-sqrt -0.0%

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

   19. sqrt-unprod 98.2%

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

   20. sqr-neg 98.2%

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

   21. sqrt-unprod 98.1%

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

6. Applied egg-rr 98.2%

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

   1. pow1/3 98.3%

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

8. Applied egg-rr 98.3%

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

   1. pow-flip 98.3%

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

   2. metadata-eval 98.3%

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

   3. pow-exp 99.3%

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

   4. *-un-lft-identity 99.3%

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

   5. exp-prod 99.4%

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

   6. associate-*l/ 99.4%

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

   7. *-un-lft-identity 99.4%

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

   8. times-frac 99.4%

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

   9. /-rgt-identity 99.4%

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

10. Applied egg-rr 99.4%

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

11. Final simplification 99.4%

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

Alternative 2: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

MATLAB

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

Derivation

1. Initial program 99.3%

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

3. Simplified 99.0%

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

5. Taylor expanded in r around inf 99.3%

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

6. Final simplification 99.3%

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

Alternative 3: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.3%

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

3. Simplified 99.0%

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

5. Taylor expanded in r around inf 99.3%

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

   1. exp-prod 99.0%

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

   2. *-lft-identity 99.0%

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

   3. associate-*l/ 99.1%

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

   4. exp-prod 99.3%

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

   5. associate-*l* 99.3%

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

   6. associate-*r/ 99.3%

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

   7. metadata-eval 99.3%

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

   8. associate-/r/ 99.3%

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

7. Simplified 99.3%

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

   1. associate-/r/ 99.3%

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

9. Applied egg-rr 99.3%

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

10. Final simplification 99.3%

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

Alternative 4: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.3%

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

3. Simplified 99.0%

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

5. Taylor expanded in r around inf 99.3%

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

   1. exp-prod 99.0%

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

   2. *-lft-identity 99.0%

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

   3. associate-*l/ 99.1%

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

   4. exp-prod 99.3%

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

   5. associate-*l* 99.3%

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

   6. associate-*r/ 99.3%

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

   7. metadata-eval 99.3%

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

   8. associate-/r/ 99.3%

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

7. Simplified 99.3%

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

8. Final simplification 99.3%

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

Alternative 5: 91.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;s \leq 0.20000000298023224:\\ \;\;\;\;\frac{\frac{\frac{0.125}{e^{\frac{r}{s}}}}{s \cdot \pi}}{r}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r} + \left(\frac{-1 - \frac{r}{s} \cdot -0.5}{s} + \frac{1}{r}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (if (<= s 0.20000000298023224)
   (/ (/ (/ 0.125 (exp (/ r s))) (* s PI)) r)
   (*
    (/ 0.125 (* s PI))
    (+
     (/ (exp (* -0.3333333333333333 (/ r s))) r)
     (+ (/ (- -1.0 (* (/ r s) -0.5)) s) (/ 1.0 r))))))

C

float code(float s, float r) {
	float tmp;
	if (s <= 0.20000000298023224f) {
		tmp = ((0.125f / expf((r / s))) / (s * ((float) M_PI))) / r;
	} else {
		tmp = (0.125f / (s * ((float) M_PI))) * ((expf((-0.3333333333333333f * (r / s))) / r) + (((-1.0f - ((r / s) * -0.5f)) / s) + (1.0f / r)));
	}
	return tmp;
}

Julia

function code(s, r)
	tmp = Float32(0.0)
	if (s <= Float32(0.20000000298023224))
		tmp = Float32(Float32(Float32(Float32(0.125) / exp(Float32(r / s))) / Float32(s * Float32(pi))) / r);
	else
		tmp = Float32(Float32(Float32(0.125) / Float32(s * Float32(pi))) * Float32(Float32(exp(Float32(Float32(-0.3333333333333333) * Float32(r / s))) / r) + Float32(Float32(Float32(Float32(-1.0) - Float32(Float32(r / s) * Float32(-0.5))) / s) + Float32(Float32(1.0) / r))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(s, r)
	tmp = single(0.0);
	if (s <= single(0.20000000298023224))
		tmp = ((single(0.125) / exp((r / s))) / (s * single(pi))) / r;
	else
		tmp = (single(0.125) / (s * single(pi))) * ((exp((single(-0.3333333333333333) * (r / s))) / r) + (((single(-1.0) - ((r / s) * single(-0.5))) / s) + (single(1.0) / r)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if s < 0.200000003

   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 99.4%

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

      1. pow-to-exp 99.4%

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

      2. rem-log-exp 99.6%

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

      3. metadata-eval 99.6%

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

      4. times-frac 99.6%

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

      5. neg-mul-1 99.6%

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

      6. distribute-frac-neg 99.6%

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

      7. *-commutative 99.6%

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

      8. exp-neg 99.4%

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

      9. add-sqr-sqrt 99.3%

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

      10. sqrt-unprod 99.4%

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

      11. sqr-neg 99.4%

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

      12. sqrt-unprod -0.0%

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

      13. add-sqr-sqrt 5.2%

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

      14. associate-/r* 5.2%

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

      15. exp-cbrt 5.2%

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

      16. frac-2neg 5.2%

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

      17. remove-double-neg 5.2%

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

      18. add-sqr-sqrt -0.0%

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

      19. sqrt-unprod 99.1%

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

      20. sqr-neg 99.1%

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

      21. sqrt-unprod 99.0%

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

   6. Applied egg-rr 99.1%

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

   7. Taylor expanded in r around 0 11.9%

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

   8. Taylor expanded in s around 0 93.8%

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

      1. associate-*r* 93.8%

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

      2. associate-*r/ 93.8%

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

      3. associate-*r* 93.8%

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

      4. *-commutative 93.8%

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

      5. associate-/r* 93.8%

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

      6. mul-1-neg 93.8%

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

      7. rec-exp 93.8%

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

      8. associate-*r/ 93.8%

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

      9. metadata-eval 93.8%

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

   10. Simplified 93.8%

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

   if 0.200000003 < s

   1. Initial program 97.2%

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

   3. Simplified 96.2%

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

   5. Taylor expanded in r around inf 97.3%

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

      1. exp-prod 96.2%

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

      2. *-lft-identity 96.2%

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

      3. associate-*l/ 96.2%

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

      4. exp-prod 97.0%

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

      5. associate-*l* 97.2%

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

      6. associate-*r/ 97.5%

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

      7. metadata-eval 97.5%

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

      8. associate-/r/ 97.5%

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

   7. Simplified 97.5%

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

      1. clear-num 97.6%

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

      2. associate-/r/ 97.3%

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

      3. clear-num 97.3%

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

   9. Applied egg-rr 97.3%

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

   10. Taylor expanded in s around -inf 66.0%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \leq 0.20000000298023224:\\ \;\;\;\;\frac{\frac{\frac{0.125}{e^{\frac{r}{s}}}}{s \cdot \pi}}{r}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r} + \left(\frac{-1 - \frac{r}{s} \cdot -0.5}{s} + \frac{1}{r}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 91.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;s \leq 0.15000000596046448:\\ \;\;\;\;\frac{\frac{\frac{0.125}{e^{\frac{r}{s}}}}{s \cdot \pi}}{r}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\frac{1}{1 + \frac{r}{s} \cdot 0.3333333333333333}}{r}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (if (<= s 0.15000000596046448)
   (/ (/ (/ 0.125 (exp (/ r s))) (* s PI)) r)
   (*
    (/ 0.125 (* s PI))
    (+
     (/ (exp (/ r (- s))) r)
     (/ (/ 1.0 (+ 1.0 (* (/ r s) 0.3333333333333333))) r)))))

C

float code(float s, float r) {
	float tmp;
	if (s <= 0.15000000596046448f) {
		tmp = ((0.125f / expf((r / s))) / (s * ((float) M_PI))) / r;
	} else {
		tmp = (0.125f / (s * ((float) M_PI))) * ((expf((r / -s)) / r) + ((1.0f / (1.0f + ((r / s) * 0.3333333333333333f))) / r));
	}
	return tmp;
}

Julia

function code(s, r)
	tmp = Float32(0.0)
	if (s <= Float32(0.15000000596046448))
		tmp = Float32(Float32(Float32(Float32(0.125) / exp(Float32(r / s))) / Float32(s * Float32(pi))) / r);
	else
		tmp = Float32(Float32(Float32(0.125) / Float32(s * Float32(pi))) * Float32(Float32(exp(Float32(r / Float32(-s))) / r) + Float32(Float32(Float32(1.0) / Float32(Float32(1.0) + Float32(Float32(r / s) * Float32(0.3333333333333333)))) / r)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(s, r)
	tmp = single(0.0);
	if (s <= single(0.15000000596046448))
		tmp = ((single(0.125) / exp((r / s))) / (s * single(pi))) / r;
	else
		tmp = (single(0.125) / (s * single(pi))) * ((exp((r / -s)) / r) + ((single(1.0) / (single(1.0) + ((r / s) * single(0.3333333333333333)))) / r));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if s < 0.150000006

   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 99.4%

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

      1. pow-to-exp 99.5%

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

      2. rem-log-exp 99.6%

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

      3. metadata-eval 99.6%

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

      4. times-frac 99.6%

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

      5. neg-mul-1 99.6%

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

      6. distribute-frac-neg 99.6%

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

      7. *-commutative 99.6%

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

      8. exp-neg 99.4%

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

      9. add-sqr-sqrt 99.3%

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

      10. sqrt-unprod 99.4%

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

      11. sqr-neg 99.4%

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

      12. sqrt-unprod -0.0%

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

      13. add-sqr-sqrt 5.2%

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

      14. associate-/r* 5.2%

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

      15. exp-cbrt 5.2%

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

      16. frac-2neg 5.2%

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

      17. remove-double-neg 5.2%

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

      18. add-sqr-sqrt -0.0%

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

      19. sqrt-unprod 99.1%

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

      20. sqr-neg 99.1%

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

      21. sqrt-unprod 99.0%

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

   6. Applied egg-rr 99.1%

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

   7. Taylor expanded in r around 0 11.9%

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

   8. Taylor expanded in s around 0 94.1%

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

      1. associate-*r* 94.1%

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

      2. associate-*r/ 94.1%

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

      3. associate-*r* 94.1%

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

      4. *-commutative 94.1%

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

      5. associate-/r* 94.1%

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

      6. mul-1-neg 94.1%

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

      7. rec-exp 94.1%

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

      8. associate-*r/ 94.1%

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

      9. metadata-eval 94.1%

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

   10. Simplified 94.1%

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

   if 0.150000006 < s

   1. Initial program 97.2%

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

   3. Simplified 96.1%

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

      1. pow-to-exp 96.3%

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

      2. rem-log-exp 97.1%

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

      3. metadata-eval 97.1%

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

      4. times-frac 97.4%

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

      5. neg-mul-1 97.4%

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

      6. distribute-frac-neg 97.4%

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

      7. *-commutative 97.4%

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

      8. exp-neg 97.1%

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

      9. add-sqr-sqrt 97.0%

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

      10. sqrt-unprod 97.1%

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

      11. sqr-neg 97.1%

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

      12. sqrt-unprod -0.0%

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

      13. add-sqr-sqrt 45.9%

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

      14. associate-/r* 45.9%

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

      15. exp-cbrt 45.8%

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

      16. frac-2neg 45.8%

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

      17. remove-double-neg 45.8%

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

      18. add-sqr-sqrt -0.0%

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

      19. sqrt-unprod 91.9%

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

      20. sqr-neg 91.9%

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

      21. sqrt-unprod 91.6%

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

   6. Applied egg-rr 91.9%

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

   7. Taylor expanded in r around 0 63.5%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \leq 0.15000000596046448:\\ \;\;\;\;\frac{\frac{\frac{0.125}{e^{\frac{r}{s}}}}{s \cdot \pi}}{r}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\frac{1}{1 + \frac{r}{s} \cdot 0.3333333333333333}}{r}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 91.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;s \leq 0.20000000298023224:\\ \;\;\;\;\frac{\frac{\frac{0.125}{e^{\frac{r}{s}}}}{s \cdot \pi}}{r}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{1 + \frac{-0.3333333333333333}{\frac{s}{r}}}{r}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (if (<= s 0.20000000298023224)
   (/ (/ (/ 0.125 (exp (/ r s))) (* s PI)) r)
   (*
    (/ 0.125 (* s PI))
    (+
     (/ (exp (/ r (- s))) r)
     (/ (+ 1.0 (/ -0.3333333333333333 (/ s r))) r)))))

C

float code(float s, float r) {
	float tmp;
	if (s <= 0.20000000298023224f) {
		tmp = ((0.125f / expf((r / s))) / (s * ((float) M_PI))) / r;
	} else {
		tmp = (0.125f / (s * ((float) M_PI))) * ((expf((r / -s)) / r) + ((1.0f + (-0.3333333333333333f / (s / r))) / r));
	}
	return tmp;
}

Julia

function code(s, r)
	tmp = Float32(0.0)
	if (s <= Float32(0.20000000298023224))
		tmp = Float32(Float32(Float32(Float32(0.125) / exp(Float32(r / s))) / Float32(s * Float32(pi))) / r);
	else
		tmp = Float32(Float32(Float32(0.125) / Float32(s * Float32(pi))) * Float32(Float32(exp(Float32(r / Float32(-s))) / r) + Float32(Float32(Float32(1.0) + Float32(Float32(-0.3333333333333333) / Float32(s / r))) / r)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(s, r)
	tmp = single(0.0);
	if (s <= single(0.20000000298023224))
		tmp = ((single(0.125) / exp((r / s))) / (s * single(pi))) / r;
	else
		tmp = (single(0.125) / (s * single(pi))) * ((exp((r / -s)) / r) + ((single(1.0) + (single(-0.3333333333333333) / (s / r))) / r));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if s < 0.200000003

   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 99.4%

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

      1. pow-to-exp 99.4%

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

      2. rem-log-exp 99.6%

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

      3. metadata-eval 99.6%

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

      4. times-frac 99.6%

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

      5. neg-mul-1 99.6%

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

      6. distribute-frac-neg 99.6%

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

      7. *-commutative 99.6%

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

      8. exp-neg 99.4%

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

      9. add-sqr-sqrt 99.3%

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

      10. sqrt-unprod 99.4%

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

      11. sqr-neg 99.4%

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

      12. sqrt-unprod -0.0%

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

      13. add-sqr-sqrt 5.2%

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

      14. associate-/r* 5.2%

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

      15. exp-cbrt 5.2%

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

      16. frac-2neg 5.2%

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

      17. remove-double-neg 5.2%

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

      18. add-sqr-sqrt -0.0%

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

      19. sqrt-unprod 99.1%

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

      20. sqr-neg 99.1%

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

      21. sqrt-unprod 99.0%

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

   6. Applied egg-rr 99.1%

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

   7. Taylor expanded in r around 0 11.9%

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

   8. Taylor expanded in s around 0 93.8%

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

      1. associate-*r* 93.8%

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

      2. associate-*r/ 93.8%

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

      3. associate-*r* 93.8%

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

      4. *-commutative 93.8%

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

      5. associate-/r* 93.8%

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

      6. mul-1-neg 93.8%

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

      7. rec-exp 93.8%

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

      8. associate-*r/ 93.8%

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

      9. metadata-eval 93.8%

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

   10. Simplified 93.8%

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

   if 0.200000003 < s

   1. Initial program 97.2%

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

   3. Simplified 96.2%

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

   5. Taylor expanded in r around 0 63.1%

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

      1. associate-*r/ 63.1%

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

      2. associate-*l/ 63.1%

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

      3. associate-/r/ 63.1%

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

   7. Simplified 63.1%

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

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \leq 0.20000000298023224:\\ \;\;\;\;\frac{\frac{\frac{0.125}{e^{\frac{r}{s}}}}{s \cdot \pi}}{r}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{1 + \frac{-0.3333333333333333}{\frac{s}{r}}}{r}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 91.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;s \leq 0.20000000298023224:\\ \;\;\;\;\frac{\frac{\frac{0.125}{e^{\frac{r}{s}}}}{s \cdot \pi}}{r}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.125 \cdot \frac{\frac{r}{\pi} \cdot 0.6111111111111112}{s} - \frac{0.16666666666666666}{\pi}}{s} + \frac{0.25}{\pi \cdot r}}{s}\\ \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (if (<= s 0.20000000298023224)
   (/ (/ (/ 0.125 (exp (/ r s))) (* s PI)) r)
   (/
    (+
     (/
      (-
       (* 0.125 (/ (* (/ r PI) 0.6111111111111112) s))
       (/ 0.16666666666666666 PI))
      s)
     (/ 0.25 (* PI r)))
    s)))

C

float code(float s, float r) {
	float tmp;
	if (s <= 0.20000000298023224f) {
		tmp = ((0.125f / expf((r / s))) / (s * ((float) M_PI))) / r;
	} else {
		tmp = ((((0.125f * (((r / ((float) M_PI)) * 0.6111111111111112f) / s)) - (0.16666666666666666f / ((float) M_PI))) / s) + (0.25f / (((float) M_PI) * r))) / s;
	}
	return tmp;
}

Julia

function code(s, r)
	tmp = Float32(0.0)
	if (s <= Float32(0.20000000298023224))
		tmp = Float32(Float32(Float32(Float32(0.125) / exp(Float32(r / s))) / Float32(s * Float32(pi))) / r);
	else
		tmp = Float32(Float32(Float32(Float32(Float32(Float32(0.125) * Float32(Float32(Float32(r / Float32(pi)) * Float32(0.6111111111111112)) / s)) - Float32(Float32(0.16666666666666666) / Float32(pi))) / s) + Float32(Float32(0.25) / Float32(Float32(pi) * r))) / s);
	end
	return tmp
end

MATLAB

function tmp_2 = code(s, r)
	tmp = single(0.0);
	if (s <= single(0.20000000298023224))
		tmp = ((single(0.125) / exp((r / s))) / (s * single(pi))) / r;
	else
		tmp = ((((single(0.125) * (((r / single(pi)) * single(0.6111111111111112)) / s)) - (single(0.16666666666666666) / single(pi))) / s) + (single(0.25) / (single(pi) * r))) / s;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if s < 0.200000003

   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 99.4%

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

      1. pow-to-exp 99.4%

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

      2. rem-log-exp 99.6%

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

      3. metadata-eval 99.6%

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

      4. times-frac 99.6%

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

      5. neg-mul-1 99.6%

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

      6. distribute-frac-neg 99.6%

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

      7. *-commutative 99.6%

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

      8. exp-neg 99.4%

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

      9. add-sqr-sqrt 99.3%

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

      10. sqrt-unprod 99.4%

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

      11. sqr-neg 99.4%

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

      12. sqrt-unprod -0.0%

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

      13. add-sqr-sqrt 5.2%

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

      14. associate-/r* 5.2%

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

      15. exp-cbrt 5.2%

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

      16. frac-2neg 5.2%

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

      17. remove-double-neg 5.2%

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

      18. add-sqr-sqrt -0.0%

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

      19. sqrt-unprod 99.1%

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

      20. sqr-neg 99.1%

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

      21. sqrt-unprod 99.0%

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

   6. Applied egg-rr 99.1%

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

   7. Taylor expanded in r around 0 11.9%

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

   8. Taylor expanded in s around 0 93.8%

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

      1. associate-*r* 93.8%

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

      2. associate-*r/ 93.8%

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

      3. associate-*r* 93.8%

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

      4. *-commutative 93.8%

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

      5. associate-/r* 93.8%

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

      6. mul-1-neg 93.8%

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

      7. rec-exp 93.8%

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

      8. associate-*r/ 93.8%

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

      9. metadata-eval 93.8%

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

   10. Simplified 93.8%

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

   if 0.200000003 < s

   1. Initial program 97.2%

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

   3. Simplified 96.2%

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

      1. pow-to-exp 96.5%

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

      2. rem-log-exp 97.3%

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

      3. metadata-eval 97.3%

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

      4. times-frac 97.5%

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

      5. neg-mul-1 97.5%

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

      6. distribute-frac-neg 97.5%

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

      7. *-commutative 97.5%

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

      8. exp-neg 97.2%

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

      9. add-sqr-sqrt 97.1%

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

      10. sqrt-unprod 97.2%

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

      11. sqr-neg 97.2%

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

      12. sqrt-unprod -0.0%

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

      13. add-sqr-sqrt 46.8%

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

      14. associate-/r* 46.8%

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

      15. exp-cbrt 46.7%

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

      16. frac-2neg 46.7%

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

      17. remove-double-neg 46.7%

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

      18. add-sqr-sqrt -0.0%

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

      19. sqrt-unprod 91.8%

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

      20. sqr-neg 91.8%

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

      21. sqrt-unprod 91.5%

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

   6. Applied egg-rr 91.8%

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

   7. Taylor expanded in r around 0 64.8%

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

   8. Taylor expanded in s around -inf 63.0%

      \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{0.125 \cdot \frac{0.5 \cdot \frac{r}{\pi} - -0.1111111111111111 \cdot \frac{r}{\pi}}{s} - 0.16666666666666666 \cdot \frac{1}{\pi}}{s} - 0.25 \cdot \frac{1}{r \cdot \pi}}{s}} \]
   9. Step-by-step derivation

      1. mul-1-neg 63.0%

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

   10. Simplified 63.0%

       \[\leadsto \color{blue}{-\frac{\left(-\frac{0.125 \cdot \frac{\frac{r}{\pi} \cdot 0.6111111111111112}{s} - \frac{0.16666666666666666}{\pi}}{s}\right) - \frac{0.25}{r \cdot \pi}}{s}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;s \leq 0.20000000298023224:\\ \;\;\;\;\frac{\frac{\frac{0.125}{e^{\frac{r}{s}}}}{s \cdot \pi}}{r}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.125 \cdot \frac{\frac{r}{\pi} \cdot 0.6111111111111112}{s} - \frac{0.16666666666666666}{\pi}}{s} + \frac{0.25}{\pi \cdot r}}{s}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 9.7% accurate, 10.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.125 \cdot \frac{\frac{r}{\pi} \cdot 0.6111111111111112}{s} - \frac{0.16666666666666666}{\pi}}{s} + \frac{0.25}{\pi \cdot r}}{s} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (/
  (+
   (/
    (-
     (* 0.125 (/ (* (/ r PI) 0.6111111111111112) s))
     (/ 0.16666666666666666 PI))
    s)
   (/ 0.25 (* PI r)))
  s))

C

float code(float s, float r) {
	return ((((0.125f * (((r / ((float) M_PI)) * 0.6111111111111112f) / s)) - (0.16666666666666666f / ((float) M_PI))) / s) + (0.25f / (((float) M_PI) * r))) / s;
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(Float32(Float32(0.125) * Float32(Float32(Float32(r / Float32(pi)) * Float32(0.6111111111111112)) / s)) - Float32(Float32(0.16666666666666666) / Float32(pi))) / s) + Float32(Float32(0.25) / Float32(Float32(pi) * r))) / s)
end

MATLAB

function tmp = code(s, r)
	tmp = ((((single(0.125) * (((r / single(pi)) * single(0.6111111111111112)) / s)) - (single(0.16666666666666666) / single(pi))) / s) + (single(0.25) / (single(pi) * r))) / s;
end

Derivation

1. Initial program 99.3%

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

3. Simplified 99.0%

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

   1. pow-to-exp 99.1%

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

   2. rem-log-exp 99.3%

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

   3. metadata-eval 99.3%

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

   4. times-frac 99.3%

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

   5. neg-mul-1 99.3%

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

   6. distribute-frac-neg 99.3%

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

   7. *-commutative 99.3%

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

   8. exp-neg 99.1%

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

   9. add-sqr-sqrt 99.0%

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

   10. sqrt-unprod 99.1%

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

   11. sqr-neg 99.1%

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

   12. sqrt-unprod -0.0%

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

   13. add-sqr-sqrt 9.9%

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

   14. associate-/r* 9.9%

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

   15. exp-cbrt 9.9%

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

   16. frac-2neg 9.9%

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

   17. remove-double-neg 9.9%

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

   18. add-sqr-sqrt -0.0%

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

   19. sqrt-unprod 98.2%

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

   20. sqr-neg 98.2%

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

   21. sqrt-unprod 98.1%

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

6. Applied egg-rr 98.2%

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

7. Taylor expanded in r around 0 17.9%

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

8. Taylor expanded in s around -inf 12.8%

   \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{0.125 \cdot \frac{0.5 \cdot \frac{r}{\pi} - -0.1111111111111111 \cdot \frac{r}{\pi}}{s} - 0.16666666666666666 \cdot \frac{1}{\pi}}{s} - 0.25 \cdot \frac{1}{r \cdot \pi}}{s}} \]
9. Step-by-step derivation

   1. mul-1-neg 12.8%

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

10. Simplified 12.8%

    \[\leadsto \color{blue}{-\frac{\left(-\frac{0.125 \cdot \frac{\frac{r}{\pi} \cdot 0.6111111111111112}{s} - \frac{0.16666666666666666}{\pi}}{s}\right) - \frac{0.25}{r \cdot \pi}}{s}} \]

11. Final simplification 12.8%

    \[\leadsto \frac{\frac{0.125 \cdot \frac{\frac{r}{\pi} \cdot 0.6111111111111112}{s} - \frac{0.16666666666666666}{\pi}}{s} + \frac{0.25}{\pi \cdot r}}{s} \]
12. Add Preprocessing

Alternative 10: 9.0% accurate, 17.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.3%

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

3. Simplified 99.0%

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

   1. pow-to-exp 99.1%

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

   2. rem-log-exp 99.3%

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

   3. metadata-eval 99.3%

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

   4. times-frac 99.3%

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

   5. neg-mul-1 99.3%

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

   6. distribute-frac-neg 99.3%

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

   7. *-commutative 99.3%

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

   8. exp-neg 99.1%

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

   9. add-sqr-sqrt 99.0%

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

   10. sqrt-unprod 99.1%

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

   11. sqr-neg 99.1%

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

   12. sqrt-unprod -0.0%

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

   13. add-sqr-sqrt 9.9%

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

   14. associate-/r* 9.9%

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

   15. exp-cbrt 9.9%

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

   16. frac-2neg 9.9%

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

   17. remove-double-neg 9.9%

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

   18. add-sqr-sqrt -0.0%

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

   19. sqrt-unprod 98.2%

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

   20. sqr-neg 98.2%

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

   21. sqrt-unprod 98.1%

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

6. Applied egg-rr 98.2%

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

7. Taylor expanded in r around 0 17.9%

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

8. Taylor expanded in s around inf 11.7%

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

   1. associate-*r/ 11.7%

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

   2. metadata-eval 11.7%

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

   3. associate-*r/ 11.7%

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

   4. metadata-eval 11.7%

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

10. Simplified 11.7%

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

11. Final simplification 11.7%

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

Alternative 11: 9.0% accurate, 25.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.3%

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

3. Simplified 99.0%

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

5. Taylor expanded in r around 0 11.7%

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

6. Taylor expanded in s around 0 11.7%

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

   1. associate-*r/ 11.7%

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

   2. times-frac 11.7%

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

   3. mul-1-neg 11.7%

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

   4. distribute-neg-frac2 11.7%

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

8. Simplified 11.7%

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

9. Taylor expanded in r around 0 11.0%

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

    1. *-commutative 11.0%

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

    2. associate-/r* 11.0%

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

11. Simplified 11.0%

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

12. Final simplification 11.0%

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

Alternative 12: 9.0% accurate, 25.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.3%

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

3. Simplified 99.0%

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

5. Taylor expanded in r around 0 11.7%

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

6. Taylor expanded in s around 0 11.7%

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

   1. associate-*r/ 11.7%

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

   2. times-frac 11.7%

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

   3. mul-1-neg 11.7%

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

   4. distribute-neg-frac2 11.7%

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

8. Simplified 11.7%

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

9. Taylor expanded in s around inf 10.9%

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

    1. associate-/r* 10.9%

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

    2. *-commutative 10.9%

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

11. Simplified 10.9%

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

    1. *-un-lft-identity 10.9%

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

    2. times-frac 11.0%

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

13. Applied egg-rr 11.0%

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

14. Final simplification 11.0%

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

Alternative 13: 8.9% accurate, 33.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.3%

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

3. Simplified 99.0%

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

5. Taylor expanded in r around 0 11.7%

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

6. Taylor expanded in s around inf 10.9%

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

7. Final simplification 10.9%

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

Alternative 14: 8.9% 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 99.3%

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

3. Simplified 99.0%

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

5. Taylor expanded in r around 0 11.7%

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

6. Taylor expanded in s around 0 11.7%

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

   1. associate-*r/ 11.7%

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

   2. times-frac 11.7%

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

   3. mul-1-neg 11.7%

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

   4. distribute-neg-frac2 11.7%

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

8. Simplified 11.7%

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

9. Taylor expanded in s around inf 10.9%

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

    1. associate-/r* 10.9%

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

    2. *-commutative 10.9%

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

11. Simplified 10.9%

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

12. Final simplification 10.9%

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

Alternative 15: 8.9% accurate, 33.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.3%

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

3. Simplified 99.0%

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

5. Taylor expanded in r around 0 11.7%

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

6. Taylor expanded in s around inf 10.9%

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

   1. *-commutative 10.9%

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

   2. associate-*l* 11.0%

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

   3. *-commutative 11.0%

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

   4. associate-/l/ 11.0%

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

8. Simplified 11.0%

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

9. Final simplification 11.0%

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

Reproduce

herbie shell --seed 2024130 
(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))))