Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.6%

Time: 9.9s

Alternatives: 12

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 99.6% 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: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

   1. times-frac 99.9%

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

   2. *-commutative 99.9%

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

   3. times-frac 99.9%

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

   4. *-commutative 99.9%

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

   5. *-commutative 99.9%

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

   6. distribute-frac-neg 99.9%

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

   7. *-commutative 99.9%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

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

Alternative 2: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

   1. associate-*l/ 98.2%

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

   2. associate-*l/ 98.2%

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

   3. associate-*l* 98.2%

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

   4. associate-*l* 98.2%

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

   5. associate-/r* 98.2%

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

   6. metadata-eval 98.2%

      \[\leadsto \frac{0.25}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} \cdot e^{\frac{-r}{s}} + \frac{\color{blue}{0.125}}{\pi \cdot \left(s \cdot r\right)} \cdot e^{\frac{-r}{3 \cdot s}} \]

   7. metadata-eval 98.2%

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

   8. associate-/r* 98.2%

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

   9. associate-*l* 98.2%

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

   10. associate-*l* 98.2%

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

   11. distribute-lft-out 98.2%

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

3. Simplified 98.1%

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

4. Taylor expanded in r around inf 99.8%

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

   1. mul-1-neg 99.8%

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

   2. distribute-frac-neg 99.8%

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

   3. associate-*r/ 99.8%

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

   4. associate-/l* 99.8%

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

   5. associate-*r* 99.8%

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

6. Simplified 99.8%

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

   1. *-un-lft-identity 99.8%

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

   2. exp-prod 99.8%

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

   3. exp-1-e 99.8%

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

   4. div-inv 99.8%

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

   5. add-sqr-sqrt 99.8%

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

   6. sqrt-unprod 99.8%

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

   7. sqr-neg 99.8%

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

   8. sqrt-unprod -0.0%

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

   9. add-sqr-sqrt 7.1%

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

   10. clear-num 7.1%

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

   11. add-sqr-sqrt -0.0%

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

   12. sqrt-unprod 99.8%

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

   13. sqr-neg 99.8%

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

   14. sqrt-unprod 99.8%

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

   15. add-sqr-sqrt 99.8%

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

8. Applied egg-rr 99.8%

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

9. Taylor expanded in s around 0 99.8%

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

10. Final simplification 99.8%

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

Alternative 3: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

   1. associate-*l/ 98.2%

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

   2. associate-*l/ 98.2%

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

   3. associate-*l* 98.2%

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

   4. associate-*l* 98.2%

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

   5. associate-/r* 98.2%

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

   6. metadata-eval 98.2%

      \[\leadsto \frac{0.25}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} \cdot e^{\frac{-r}{s}} + \frac{\color{blue}{0.125}}{\pi \cdot \left(s \cdot r\right)} \cdot e^{\frac{-r}{3 \cdot s}} \]

   7. metadata-eval 98.2%

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

   8. associate-/r* 98.2%

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

   9. associate-*l* 98.2%

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

   10. associate-*l* 98.2%

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

   11. distribute-lft-out 98.2%

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

3. Simplified 98.1%

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

4. Taylor expanded in r around inf 99.8%

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

   1. mul-1-neg 99.8%

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

   2. distribute-frac-neg 99.8%

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

   3. associate-*r/ 99.8%

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

   4. associate-/l* 99.8%

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

   5. associate-*r* 99.8%

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

6. Simplified 99.8%

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

   1. *-un-lft-identity 99.8%

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

   2. exp-prod 99.8%

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

   3. exp-1-e 99.8%

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

   4. div-inv 99.8%

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

   5. add-sqr-sqrt 99.8%

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

   6. sqrt-unprod 99.8%

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

   7. sqr-neg 99.8%

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

   8. sqrt-unprod -0.0%

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

   9. add-sqr-sqrt 7.1%

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

   10. clear-num 7.1%

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

   11. add-sqr-sqrt -0.0%

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

   12. sqrt-unprod 99.8%

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

   13. sqr-neg 99.8%

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

   14. sqrt-unprod 99.8%

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

   15. add-sqr-sqrt 99.8%

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

8. Applied egg-rr 99.8%

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

9. Final simplification 99.8%

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

Alternative 4: 99.6% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

   1. associate-*l/ 98.2%

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

   2. associate-*l/ 98.2%

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

   3. associate-*l* 98.2%

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

   4. associate-*l* 98.2%

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

   5. associate-/r* 98.2%

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

   6. metadata-eval 98.2%

      \[\leadsto \frac{0.25}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} \cdot e^{\frac{-r}{s}} + \frac{\color{blue}{0.125}}{\pi \cdot \left(s \cdot r\right)} \cdot e^{\frac{-r}{3 \cdot s}} \]

   7. metadata-eval 98.2%

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

   8. associate-/r* 98.2%

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

   9. associate-*l* 98.2%

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

   10. associate-*l* 98.2%

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

   11. distribute-lft-out 98.2%

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

3. Simplified 98.1%

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

4. Taylor expanded in r around inf 99.8%

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

   1. mul-1-neg 99.8%

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

   2. distribute-frac-neg 99.8%

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

   3. associate-*r/ 99.8%

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

   4. associate-/l* 99.8%

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

   5. associate-*r* 99.8%

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

6. Simplified 99.8%

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

7. Taylor expanded in s around 0 99.8%

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

8. Final simplification 99.8%

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

Alternative 5: 99.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

   1. associate-*l/ 98.2%

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

   2. associate-*l/ 98.2%

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

   3. associate-*l* 98.2%

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

   4. associate-*l* 98.2%

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

   5. associate-/r* 98.2%

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

   6. metadata-eval 98.2%

      \[\leadsto \frac{0.25}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} \cdot e^{\frac{-r}{s}} + \frac{\color{blue}{0.125}}{\pi \cdot \left(s \cdot r\right)} \cdot e^{\frac{-r}{3 \cdot s}} \]

   7. metadata-eval 98.2%

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

   8. associate-/r* 98.2%

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

   9. associate-*l* 98.2%

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

   10. associate-*l* 98.2%

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

   11. distribute-lft-out 98.2%

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

3. Simplified 98.1%

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

4. Taylor expanded in r around inf 99.8%

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

   1. mul-1-neg 99.8%

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

   2. distribute-frac-neg 99.8%

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

   3. associate-*r/ 99.8%

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

   4. associate-/l* 99.8%

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

   5. associate-*r* 99.8%

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

6. Simplified 99.8%

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

7. Final simplification 99.8%

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

Alternative 6: 43.6% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.8%

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

   1. associate-*l/ 98.2%

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

   2. associate-*l/ 98.2%

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

   3. associate-*l* 98.2%

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

   4. associate-*l* 98.2%

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

   5. associate-/r* 98.2%

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

   6. metadata-eval 98.2%

      \[\leadsto \frac{0.25}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} \cdot e^{\frac{-r}{s}} + \frac{\color{blue}{0.125}}{\pi \cdot \left(s \cdot r\right)} \cdot e^{\frac{-r}{3 \cdot s}} \]

   7. metadata-eval 98.2%

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

   8. associate-/r* 98.2%

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

   9. associate-*l* 98.2%

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

   10. associate-*l* 98.2%

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

   11. distribute-lft-out 98.2%

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

3. Simplified 98.1%

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

4. Taylor expanded in r around 0 8.2%

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

   1. log1p-expm1-u 45.8%

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

   2. *-commutative 45.8%

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

6. Applied egg-rr 45.8%

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

7. Final simplification 45.8%

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

Alternative 7: 9.7% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

   1. times-frac 99.9%

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

   2. times-frac 99.9%

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

   3. associate-*l* 99.8%

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

   4. associate-/r* 99.8%

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

   5. metadata-eval 99.8%

      \[\leadsto \frac{0.25}{\left(2 \cdot \pi\right) \cdot s} \cdot \frac{e^{\frac{-r}{s}}}{r} + \frac{\color{blue}{0.125}}{\pi \cdot s} \cdot \frac{e^{\frac{-r}{3 \cdot s}}}{r} \]

   6. metadata-eval 99.8%

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

   7. associate-/r* 99.8%

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

   8. associate-*l* 99.8%

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

   9. distribute-lft-out 99.8%

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

3. Simplified 99.7%

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

4. Taylor expanded in r around 0 9.2%

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

   1. associate-*r/ 9.2%

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

   2. associate-/l* 9.2%

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

6. Simplified 9.2%

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

7. Final simplification 9.2%

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

Alternative 8: 9.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

   1. times-frac 99.9%

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

   2. times-frac 99.9%

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

   3. associate-*l* 99.8%

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

   4. associate-/r* 99.8%

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

   5. metadata-eval 99.8%

      \[\leadsto \frac{0.25}{\left(2 \cdot \pi\right) \cdot s} \cdot \frac{e^{\frac{-r}{s}}}{r} + \frac{\color{blue}{0.125}}{\pi \cdot s} \cdot \frac{e^{\frac{-r}{3 \cdot s}}}{r} \]

   6. metadata-eval 99.8%

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

   7. associate-/r* 99.8%

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

   8. associate-*l* 99.8%

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

   9. distribute-lft-out 99.8%

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

3. Simplified 99.7%

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

4. Taylor expanded in r around 0 9.2%

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

   1. associate-*r/ 9.2%

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

   2. associate-/l* 9.2%

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

6. Simplified 9.2%

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

7. Taylor expanded in s around 0 9.2%

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

   1. associate-*r/ 9.2%

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

   2. metadata-eval 9.2%

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

9. Simplified 9.2%

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

10. Final simplification 9.2%

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

Alternative 9: 9.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

   1. associate-*l/ 98.2%

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

   2. associate-*l/ 98.2%

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

   3. associate-*l* 98.2%

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

   4. associate-*l* 98.2%

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

   5. associate-/r* 98.2%

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

   6. metadata-eval 98.2%

      \[\leadsto \frac{0.25}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} \cdot e^{\frac{-r}{s}} + \frac{\color{blue}{0.125}}{\pi \cdot \left(s \cdot r\right)} \cdot e^{\frac{-r}{3 \cdot s}} \]

   7. metadata-eval 98.2%

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

   8. associate-/r* 98.2%

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

   9. associate-*l* 98.2%

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

   10. associate-*l* 98.2%

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

   11. distribute-lft-out 98.2%

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

3. Simplified 98.1%

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

4. Taylor expanded in r around inf 99.8%

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

   1. mul-1-neg 99.8%

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

   2. distribute-frac-neg 99.8%

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

   3. associate-*r/ 99.8%

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

   4. associate-/l* 99.8%

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

   5. associate-*r* 99.8%

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

6. Simplified 99.8%

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

7. Taylor expanded in r around 0 8.7%

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

   1. mul-1-neg 8.6%

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

   2. unsub-neg 8.6%

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

9. Simplified 8.7%

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

10. Final simplification 8.7%

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

Alternative 10: 9.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

   1. associate-*l/ 98.2%

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

   2. associate-*l/ 98.2%

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

   3. associate-*l* 98.2%

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

   4. associate-*l* 98.2%

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

   5. associate-/r* 98.2%

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

   6. metadata-eval 98.2%

      \[\leadsto \frac{0.25}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} \cdot e^{\frac{-r}{s}} + \frac{\color{blue}{0.125}}{\pi \cdot \left(s \cdot r\right)} \cdot e^{\frac{-r}{3 \cdot s}} \]

   7. metadata-eval 98.2%

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

   8. associate-/r* 98.2%

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

   9. associate-*l* 98.2%

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

   10. associate-*l* 98.2%

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

   11. distribute-lft-out 98.2%

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

3. Simplified 98.1%

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

4. Taylor expanded in r around 0 8.5%

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

   1. associate-*r/ 8.6%

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

   2. metadata-eval 8.6%

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

   3. associate-*r* 8.6%

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

   4. associate-*r/ 8.6%

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

   5. metadata-eval 8.6%

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

   6. *-commutative 8.6%

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

   7. unpow2 8.6%

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

6. Simplified 8.6%

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

7. Taylor expanded in s around 0 8.5%

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

   1. associate-*r/ 8.6%

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

   2. metadata-eval 8.6%

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

   3. associate-*r* 8.6%

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

   4. associate-*r/ 8.6%

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

   5. metadata-eval 8.6%

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

   6. unpow2 8.6%

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

   7. *-commutative 8.6%

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

   8. associate-/r* 8.6%

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

9. Simplified 8.6%

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

    1. associate-*r* 8.6%

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

    2. *-commutative 8.6%

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

    3. associate-/r* 8.6%

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

    4. associate-/r* 8.6%

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

    5. associate-/r* 8.6%

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

    6. frac-sub 8.7%

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

11. Applied egg-rr 8.7%

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

12. Final simplification 8.7%

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

Alternative 11: 9.1% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

   1. times-frac 99.9%

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

   2. times-frac 99.9%

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

   3. associate-*l* 99.8%

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

   4. associate-/r* 99.8%

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

   5. metadata-eval 99.8%

      \[\leadsto \frac{0.25}{\left(2 \cdot \pi\right) \cdot s} \cdot \frac{e^{\frac{-r}{s}}}{r} + \frac{\color{blue}{0.125}}{\pi \cdot s} \cdot \frac{e^{\frac{-r}{3 \cdot s}}}{r} \]

   6. metadata-eval 99.8%

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

   7. associate-/r* 99.8%

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

   8. associate-*l* 99.8%

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

   9. distribute-lft-out 99.8%

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

3. Simplified 99.7%

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

4. Taylor expanded in r around 0 9.2%

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

   1. associate-*r/ 9.2%

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

   2. associate-/l* 9.2%

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

6. Simplified 9.2%

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

7. Taylor expanded in s around 0 9.2%

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

   1. associate-*r/ 9.2%

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

   2. metadata-eval 9.2%

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

9. Simplified 9.2%

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

10. Taylor expanded in r around 0 8.6%

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

    1. mul-1-neg 8.6%

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

    2. unsub-neg 8.6%

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

12. Simplified 8.6%

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

13. Final simplification 8.6%

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

Alternative 12: 9.0% accurate, 4.0× speedup

Math

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

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

   1. associate-*l/ 98.2%

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

   2. associate-*l/ 98.2%

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

   3. associate-*l* 98.2%

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

   4. associate-*l* 98.2%

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

   5. associate-/r* 98.2%

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

   6. metadata-eval 98.2%

      \[\leadsto \frac{0.25}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} \cdot e^{\frac{-r}{s}} + \frac{\color{blue}{0.125}}{\pi \cdot \left(s \cdot r\right)} \cdot e^{\frac{-r}{3 \cdot s}} \]

   7. metadata-eval 98.2%

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

   8. associate-/r* 98.2%

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

   9. associate-*l* 98.2%

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

   10. associate-*l* 98.2%

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

   11. distribute-lft-out 98.2%

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

3. Simplified 98.1%

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

4. Taylor expanded in r around 0 8.2%

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

5. Final simplification 8.2%

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

Reproduce

herbie shell --seed 2023257 
(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))))