Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.5%

Time: 12.3s

Alternatives: 16

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

\[\begin{array}{l} \\ \frac{\frac{0.125}{s}}{\pi} \cdot \frac{e^{\frac{-r}{s}}}{r} + \left(0.75 \cdot \frac{0.16666666666666666}{s \cdot \pi}\right) \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (* (/ (/ 0.125 s) PI) (/ (exp (/ (- r) s)) r))
  (*
   (* 0.75 (/ 0.16666666666666666 (* s PI)))
   (/ (exp (/ (- r) (* s 3.0))) r))))

C

float code(float s, float r) {
	return (((0.125f / s) / ((float) M_PI)) * (expf((-r / s)) / r)) + ((0.75f * (0.16666666666666666f / (s * ((float) M_PI)))) * (expf((-r / (s * 3.0f))) / r));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(Float32(0.125) / s) / Float32(pi)) * Float32(exp(Float32(Float32(-r) / s)) / r)) + Float32(Float32(Float32(0.75) * Float32(Float32(0.16666666666666666) / Float32(s * Float32(pi)))) * Float32(exp(Float32(Float32(-r) / Float32(s * Float32(3.0)))) / r)))
end

MATLAB

function tmp = code(s, r)
	tmp = (((single(0.125) / s) / single(pi)) * (exp((-r / s)) / r)) + ((single(0.75) * (single(0.16666666666666666) / (s * single(pi)))) * (exp((-r / (s * single(3.0)))) / r));
end

Derivation

1. Initial program 99.6%

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

   1. times-frac 99.6%

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

   2. *-commutative 99.6%

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

   3. times-frac 99.6%

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

   4. *-commutative 99.6%

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

   5. *-commutative 99.6%

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

   6. distribute-frac-neg 99.6%

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

   7. *-commutative 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in s around 0 99.6%

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

   1. associate-/r* 99.6%

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

6. Simplified 99.6%

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

   1. div-inv 99.6%

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

8. Applied egg-rr 99.6%

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

9. Taylor expanded in s around 0 99.6%

   \[\leadsto \frac{\frac{0.125}{s}}{\pi} \cdot \frac{e^{\frac{-r}{s}}}{r} + \left(0.75 \cdot \color{blue}{\frac{0.16666666666666666}{s \cdot \pi}}\right) \cdot \frac{e^{-\frac{r}{s \cdot 3}}}{r} \]

10. Final simplification 99.6%

    \[\leadsto \frac{\frac{0.125}{s}}{\pi} \cdot \frac{e^{\frac{-r}{s}}}{r} + \left(0.75 \cdot \frac{0.16666666666666666}{s \cdot \pi}\right) \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r} \]

Alternative 2: 99.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Derivation

1. Initial program 99.6%

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

   1. associate-*l/ 97.6%

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

   2. associate-*l/ 97.6%

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

   3. associate-*l* 97.6%

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

   4. associate-*l* 97.6%

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

   5. associate-/r* 97.6%

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

   6. metadata-eval 97.6%

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

   7. metadata-eval 97.6%

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

   8. associate-/r* 97.6%

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

   9. associate-*l* 97.6%

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

   10. associate-*l* 97.6%

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

   11. distribute-lft-out 97.6%

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

3. Simplified 97.3%

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

4. Taylor expanded in r around inf 99.5%

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

   1. mul-1-neg 99.5%

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

   2. exp-neg 99.5%

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

6. Applied egg-rr 99.5%

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

   1. rec-exp 99.5%

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

   2. distribute-neg-frac 99.5%

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

8. Simplified 99.5%

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

9. Final simplification 99.5%

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

Alternative 3: 43.9% 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 99.6%

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

   1. associate-*l/ 97.6%

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

   2. associate-*l/ 97.6%

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

   3. associate-*l* 97.6%

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

   4. associate-*l* 97.6%

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

   5. associate-/r* 97.6%

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

   6. metadata-eval 97.6%

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

   7. metadata-eval 97.6%

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

   8. associate-/r* 97.6%

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

   9. associate-*l* 97.6%

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

   10. associate-*l* 97.6%

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

   11. distribute-lft-out 97.6%

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

3. Simplified 97.3%

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

4. Taylor expanded in r around 0 10.5%

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

   1. *-commutative 10.5%

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

6. Simplified 10.5%

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

   1. log1p-expm1-u 43.5%

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

8. Applied egg-rr 43.5%

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

9. Final simplification 43.5%

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

Alternative 4: 10.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.125}{\pi \cdot r}}{s} \cdot \left(\left(0.5 \cdot {\left(\frac{r}{s}\right)}^{2} + \left(1 - \frac{r}{s}\right)\right) + \left(-0.3333333333333333 \cdot \frac{r}{s} + 1\right)\right) \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (*
  (/ (/ 0.125 (* PI r)) s)
  (+
   (+ (* 0.5 (pow (/ r s) 2.0)) (- 1.0 (/ r s)))
   (+ (* -0.3333333333333333 (/ r s)) 1.0))))

C

float code(float s, float r) {
	return ((0.125f / (((float) M_PI) * r)) / s) * (((0.5f * powf((r / s), 2.0f)) + (1.0f - (r / s))) + ((-0.3333333333333333f * (r / s)) + 1.0f));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.125) / Float32(Float32(pi) * r)) / s) * Float32(Float32(Float32(Float32(0.5) * (Float32(r / s) ^ Float32(2.0))) + Float32(Float32(1.0) - Float32(r / s))) + Float32(Float32(Float32(-0.3333333333333333) * Float32(r / s)) + Float32(1.0))))
end

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.125) / (single(pi) * r)) / s) * (((single(0.5) * ((r / s) ^ single(2.0))) + (single(1.0) - (r / s))) + ((single(-0.3333333333333333) * (r / s)) + single(1.0)));
end

Derivation

1. Initial program 99.6%

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

   1. associate-*l/ 97.6%

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

   2. associate-*l/ 97.6%

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

   3. associate-*l* 97.6%

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

   4. associate-*l* 97.6%

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

   5. associate-/r* 97.6%

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

   6. metadata-eval 97.6%

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

   7. metadata-eval 97.6%

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

   8. associate-/r* 97.6%

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

   9. associate-*l* 97.6%

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

   10. associate-*l* 97.6%

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

   11. distribute-lft-out 97.6%

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

3. Simplified 97.3%

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

4. Taylor expanded in r around 0 11.7%

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

5. Taylor expanded in r around 0 11.9%

   \[\leadsto \frac{\frac{0.125}{r \cdot \pi}}{s} \cdot \left(\color{blue}{\left(0.5 \cdot \frac{{r}^{2}}{{s}^{2}} + \left(1 + -1 \cdot \frac{r}{s}\right)\right)} + \left(1 + -0.3333333333333333 \cdot \frac{r}{s}\right)\right) \]
6. Step-by-step derivation

   1. fma-def 11.9%

      \[\leadsto \frac{\frac{0.125}{r \cdot \pi}}{s} \cdot \left(\color{blue}{\mathsf{fma}\left(0.5, \frac{{r}^{2}}{{s}^{2}}, 1 + -1 \cdot \frac{r}{s}\right)} + \left(1 + -0.3333333333333333 \cdot \frac{r}{s}\right)\right) \]

   2. unpow2 11.9%

      \[\leadsto \frac{\frac{0.125}{r \cdot \pi}}{s} \cdot \left(\mathsf{fma}\left(0.5, \frac{\color{blue}{r \cdot r}}{{s}^{2}}, 1 + -1 \cdot \frac{r}{s}\right) + \left(1 + -0.3333333333333333 \cdot \frac{r}{s}\right)\right) \]

   3. unpow2 11.9%

      \[\leadsto \frac{\frac{0.125}{r \cdot \pi}}{s} \cdot \left(\mathsf{fma}\left(0.5, \frac{r \cdot r}{\color{blue}{s \cdot s}}, 1 + -1 \cdot \frac{r}{s}\right) + \left(1 + -0.3333333333333333 \cdot \frac{r}{s}\right)\right) \]

   4. mul-1-neg 11.9%

      \[\leadsto \frac{\frac{0.125}{r \cdot \pi}}{s} \cdot \left(\mathsf{fma}\left(0.5, \frac{r \cdot r}{s \cdot s}, 1 + \color{blue}{\left(-\frac{r}{s}\right)}\right) + \left(1 + -0.3333333333333333 \cdot \frac{r}{s}\right)\right) \]

   5. distribute-frac-neg 11.9%

      \[\leadsto \frac{\frac{0.125}{r \cdot \pi}}{s} \cdot \left(\mathsf{fma}\left(0.5, \frac{r \cdot r}{s \cdot s}, 1 + \color{blue}{\frac{-r}{s}}\right) + \left(1 + -0.3333333333333333 \cdot \frac{r}{s}\right)\right) \]

7. Simplified 11.9%

   \[\leadsto \frac{\frac{0.125}{r \cdot \pi}}{s} \cdot \left(\color{blue}{\mathsf{fma}\left(0.5, \frac{r \cdot r}{s \cdot s}, 1 + \frac{-r}{s}\right)} + \left(1 + -0.3333333333333333 \cdot \frac{r}{s}\right)\right) \]
8. Step-by-step derivation

   1. fma-udef 11.9%

      \[\leadsto \frac{\frac{0.125}{r \cdot \pi}}{s} \cdot \left(\color{blue}{\left(0.5 \cdot \frac{r \cdot r}{s \cdot s} + \left(1 + \frac{-r}{s}\right)\right)} + \left(1 + -0.3333333333333333 \cdot \frac{r}{s}\right)\right) \]

   2. times-frac 11.9%

      \[\leadsto \frac{\frac{0.125}{r \cdot \pi}}{s} \cdot \left(\left(0.5 \cdot \color{blue}{\left(\frac{r}{s} \cdot \frac{r}{s}\right)} + \left(1 + \frac{-r}{s}\right)\right) + \left(1 + -0.3333333333333333 \cdot \frac{r}{s}\right)\right) \]

   3. pow2 11.9%

      \[\leadsto \frac{\frac{0.125}{r \cdot \pi}}{s} \cdot \left(\left(0.5 \cdot \color{blue}{{\left(\frac{r}{s}\right)}^{2}} + \left(1 + \frac{-r}{s}\right)\right) + \left(1 + -0.3333333333333333 \cdot \frac{r}{s}\right)\right) \]

   4. +-commutative 11.9%

      \[\leadsto \frac{\frac{0.125}{r \cdot \pi}}{s} \cdot \left(\left(0.5 \cdot {\left(\frac{r}{s}\right)}^{2} + \color{blue}{\left(\frac{-r}{s} + 1\right)}\right) + \left(1 + -0.3333333333333333 \cdot \frac{r}{s}\right)\right) \]

9. Applied egg-rr 11.9%

   \[\leadsto \frac{\frac{0.125}{r \cdot \pi}}{s} \cdot \left(\color{blue}{\left(0.5 \cdot {\left(\frac{r}{s}\right)}^{2} + \left(\frac{-r}{s} + 1\right)\right)} + \left(1 + -0.3333333333333333 \cdot \frac{r}{s}\right)\right) \]

10. Final simplification 11.9%

    \[\leadsto \frac{\frac{0.125}{\pi \cdot r}}{s} \cdot \left(\left(0.5 \cdot {\left(\frac{r}{s}\right)}^{2} + \left(1 - \frac{r}{s}\right)\right) + \left(-0.3333333333333333 \cdot \frac{r}{s} + 1\right)\right) \]

Alternative 5: 9.9% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Derivation

1. Initial program 99.6%

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

   1. times-frac 99.6%

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

   2. times-frac 99.6%

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

   3. associate-*l* 99.6%

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

   4. associate-/r* 99.6%

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

   5. metadata-eval 99.6%

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

   6. metadata-eval 99.6%

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

   7. associate-/r* 99.6%

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

   8. associate-*l* 99.6%

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

   9. distribute-lft-out 99.6%

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

3. Simplified 99.3%

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

4. Taylor expanded in r around 0 11.7%

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

5. Final simplification 11.7%

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

Alternative 6: 9.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

   1. associate-*l/ 97.6%

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

   2. associate-*l/ 97.6%

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

   3. associate-*l* 97.6%

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

   4. associate-*l* 97.6%

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

   5. associate-/r* 97.6%

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

   6. metadata-eval 97.6%

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

   7. metadata-eval 97.6%

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

   8. associate-/r* 97.6%

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

   9. associate-*l* 97.6%

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

   10. associate-*l* 97.6%

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

   11. distribute-lft-out 97.6%

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

3. Simplified 97.3%

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

4. Taylor expanded in r around 0 11.7%

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

5. Taylor expanded in r around 0 11.7%

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

   1. *-commutative 11.7%

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

7. Simplified 11.7%

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

8. Final simplification 11.7%

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

Alternative 7: 9.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

   1. associate-*l/ 97.6%

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

   2. associate-*l/ 97.6%

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

   3. associate-*l* 97.6%

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

   4. associate-*l* 97.6%

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

   5. associate-/r* 97.6%

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

   6. metadata-eval 97.6%

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

   7. metadata-eval 97.6%

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

   8. associate-/r* 97.6%

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

   9. associate-*l* 97.6%

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

   10. associate-*l* 97.6%

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

   11. distribute-lft-out 97.6%

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

3. Simplified 97.3%

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

4. Taylor expanded in r around 0 11.7%

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

5. Final simplification 11.7%

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

Alternative 8: 9.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

   1. associate-*l/ 97.6%

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

   2. associate-*l/ 97.6%

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

   3. associate-*l* 97.6%

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

   4. associate-*l* 97.6%

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

   5. associate-/r* 97.6%

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

   6. metadata-eval 97.6%

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

   7. metadata-eval 97.6%

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

   8. associate-/r* 97.6%

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

   9. associate-*l* 97.6%

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

   10. associate-*l* 97.6%

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

   11. distribute-lft-out 97.6%

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

3. Simplified 97.3%

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

4. Taylor expanded in r around 0 11.7%

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

   1. expm1-log1p-u 11.6%

      \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.125}{r \cdot \pi}\right)\right)}}{s} \cdot \left(e^{\frac{-r}{s}} + \left(1 + -0.3333333333333333 \cdot \frac{r}{s}\right)\right) \]

   2. expm1-udef 12.7%

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

   3. *-commutative 12.7%

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

6. Applied egg-rr 12.7%

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

   1. expm1-def 11.6%

      \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{0.125}{\pi \cdot r}\right)\right)}}{s} \cdot \left(e^{\frac{-r}{s}} + \left(1 + -0.3333333333333333 \cdot \frac{r}{s}\right)\right) \]

   2. expm1-log1p 11.7%

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

   3. associate-/r* 11.7%

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

8. Simplified 11.7%

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

9. Final simplification 11.7%

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

Alternative 9: 9.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

   1. times-frac 99.6%

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

   2. times-frac 99.6%

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

   3. associate-*l* 99.6%

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

   4. associate-/r* 99.6%

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

   5. metadata-eval 99.6%

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

   6. metadata-eval 99.6%

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

   7. associate-/r* 99.6%

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

   8. associate-*l* 99.6%

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

   9. distribute-lft-out 99.6%

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

3. Simplified 99.3%

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

4. Taylor expanded in r around 0 11.1%

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

5. Taylor expanded in r around inf 11.1%

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

   1. mul-1-neg 11.1%

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

   2. distribute-frac-neg 11.1%

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

7. Simplified 11.1%

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

8. Final simplification 11.1%

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

Alternative 10: 9.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

   1. associate-*l/ 97.6%

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

   2. associate-*l/ 97.6%

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

   3. associate-*l* 97.6%

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

   4. associate-*l* 97.6%

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

   5. associate-/r* 97.6%

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

   6. metadata-eval 97.6%

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

   7. metadata-eval 97.6%

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

   8. associate-/r* 97.6%

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

   9. associate-*l* 97.6%

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

   10. associate-*l* 97.6%

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

   11. distribute-lft-out 97.6%

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

3. Simplified 97.3%

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

4. Taylor expanded in r around 0 11.1%

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

5. Final simplification 11.1%

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

Alternative 11: 9.3% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

   1. associate-*l/ 97.6%

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

   2. associate-*l/ 97.6%

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

   3. associate-*l* 97.6%

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

   4. associate-*l* 97.6%

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

   5. associate-/r* 97.6%

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

   6. metadata-eval 97.6%

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

   7. metadata-eval 97.6%

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

   8. associate-/r* 97.6%

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

   9. associate-*l* 97.6%

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

   10. associate-*l* 97.6%

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

   11. distribute-lft-out 97.6%

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

3. Simplified 97.3%

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

4. Taylor expanded in r around 0 11.7%

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

5. Taylor expanded in r around 0 11.0%

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

   1. mul-1-neg 11.0%

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

   2. unsub-neg 11.0%

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

7. Simplified 11.0%

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

8. Final simplification 11.0%

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

Alternative 12: 9.3% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

   1. associate-*l/ 97.6%

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

   2. associate-*l/ 97.6%

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

   3. associate-*l* 97.6%

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

   4. associate-*l* 97.6%

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

   5. associate-/r* 97.6%

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

   6. metadata-eval 97.6%

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

   7. metadata-eval 97.6%

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

   8. associate-/r* 97.6%

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

   9. associate-*l* 97.6%

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

   10. associate-*l* 97.6%

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

   11. distribute-lft-out 97.6%

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

3. Simplified 97.3%

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

4. Taylor expanded in r around 0 11.7%

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

5. Taylor expanded in r around 0 11.9%

   \[\leadsto \frac{\frac{0.125}{r \cdot \pi}}{s} \cdot \left(\color{blue}{\left(0.5 \cdot \frac{{r}^{2}}{{s}^{2}} + \left(1 + -1 \cdot \frac{r}{s}\right)\right)} + \left(1 + -0.3333333333333333 \cdot \frac{r}{s}\right)\right) \]
6. Step-by-step derivation

   1. fma-def 11.9%

      \[\leadsto \frac{\frac{0.125}{r \cdot \pi}}{s} \cdot \left(\color{blue}{\mathsf{fma}\left(0.5, \frac{{r}^{2}}{{s}^{2}}, 1 + -1 \cdot \frac{r}{s}\right)} + \left(1 + -0.3333333333333333 \cdot \frac{r}{s}\right)\right) \]

   2. unpow2 11.9%

      \[\leadsto \frac{\frac{0.125}{r \cdot \pi}}{s} \cdot \left(\mathsf{fma}\left(0.5, \frac{\color{blue}{r \cdot r}}{{s}^{2}}, 1 + -1 \cdot \frac{r}{s}\right) + \left(1 + -0.3333333333333333 \cdot \frac{r}{s}\right)\right) \]

   3. unpow2 11.9%

      \[\leadsto \frac{\frac{0.125}{r \cdot \pi}}{s} \cdot \left(\mathsf{fma}\left(0.5, \frac{r \cdot r}{\color{blue}{s \cdot s}}, 1 + -1 \cdot \frac{r}{s}\right) + \left(1 + -0.3333333333333333 \cdot \frac{r}{s}\right)\right) \]

   4. mul-1-neg 11.9%

      \[\leadsto \frac{\frac{0.125}{r \cdot \pi}}{s} \cdot \left(\mathsf{fma}\left(0.5, \frac{r \cdot r}{s \cdot s}, 1 + \color{blue}{\left(-\frac{r}{s}\right)}\right) + \left(1 + -0.3333333333333333 \cdot \frac{r}{s}\right)\right) \]

   5. distribute-frac-neg 11.9%

      \[\leadsto \frac{\frac{0.125}{r \cdot \pi}}{s} \cdot \left(\mathsf{fma}\left(0.5, \frac{r \cdot r}{s \cdot s}, 1 + \color{blue}{\frac{-r}{s}}\right) + \left(1 + -0.3333333333333333 \cdot \frac{r}{s}\right)\right) \]

7. Simplified 11.9%

   \[\leadsto \frac{\frac{0.125}{r \cdot \pi}}{s} \cdot \left(\color{blue}{\mathsf{fma}\left(0.5, \frac{r \cdot r}{s \cdot s}, 1 + \frac{-r}{s}\right)} + \left(1 + -0.3333333333333333 \cdot \frac{r}{s}\right)\right) \]

8. Taylor expanded in r around 0 10.9%

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

9. Final simplification 10.9%

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

Alternative 13: 9.1% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

   1. associate-*l/ 97.6%

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

   2. associate-*l/ 97.6%

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

   3. associate-*l* 97.6%

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

   4. associate-*l* 97.6%

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

   5. associate-/r* 97.6%

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

   6. metadata-eval 97.6%

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

   7. metadata-eval 97.6%

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

   8. associate-/r* 97.6%

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

   9. associate-*l* 97.6%

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

   10. associate-*l* 97.6%

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

   11. distribute-lft-out 97.6%

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

3. Simplified 97.3%

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

4. Taylor expanded in r around 0 10.5%

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

   1. *-commutative 10.5%

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

6. Simplified 10.5%

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

   1. frac-2neg 10.5%

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

   2. div-inv 10.5%

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

   3. metadata-eval 10.5%

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

   4. *-commutative 10.5%

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

   5. distribute-rgt-neg-in 10.5%

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

   6. *-commutative 10.5%

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

   7. distribute-rgt-neg-in 10.5%

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

8. Applied egg-rr 10.5%

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

   1. associate-*r/ 10.5%

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

   2. metadata-eval 10.5%

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

   3. associate-*r* 10.5%

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

   4. *-commutative 10.5%

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

   5. associate-*l* 10.5%

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

10. Simplified 10.5%

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

    1. expm1-log1p-u 10.5%

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

    2. expm1-udef 10.8%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-0.25}{\pi \cdot \left(s \cdot \left(-r\right)\right)}\right)} - 1} \]

12. Applied egg-rr 10.8%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-0.25}{\pi \cdot \left(s \cdot \left(-r\right)\right)}\right)} - 1} \]
13. Step-by-step derivation

    1. expm1-def 10.5%

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

    2. expm1-log1p 10.5%

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

    3. associate-/r* 10.5%

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

14. Simplified 10.5%

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

15. Final simplification 10.5%

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

Alternative 14: 9.1% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

   1. associate-*l/ 97.6%

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

   2. associate-*l/ 97.6%

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

   3. associate-*l* 97.6%

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

   4. associate-*l* 97.6%

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

   5. associate-/r* 97.6%

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

   6. metadata-eval 97.6%

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

   7. metadata-eval 97.6%

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

   8. associate-/r* 97.6%

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

   9. associate-*l* 97.6%

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

   10. associate-*l* 97.6%

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

   11. distribute-lft-out 97.6%

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

3. Simplified 97.3%

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

4. Taylor expanded in r around 0 10.5%

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

   1. *-commutative 10.5%

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

6. Simplified 10.5%

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

   1. frac-2neg 10.5%

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

   2. div-inv 10.5%

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

   3. metadata-eval 10.5%

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

   4. *-commutative 10.5%

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

   5. distribute-rgt-neg-in 10.5%

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

   6. *-commutative 10.5%

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

   7. distribute-rgt-neg-in 10.5%

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

8. Applied egg-rr 10.5%

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

   1. associate-*r/ 10.5%

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

   2. metadata-eval 10.5%

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

   3. associate-*r* 10.5%

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

   4. *-commutative 10.5%

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

   5. associate-*l* 10.5%

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

10. Simplified 10.5%

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

11. Taylor expanded in s around 0 10.5%

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

    1. *-commutative 10.5%

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

    2. associate-*l* 10.5%

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

    3. *-commutative 10.5%

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

13. Simplified 10.5%

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

14. Final simplification 10.5%

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

Alternative 15: 9.1% 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 99.6%

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

   1. associate-*l/ 97.6%

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

   2. associate-*l/ 97.6%

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

   3. associate-*l* 97.6%

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

   4. associate-*l* 97.6%

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

   5. associate-/r* 97.6%

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

   6. metadata-eval 97.6%

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

   7. metadata-eval 97.6%

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

   8. associate-/r* 97.6%

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

   9. associate-*l* 97.6%

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

   10. associate-*l* 97.6%

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

   11. distribute-lft-out 97.6%

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

3. Simplified 97.3%

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

4. Taylor expanded in r around 0 10.5%

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

   1. *-commutative 10.5%

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

6. Simplified 10.5%

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

7. Final simplification 10.5%

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

Alternative 16: 9.1% accurate, 4.0× speedup

Math

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

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(Float32(0.25) / 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 99.6%

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

   1. associate-*l/ 97.6%

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

   2. associate-*l/ 97.6%

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

   3. associate-*l* 97.6%

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

   4. associate-*l* 97.6%

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

   5. associate-/r* 97.6%

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

   6. metadata-eval 97.6%

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

   7. metadata-eval 97.6%

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

   8. associate-/r* 97.6%

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

   9. associate-*l* 97.6%

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

   10. associate-*l* 97.6%

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

   11. distribute-lft-out 97.6%

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

3. Simplified 97.3%

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

4. Taylor expanded in r around 0 10.5%

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

   1. *-commutative 10.5%

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

6. Simplified 10.5%

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

   1. frac-2neg 10.5%

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

   2. div-inv 10.5%

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

   3. metadata-eval 10.5%

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

   4. *-commutative 10.5%

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

   5. distribute-rgt-neg-in 10.5%

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

   6. *-commutative 10.5%

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

   7. distribute-rgt-neg-in 10.5%

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

8. Applied egg-rr 10.5%

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

   1. associate-*r/ 10.5%

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

   2. metadata-eval 10.5%

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

   3. associate-*r* 10.5%

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

   4. *-commutative 10.5%

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

   5. associate-*l* 10.5%

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

10. Simplified 10.5%

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

11. Taylor expanded in s around 0 10.5%

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

    1. associate-/r* 10.5%

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

13. Simplified 10.5%

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

14. Final simplification 10.5%

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

Reproduce

herbie shell --seed 2023207 
(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))))