Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.5%

Time: 21.5s

Alternatives: 15

Speedup: N/A×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

   1. times-frac 99.4%

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

   2. times-frac 99.4%

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

   3. associate-*l* 99.4%

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

   4. associate-/r* 99.4%

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

   5. metadata-eval 99.4%

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

   6. metadata-eval 99.4%

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

   7. associate-/r* 99.4%

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

   8. associate-*l* 99.4%

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

   9. distribute-lft-out 99.4%

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

3. Simplified 99.3%

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

   1. *-commutative 99.3%

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

   2. frac-2neg 99.3%

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

   3. div-inv 99.3%

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

   4. metadata-eval 99.3%

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

   5. *-commutative 99.3%

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

   6. distribute-rgt-neg-in 99.3%

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

5. Applied egg-rr 99.3%

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

   1. associate-*r/ 99.3%

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

   2. metadata-eval 99.3%

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

   3. associate-/r* 99.3%

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

7. Simplified 99.3%

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

8. Taylor expanded in r around inf 99.8%

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

   1. *-commutative 99.8%

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

   2. mul-1-neg 99.8%

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

   3. distribute-neg-frac 99.8%

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

   4. associate-*r/ 99.8%

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

   5. *-commutative 99.8%

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

10. Simplified 99.8%

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

11. Final simplification 99.8%

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

Alternative 2: 99.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

   1. times-frac 99.4%

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

   2. times-frac 99.4%

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

   3. associate-*l* 99.4%

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

   4. associate-/r* 99.4%

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

   5. metadata-eval 99.4%

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

   6. metadata-eval 99.4%

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

   7. associate-/r* 99.4%

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

   8. associate-*l* 99.4%

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

   9. distribute-lft-out 99.4%

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

3. Simplified 99.3%

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

   1. *-commutative 99.3%

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

   2. frac-2neg 99.3%

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

   3. div-inv 99.3%

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

   4. metadata-eval 99.3%

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

   5. *-commutative 99.3%

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

   6. distribute-rgt-neg-in 99.3%

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

5. Applied egg-rr 99.3%

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

   1. associate-*r/ 99.3%

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

   2. metadata-eval 99.3%

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

   3. associate-/r* 99.3%

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

7. Simplified 99.3%

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

8. Taylor expanded in r around inf 99.8%

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

   1. *-commutative 99.8%

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

   2. mul-1-neg 99.8%

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

   3. distribute-neg-frac 99.8%

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

   4. associate-*r/ 99.8%

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

   5. *-commutative 99.8%

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

10. Simplified 99.8%

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

    1. *-commutative 99.8%

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

    2. associate-*l/ 99.8%

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

12. Applied egg-rr 99.8%

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

13. Final simplification 99.8%

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

Alternative 3: 99.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = single(0.125) * ((exp((-r / s)) + exp((r * (single(-0.3333333333333333) / s)))) / (single(pi) * (r * 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 99.4%

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

   2. times-frac 99.4%

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

   3. associate-*l* 99.4%

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

   4. associate-/r* 99.4%

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

   5. metadata-eval 99.4%

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

   6. metadata-eval 99.4%

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

   7. associate-/r* 99.4%

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

   8. associate-*l* 99.4%

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

   9. distribute-lft-out 99.4%

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

3. Simplified 99.3%

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

   1. *-commutative 99.3%

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

   2. frac-2neg 99.3%

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

   3. div-inv 99.3%

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

   4. metadata-eval 99.3%

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

   5. *-commutative 99.3%

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

   6. distribute-rgt-neg-in 99.3%

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

5. Applied egg-rr 99.3%

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

   1. associate-*r/ 99.3%

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

   2. metadata-eval 99.3%

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

   3. associate-/r* 99.3%

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

7. Simplified 99.3%

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

8. Taylor expanded in r around inf 99.8%

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

   1. *-commutative 99.8%

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

   2. mul-1-neg 99.8%

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

   3. distribute-neg-frac 99.8%

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

   4. associate-*r/ 99.8%

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

   5. *-commutative 99.8%

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

10. Simplified 99.8%

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

    1. *-commutative 99.8%

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

    2. associate-*l/ 99.8%

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

12. Applied egg-rr 99.8%

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

13. Taylor expanded in r around 0 99.8%

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

    1. associate-*r* 99.8%

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

    2. *-commutative 99.8%

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

15. Simplified 99.8%

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

16. Final simplification 99.8%

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

Alternative 4: 43.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;r \leq 9:\\ \;\;\;\;\frac{\frac{-0.125}{s}}{-\pi} \cdot \left(\frac{1 + \frac{-0.3333333333333333}{\frac{s}{r}}}{r} + \frac{e^{\frac{r}{-s}}}{r}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.25}{s \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(r \cdot \pi\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (if (<= r 9.0)
   (*
    (/ (/ -0.125 s) (- PI))
    (+ (/ (+ 1.0 (/ -0.3333333333333333 (/ s r))) r) (/ (exp (/ r (- s))) r)))
   (/ -0.25 (* s (log1p (expm1 (* r PI)))))))

C

float code(float s, float r) {
	float tmp;
	if (r <= 9.0f) {
		tmp = ((-0.125f / s) / -((float) M_PI)) * (((1.0f + (-0.3333333333333333f / (s / r))) / r) + (expf((r / -s)) / r));
	} else {
		tmp = -0.25f / (s * log1pf(expm1f((r * ((float) M_PI)))));
	}
	return tmp;
}

Julia

function code(s, r)
	tmp = Float32(0.0)
	if (r <= Float32(9.0))
		tmp = Float32(Float32(Float32(Float32(-0.125) / s) / Float32(-Float32(pi))) * Float32(Float32(Float32(Float32(1.0) + Float32(Float32(-0.3333333333333333) / Float32(s / r))) / r) + Float32(exp(Float32(r / Float32(-s))) / r)));
	else
		tmp = Float32(Float32(-0.25) / Float32(s * log1p(expm1(Float32(r * Float32(pi))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if r < 9

   1. Initial program 99.7%

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

      1. times-frac 99.1%

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

      2. times-frac 99.1%

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

      3. associate-*l* 99.0%

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

      4. associate-/r* 99.1%

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

      5. metadata-eval 99.1%

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

      6. metadata-eval 99.1%

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

      7. associate-/r* 99.1%

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

      8. associate-*l* 99.1%

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

      9. distribute-lft-out 99.1%

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

   3. Simplified 98.9%

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

   4. Taylor expanded in r around 0 12.9%

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

      1. associate-*r/ 12.9%

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

      2. associate-/l* 12.9%

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

   6. Simplified 12.9%

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

      1. *-commutative 98.9%

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

      2. frac-2neg 98.9%

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

      3. div-inv 98.9%

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

      4. metadata-eval 98.9%

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

      5. *-commutative 98.9%

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

      6. distribute-rgt-neg-in 98.9%

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

   8. Applied egg-rr 12.9%

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

      1. associate-*r/ 98.9%

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

      2. metadata-eval 98.9%

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

      3. associate-/r* 98.9%

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

   10. Simplified 12.9%

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

   if 9 < r

   1. Initial program 99.9%

      \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
   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* 99.9%

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

      4. associate-/r* 99.9%

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

      5. metadata-eval 99.9%

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

      6. metadata-eval 99.9%

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

      7. associate-/r* 99.9%

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

      8. associate-*l* 99.9%

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

      9. distribute-lft-out 99.9%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in r around 0 4.9%

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

   5. Taylor expanded in s around inf 4.9%

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

      1. *-commutative 4.9%

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

   7. Simplified 4.9%

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

      1. frac-2neg 4.9%

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

      2. div-inv 4.9%

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

      3. metadata-eval 4.9%

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

      4. distribute-lft-neg-in 4.9%

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

      5. add-sqr-sqrt -0.0%

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

      6. sqrt-unprod 4.8%

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

      7. sqr-neg 4.8%

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

      8. sqrt-unprod 4.8%

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

      9. add-sqr-sqrt 4.8%

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

      10. associate-*r* 4.8%

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

      11. *-commutative 4.8%

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

   9. Applied egg-rr 4.8%

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

       1. associate-*r/ 4.8%

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

       2. metadata-eval 4.8%

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

   11. Simplified 4.8%

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

       1. log1p-expm1-u 88.4%

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

   13. Applied egg-rr 88.4%

       \[\leadsto \frac{-0.25}{s \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(r \cdot \pi\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 41.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;r \leq 9:\\ \;\;\;\;\frac{\frac{-0.125}{s}}{-\pi} \cdot \left(\frac{1 + \frac{-0.3333333333333333}{\frac{s}{r}}}{r} + \frac{e^{\frac{r}{-s}}}{r}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.25}{s \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(r \cdot \pi\right)\right)}\\ \end{array} \]

Alternative 5: 9.4% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = ((single(-0.125) / s) / -single(pi)) * (((single(1.0) + (single(-0.3333333333333333) / (s / r))) / r) + (exp((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 99.4%

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

   2. times-frac 99.4%

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

   3. associate-*l* 99.4%

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

   4. associate-/r* 99.4%

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

   5. metadata-eval 99.4%

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

   6. metadata-eval 99.4%

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

   7. associate-/r* 99.4%

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

   8. associate-*l* 99.4%

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

   9. distribute-lft-out 99.4%

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

3. Simplified 99.3%

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

4. Taylor expanded in r around 0 9.4%

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

   1. associate-*r/ 9.4%

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

   2. associate-/l* 9.4%

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

6. Simplified 9.4%

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

   1. *-commutative 99.3%

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

   2. frac-2neg 99.3%

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

   3. div-inv 99.3%

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

   4. metadata-eval 99.3%

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

   5. *-commutative 99.3%

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

   6. distribute-rgt-neg-in 99.3%

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

8. Applied egg-rr 9.4%

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

   1. associate-*r/ 99.3%

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

   2. metadata-eval 99.3%

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

   3. associate-/r* 99.3%

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

10. Simplified 9.4%

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

11. Final simplification 9.4%

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

Alternative 6: 9.4% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.3%

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

4. Taylor expanded in r around 0 9.4%

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

   1. associate-*r/ 9.4%

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

   2. associate-/l* 9.4%

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

6. Simplified 9.4%

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

7. Final simplification 9.4%

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

Alternative 7: 9.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = ((exp((r / -s)) / r) + ((single(-0.3333333333333333) / s) + (single(1.0) / r))) * (single(0.125) / (s * single(pi)));
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 99.4%

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

   2. times-frac 99.4%

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

   3. associate-*l* 99.4%

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

   4. associate-/r* 99.4%

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

   5. metadata-eval 99.4%

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

   6. metadata-eval 99.4%

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

   7. associate-/r* 99.4%

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

   8. associate-*l* 99.4%

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

   9. distribute-lft-out 99.4%

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

3. Simplified 99.3%

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

4. Taylor expanded in r around 0 9.4%

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

   1. associate-*r/ 9.4%

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

   2. associate-/l* 9.4%

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

6. Simplified 9.4%

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

7. Taylor expanded in s around 0 9.4%

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

   1. cancel-sign-sub-inv 8.8%

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

   2. metadata-eval 8.8%

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

   3. associate-*r/ 8.8%

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

   4. metadata-eval 8.8%

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

9. Simplified 9.4%

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

10. Final simplification 9.4%

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

Alternative 8: 9.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.3%

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

4. Taylor expanded in r around 0 9.4%

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

   1. associate-*r/ 9.4%

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

   2. associate-/l* 9.4%

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

6. Simplified 9.4%

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

7. Taylor expanded in s around 0 9.4%

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

   1. cancel-sign-sub-inv 8.8%

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

   2. metadata-eval 8.8%

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

   3. associate-*r/ 8.8%

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

   4. metadata-eval 8.8%

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

9. Simplified 9.4%

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

10. Final simplification 9.4%

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

Alternative 9: 9.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = single(0.125) * ((exp((-r / s)) + single(1.0)) / (single(pi) * (r * 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 99.4%

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

   2. times-frac 99.4%

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

   3. associate-*l* 99.4%

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

   4. associate-/r* 99.4%

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

   5. metadata-eval 99.4%

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

   6. metadata-eval 99.4%

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

   7. associate-/r* 99.4%

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

   8. associate-*l* 99.4%

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

   9. distribute-lft-out 99.4%

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

3. Simplified 99.3%

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

4. Taylor expanded in r around 0 8.8%

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

   1. *-commutative 99.3%

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

   2. frac-2neg 99.3%

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

   3. div-inv 99.3%

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

   4. metadata-eval 99.3%

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

   5. *-commutative 99.3%

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

   6. distribute-rgt-neg-in 99.3%

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

6. Applied egg-rr 8.8%

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

   1. associate-*r/ 99.3%

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

   2. metadata-eval 99.3%

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

   3. associate-/r* 99.3%

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

8. Simplified 8.8%

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

9. Taylor expanded in r around inf 8.9%

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

    1. mul-1-neg 8.9%

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

    2. associate-*r* 8.9%

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

    3. *-commutative 8.9%

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

11. Simplified 8.9%

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

12. Final simplification 8.9%

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

Alternative 10: 9.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

   1. times-frac 99.4%

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

   2. times-frac 99.4%

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

   3. associate-*l* 99.4%

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

   4. associate-/r* 99.4%

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

   5. metadata-eval 99.4%

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

   6. metadata-eval 99.4%

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

   7. associate-/r* 99.4%

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

   8. associate-*l* 99.4%

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

   9. distribute-lft-out 99.4%

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

3. Simplified 99.3%

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

4. Taylor expanded in r around 0 8.8%

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

5. Taylor expanded in r around inf 8.9%

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

   1. associate-*r/ 8.9%

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

   2. mul-1-neg 8.9%

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

   3. distribute-neg-frac 8.9%

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

   4. *-commutative 8.9%

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

7. Simplified 8.9%

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

8. Final simplification 8.9%

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

Alternative 11: 8.8% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = (single(0.125) / (s * single(pi))) * (((single(1.0) + (single(-0.3333333333333333) / (s / r))) / r) + ((single(1.0) / r) + (single(-1.0) / 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 99.4%

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

   2. times-frac 99.4%

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

   3. associate-*l* 99.4%

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

   4. associate-/r* 99.4%

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

   5. metadata-eval 99.4%

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

   6. metadata-eval 99.4%

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

   7. associate-/r* 99.4%

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

   8. associate-*l* 99.4%

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

   9. distribute-lft-out 99.4%

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

3. Simplified 99.3%

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

4. Taylor expanded in r around 0 9.4%

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

   1. associate-*r/ 9.4%

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

   2. associate-/l* 9.4%

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

6. Simplified 9.4%

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

7. Taylor expanded in r around 0 8.8%

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

   1. mul-1-neg 8.8%

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

   2. distribute-neg-frac 8.8%

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

9. Simplified 8.8%

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

10. Taylor expanded in r around 0 8.8%

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

11. Final simplification 8.8%

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

Alternative 12: 8.8% accurate, 3.5× speedup

Math

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

FPCore

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

C

float code(float s, float r) {
	return (0.125f / (s * ((float) M_PI))) * (((1.0f + (-0.3333333333333333f / (s / r))) / r) + ((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) + Float32(Float32(-0.3333333333333333) / Float32(s / r))) / r) + 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) + (single(-0.3333333333333333) / (s / r))) / r) + ((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 99.4%

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

   2. times-frac 99.4%

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

   3. associate-*l* 99.4%

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

   4. associate-/r* 99.4%

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

   5. metadata-eval 99.4%

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

   6. metadata-eval 99.4%

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

   7. associate-/r* 99.4%

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

   8. associate-*l* 99.4%

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

   9. distribute-lft-out 99.4%

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

3. Simplified 99.3%

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

4. Taylor expanded in r around 0 9.4%

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

   1. associate-*r/ 9.4%

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

   2. associate-/l* 9.4%

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

6. Simplified 9.4%

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

7. Taylor expanded in r around 0 8.8%

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

   1. mul-1-neg 8.8%

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

   2. distribute-neg-frac 8.8%

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

9. Simplified 8.8%

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

10. Final simplification 8.8%

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

Alternative 13: 8.8% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = (single(0.125) / (s * single(pi))) * (((single(-0.3333333333333333) / s) + (single(1.0) / r)) + ((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 99.4%

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

   2. times-frac 99.4%

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

   3. associate-*l* 99.4%

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

   4. associate-/r* 99.4%

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

   5. metadata-eval 99.4%

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

   6. metadata-eval 99.4%

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

   7. associate-/r* 99.4%

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

   8. associate-*l* 99.4%

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

   9. distribute-lft-out 99.4%

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

3. Simplified 99.3%

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

4. Taylor expanded in r around 0 9.4%

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

   1. associate-*r/ 9.4%

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

   2. associate-/l* 9.4%

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

6. Simplified 9.4%

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

7. Taylor expanded in r around 0 8.8%

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

   1. mul-1-neg 8.8%

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

   2. distribute-neg-frac 8.8%

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

9. Simplified 8.8%

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

10. Taylor expanded in s around 0 8.8%

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

    1. cancel-sign-sub-inv 8.8%

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

    2. metadata-eval 8.8%

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

    3. associate-*r/ 8.8%

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

    4. metadata-eval 8.8%

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

12. Simplified 8.8%

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

13. Final simplification 8.8%

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

Alternative 14: 4.4% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = single(-0.25) / (s * (r * single(pi)));
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 99.4%

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

   2. times-frac 99.4%

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

   3. associate-*l* 99.4%

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

   4. associate-/r* 99.4%

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

   5. metadata-eval 99.4%

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

   6. metadata-eval 99.4%

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

   7. associate-/r* 99.4%

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

   8. associate-*l* 99.4%

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

   9. distribute-lft-out 99.4%

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

3. Simplified 99.3%

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

4. Taylor expanded in r around 0 8.8%

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

5. Taylor expanded in s around inf 8.4%

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

   1. *-commutative 8.4%

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

7. Simplified 8.4%

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

   1. frac-2neg 8.4%

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

   2. div-inv 8.4%

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

   3. metadata-eval 8.4%

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

   4. distribute-lft-neg-in 8.4%

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

   5. add-sqr-sqrt -0.0%

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

   6. sqrt-unprod 4.3%

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

   7. sqr-neg 4.3%

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

   8. sqrt-unprod 4.3%

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

   9. add-sqr-sqrt 4.3%

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

   10. associate-*r* 4.3%

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

   11. *-commutative 4.3%

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

9. Applied egg-rr 4.3%

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

    1. associate-*r/ 4.3%

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

    2. metadata-eval 4.3%

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

11. Simplified 4.3%

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

12. Final simplification 4.3%

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

Alternative 15: 8.8% 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 99.8%

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

   1. times-frac 99.4%

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

   2. times-frac 99.4%

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

   3. associate-*l* 99.4%

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

   4. associate-/r* 99.4%

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

   5. metadata-eval 99.4%

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

   6. metadata-eval 99.4%

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

   7. associate-/r* 99.4%

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

   8. associate-*l* 99.4%

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

   9. distribute-lft-out 99.4%

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

3. Simplified 99.3%

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

4. Taylor expanded in r around 0 8.8%

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

5. Taylor expanded in s around inf 8.4%

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

   1. *-commutative 8.4%

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

7. Simplified 8.4%

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

8. Final simplification 8.4%

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

Reproduce

herbie shell --seed 2023301 
(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))))