Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.1%

Time: 19.3s

Alternatives: 18

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 99.6% 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: 99.1% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = single(0.125) * ((sqrt((exp(single(-0.6666666666666666)) ^ (r / s))) + exp((-r / s))) / (r * (s * single(pi))));
end

Derivation

1. Initial program 99.8%

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

3. Simplified 99.5%

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

   1. add-sqr-sqrt 99.5%

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

   2. sqrt-unprod 99.5%

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

   3. pow-prod-down 99.5%

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

   4. prod-exp 99.8%

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

   5. metadata-eval 99.8%

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

6. Applied egg-rr 99.8%

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

7. Taylor expanded in r around inf 99.7%

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

   1. +-commutative 99.7%

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

   2. exp-prod 99.9%

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

   3. neg-mul-1 99.9%

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

   4. distribute-frac-neg 99.9%

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

9. Simplified 99.9%

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

10. Final simplification 99.9%

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

Alternative 2: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in s around 0 99.8%

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

   1. associate-*r* 99.8%

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

   2. *-commutative 99.8%

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

   3. associate-*r* 99.8%

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

   4. *-commutative 99.8%

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

   5. *-commutative 99.8%

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

   6. associate-*l* 99.8%

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

5. Simplified 99.8%

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

6. Final simplification 99.8%

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

Alternative 3: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{e^{\frac{-r}{s}} \cdot 0.25}{r \cdot \left(s \cdot \left(\pi \cdot 2\right)\right)} + \frac{0.75 \cdot e^{\frac{-0.3333333333333333}{\frac{s}{r}}}}{r \cdot \left(s \cdot \left(\pi \cdot 6\right)\right)} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/ (* (exp (/ (- r) s)) 0.25) (* r (* s (* PI 2.0))))
  (/ (* 0.75 (exp (/ -0.3333333333333333 (/ s r)))) (* r (* s (* PI 6.0))))))

C

float code(float s, float r) {
	return ((expf((-r / s)) * 0.25f) / (r * (s * (((float) M_PI) * 2.0f)))) + ((0.75f * expf((-0.3333333333333333f / (s / r)))) / (r * (s * (((float) M_PI) * 6.0f))));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(exp(Float32(Float32(-r) / s)) * Float32(0.25)) / Float32(r * Float32(s * Float32(Float32(pi) * Float32(2.0))))) + Float32(Float32(Float32(0.75) * exp(Float32(Float32(-0.3333333333333333) / Float32(s / r)))) / Float32(r * Float32(s * Float32(Float32(pi) * Float32(6.0))))))
end

MATLAB

function tmp = code(s, r)
	tmp = ((exp((-r / s)) * single(0.25)) / (r * (s * (single(pi) * single(2.0))))) + ((single(0.75) * exp((single(-0.3333333333333333) / (s / r)))) / (r * (s * (single(pi) * single(6.0)))));
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. Add Preprocessing

3. Taylor expanded in r around 0 99.7%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\color{blue}{-0.3333333333333333 \cdot \frac{r}{s}}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
4. Step-by-step derivation

   1. associate-*r/ 99.8%

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\color{blue}{\frac{-0.3333333333333333 \cdot r}{s}}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

   2. associate-/l* 99.8%

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\color{blue}{\frac{-0.3333333333333333}{\frac{s}{r}}}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

5. Simplified 99.8%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\color{blue}{\frac{-0.3333333333333333}{\frac{s}{r}}}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

6. Final simplification 99.8%

   \[\leadsto \frac{e^{\frac{-r}{s}} \cdot 0.25}{r \cdot \left(s \cdot \left(\pi \cdot 2\right)\right)} + \frac{0.75 \cdot e^{\frac{-0.3333333333333333}{\frac{s}{r}}}}{r \cdot \left(s \cdot \left(\pi \cdot 6\right)\right)} \]
7. Add Preprocessing

Alternative 4: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{0.25 \cdot \frac{0.5}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{e^{\frac{r}{s \cdot -3}}}{r}\right) \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (*
  (/ (* 0.25 (/ 0.5 PI)) s)
  (+ (/ (exp (/ r (- s))) r) (/ (exp (/ r (* s -3.0))) r))))

C

float code(float s, float r) {
	return ((0.25f * (0.5f / ((float) M_PI))) / s) * ((expf((r / -s)) / r) + (expf((r / (s * -3.0f))) / r));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.25) * Float32(Float32(0.5) / Float32(pi))) / s) * Float32(Float32(exp(Float32(r / Float32(-s))) / r) + Float32(exp(Float32(r / Float32(s * Float32(-3.0)))) / r)))
end

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.25) * (single(0.5) / single(pi))) / s) * ((exp((r / -s)) / r) + (exp((r / (s * single(-3.0)))) / r));
end

Derivation

1. Initial program 99.8%

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

3. Simplified 99.5%

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

   1. metadata-eval 8.6%

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

   2. associate-/r* 8.6%

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

   3. *-commutative 8.6%

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

   4. associate-*l* 8.6%

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

   5. associate-/l/ 8.6%

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

   6. div-inv 8.6%

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

   7. *-commutative 8.6%

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

6. Applied egg-rr 99.2%

   \[\leadsto \color{blue}{\left(\frac{0.25}{s} \cdot \frac{1}{\pi \cdot 2}\right)} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{{\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}}{r}\right) \]
7. Step-by-step derivation

   1. associate-*l/ 8.6%

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

   2. *-commutative 8.6%

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

   3. associate-/r* 8.6%

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

   4. metadata-eval 8.6%

      \[\leadsto \frac{0.25 \cdot \frac{\color{blue}{0.5}}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{1}{r}\right) \]

8. Simplified 99.6%

   \[\leadsto \color{blue}{\frac{0.25 \cdot \frac{0.5}{\pi}}{s}} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{{\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}}{r}\right) \]

9. Taylor expanded in r around inf 99.7%

   \[\leadsto \frac{0.25 \cdot \frac{0.5}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\color{blue}{e^{-0.3333333333333333 \cdot \frac{r}{s}}}}{r}\right) \]
10. Step-by-step derivation

    1. metadata-eval 99.7%

       \[\leadsto \frac{0.25 \cdot \frac{0.5}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{e^{\color{blue}{\frac{-1}{3}} \cdot \frac{r}{s}}}{r}\right) \]

    2. times-frac 99.8%

       \[\leadsto \frac{0.25 \cdot \frac{0.5}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{e^{\color{blue}{\frac{-1 \cdot r}{3 \cdot s}}}}{r}\right) \]

    3. neg-mul-1 99.8%

       \[\leadsto \frac{0.25 \cdot \frac{0.5}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{e^{\frac{\color{blue}{-r}}{3 \cdot s}}}{r}\right) \]

    4. frac-2neg 99.8%

       \[\leadsto \frac{0.25 \cdot \frac{0.5}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{e^{\color{blue}{\frac{-\left(-r\right)}{-3 \cdot s}}}}{r}\right) \]

    5. remove-double-neg 99.8%

       \[\leadsto \frac{0.25 \cdot \frac{0.5}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{e^{\frac{\color{blue}{r}}{-3 \cdot s}}}{r}\right) \]

    6. *-commutative 99.8%

       \[\leadsto \frac{0.25 \cdot \frac{0.5}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{e^{\frac{r}{-\color{blue}{s \cdot 3}}}}{r}\right) \]

    7. distribute-rgt-neg-in 99.8%

       \[\leadsto \frac{0.25 \cdot \frac{0.5}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{e^{\frac{r}{\color{blue}{s \cdot \left(-3\right)}}}}{r}\right) \]

    8. metadata-eval 99.8%

       \[\leadsto \frac{0.25 \cdot \frac{0.5}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{e^{\frac{r}{s \cdot \color{blue}{-3}}}}{r}\right) \]

11. Applied egg-rr 99.8%

    \[\leadsto \frac{0.25 \cdot \frac{0.5}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{e^{\color{blue}{\frac{r}{s \cdot -3}}}}{r}\right) \]

12. Final simplification 99.8%

    \[\leadsto \frac{0.25 \cdot \frac{0.5}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{e^{\frac{r}{s \cdot -3}}}{r}\right) \]
13. Add Preprocessing

Alternative 5: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.5%

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

   1. metadata-eval 8.6%

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

   2. associate-/r* 8.6%

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

   3. *-commutative 8.6%

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

   4. associate-*l* 8.6%

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

   5. associate-/l/ 8.6%

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

   6. div-inv 8.6%

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

   7. *-commutative 8.6%

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

6. Applied egg-rr 99.2%

   \[\leadsto \color{blue}{\left(\frac{0.25}{s} \cdot \frac{1}{\pi \cdot 2}\right)} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{{\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}}{r}\right) \]
7. Step-by-step derivation

   1. associate-*l/ 8.6%

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

   2. *-commutative 8.6%

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

   3. associate-/r* 8.6%

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

   4. metadata-eval 8.6%

      \[\leadsto \frac{0.25 \cdot \frac{\color{blue}{0.5}}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{1}{r}\right) \]

8. Simplified 99.6%

   \[\leadsto \color{blue}{\frac{0.25 \cdot \frac{0.5}{\pi}}{s}} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{{\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}}{r}\right) \]

9. Taylor expanded in r around inf 99.7%

   \[\leadsto \frac{0.25 \cdot \frac{0.5}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\color{blue}{e^{-0.3333333333333333 \cdot \frac{r}{s}}}}{r}\right) \]

10. Taylor expanded in r around inf 99.7%

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

11. Final simplification 99.7%

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

Alternative 6: 12.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.5%

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

5. Taylor expanded in r around 0 8.6%

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

6. Taylor expanded in s around inf 8.2%

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

   1. associate-*r* 8.2%

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

   2. log1p-expm1-u 11.8%

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

8. Applied egg-rr 11.8%

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

9. Final simplification 11.8%

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

Alternative 7: 43.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.5%

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

5. Taylor expanded in r around 0 8.6%

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

6. Taylor expanded in s around inf 8.2%

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

   1. *-commutative 8.2%

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

   2. associate-*r* 8.2%

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

8. Simplified 8.2%

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

   1. log1p-expm1-u 41.6%

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

   2. *-commutative 41.6%

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

10. Applied egg-rr 41.6%

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

11. Final simplification 41.6%

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

Alternative 8: 10.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \frac{0.25 \cdot \frac{0.5}{\pi}}{s} \cdot \left(\frac{\left(1 - \frac{r}{s}\right) + 0.5 \cdot {\left(\frac{r}{s}\right)}^{2}}{r} + \frac{1 + \frac{r}{\frac{s}{-0.3333333333333333}}}{r}\right) \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (*
  (/ (* 0.25 (/ 0.5 PI)) s)
  (+
   (/ (+ (- 1.0 (/ r s)) (* 0.5 (pow (/ r s) 2.0))) r)
   (/ (+ 1.0 (/ r (/ s -0.3333333333333333))) r))))

C

float code(float s, float r) {
	return ((0.25f * (0.5f / ((float) M_PI))) / s) * ((((1.0f - (r / s)) + (0.5f * powf((r / s), 2.0f))) / r) + ((1.0f + (r / (s / -0.3333333333333333f))) / r));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.25) * Float32(Float32(0.5) / Float32(pi))) / s) * Float32(Float32(Float32(Float32(Float32(1.0) - Float32(r / s)) + Float32(Float32(0.5) * (Float32(r / s) ^ Float32(2.0)))) / r) + Float32(Float32(Float32(1.0) + Float32(r / Float32(s / Float32(-0.3333333333333333)))) / r)))
end

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.25) * (single(0.5) / single(pi))) / s) * ((((single(1.0) - (r / s)) + (single(0.5) * ((r / s) ^ single(2.0)))) / r) + ((single(1.0) + (r / (s / single(-0.3333333333333333)))) / 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 99.5%

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

   1. metadata-eval 8.6%

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

   2. associate-/r* 8.6%

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

   3. *-commutative 8.6%

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

   4. associate-*l* 8.6%

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

   5. associate-/l/ 8.6%

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

   6. div-inv 8.6%

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

   7. *-commutative 8.6%

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

6. Applied egg-rr 99.2%

   \[\leadsto \color{blue}{\left(\frac{0.25}{s} \cdot \frac{1}{\pi \cdot 2}\right)} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{{\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}}{r}\right) \]
7. Step-by-step derivation

   1. associate-*l/ 8.6%

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

   2. *-commutative 8.6%

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

   3. associate-/r* 8.6%

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

   4. metadata-eval 8.6%

      \[\leadsto \frac{0.25 \cdot \frac{\color{blue}{0.5}}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{1}{r}\right) \]

8. Simplified 99.6%

   \[\leadsto \color{blue}{\frac{0.25 \cdot \frac{0.5}{\pi}}{s}} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{{\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}}{r}\right) \]

9. Taylor expanded in r around 0 8.8%

   \[\leadsto \frac{0.25 \cdot \frac{0.5}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\color{blue}{1 + -0.3333333333333333 \cdot \frac{r}{s}}}{r}\right) \]
10. Step-by-step derivation

    1. *-commutative 8.8%

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

    2. associate-/r/ 8.8%

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

11. Simplified 8.8%

    \[\leadsto \frac{0.25 \cdot \frac{0.5}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\color{blue}{1 + \frac{r}{\frac{s}{-0.3333333333333333}}}}{r}\right) \]

12. Taylor expanded in r around 0 8.9%

    \[\leadsto \frac{0.25 \cdot \frac{0.5}{\pi}}{s} \cdot \left(\frac{\color{blue}{1 + \left(-1 \cdot \frac{r}{s} + 0.5 \cdot \frac{{r}^{2}}{{s}^{2}}\right)}}{r} + \frac{1 + \frac{r}{\frac{s}{-0.3333333333333333}}}{r}\right) \]
13. Step-by-step derivation

    1. associate-+r+ 8.9%

       \[\leadsto \frac{0.25 \cdot \frac{0.5}{\pi}}{s} \cdot \left(\frac{\color{blue}{\left(1 + -1 \cdot \frac{r}{s}\right) + 0.5 \cdot \frac{{r}^{2}}{{s}^{2}}}}{r} + \frac{1 + \frac{r}{\frac{s}{-0.3333333333333333}}}{r}\right) \]

    2. mul-1-neg 8.9%

       \[\leadsto \frac{0.25 \cdot \frac{0.5}{\pi}}{s} \cdot \left(\frac{\left(1 + \color{blue}{\left(-\frac{r}{s}\right)}\right) + 0.5 \cdot \frac{{r}^{2}}{{s}^{2}}}{r} + \frac{1 + \frac{r}{\frac{s}{-0.3333333333333333}}}{r}\right) \]

    3. unsub-neg 8.9%

       \[\leadsto \frac{0.25 \cdot \frac{0.5}{\pi}}{s} \cdot \left(\frac{\color{blue}{\left(1 - \frac{r}{s}\right)} + 0.5 \cdot \frac{{r}^{2}}{{s}^{2}}}{r} + \frac{1 + \frac{r}{\frac{s}{-0.3333333333333333}}}{r}\right) \]

    4. unpow2 8.9%

       \[\leadsto \frac{0.25 \cdot \frac{0.5}{\pi}}{s} \cdot \left(\frac{\left(1 - \frac{r}{s}\right) + 0.5 \cdot \frac{\color{blue}{r \cdot r}}{{s}^{2}}}{r} + \frac{1 + \frac{r}{\frac{s}{-0.3333333333333333}}}{r}\right) \]

    5. unpow2 8.9%

       \[\leadsto \frac{0.25 \cdot \frac{0.5}{\pi}}{s} \cdot \left(\frac{\left(1 - \frac{r}{s}\right) + 0.5 \cdot \frac{r \cdot r}{\color{blue}{s \cdot s}}}{r} + \frac{1 + \frac{r}{\frac{s}{-0.3333333333333333}}}{r}\right) \]

    6. times-frac 8.9%

       \[\leadsto \frac{0.25 \cdot \frac{0.5}{\pi}}{s} \cdot \left(\frac{\left(1 - \frac{r}{s}\right) + 0.5 \cdot \color{blue}{\left(\frac{r}{s} \cdot \frac{r}{s}\right)}}{r} + \frac{1 + \frac{r}{\frac{s}{-0.3333333333333333}}}{r}\right) \]

    7. unpow2 8.9%

       \[\leadsto \frac{0.25 \cdot \frac{0.5}{\pi}}{s} \cdot \left(\frac{\left(1 - \frac{r}{s}\right) + 0.5 \cdot \color{blue}{{\left(\frac{r}{s}\right)}^{2}}}{r} + \frac{1 + \frac{r}{\frac{s}{-0.3333333333333333}}}{r}\right) \]

14. Simplified 8.9%

    \[\leadsto \frac{0.25 \cdot \frac{0.5}{\pi}}{s} \cdot \left(\frac{\color{blue}{\left(1 - \frac{r}{s}\right) + 0.5 \cdot {\left(\frac{r}{s}\right)}^{2}}}{r} + \frac{1 + \frac{r}{\frac{s}{-0.3333333333333333}}}{r}\right) \]

15. Final simplification 8.9%

    \[\leadsto \frac{0.25 \cdot \frac{0.5}{\pi}}{s} \cdot \left(\frac{\left(1 - \frac{r}{s}\right) + 0.5 \cdot {\left(\frac{r}{s}\right)}^{2}}{r} + \frac{1 + \frac{r}{\frac{s}{-0.3333333333333333}}}{r}\right) \]
16. Add Preprocessing

Alternative 9: 9.9% accurate, 1.9× speedup

Math

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

FPCore

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

C

float code(float s, float r) {
	return ((expf((r / -s)) / r) + ((1.0f + (r / (s / -0.3333333333333333f))) / 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(1.0) + Float32(r / Float32(s / Float32(-0.3333333333333333)))) / r)) * Float32(Float32(0.125) / Float32(s * Float32(pi))))
end

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.5%

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

5. Taylor expanded in r around 0 8.8%

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

   1. *-commutative 8.8%

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

   2. associate-/r/ 8.8%

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

7. Simplified 8.8%

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

8. Final simplification 8.8%

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

Alternative 10: 9.9% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.125) / s) / single(pi)) * ((exp((r / -s)) / r) + ((single(1.0) + (r / (s / single(-0.3333333333333333)))) / 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 99.5%

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

   1. metadata-eval 8.6%

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

   2. associate-/r* 8.6%

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

   3. *-commutative 8.6%

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

   4. associate-*l* 8.6%

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

   5. associate-/l/ 8.6%

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

   6. div-inv 8.6%

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

   7. *-commutative 8.6%

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

6. Applied egg-rr 99.2%

   \[\leadsto \color{blue}{\left(\frac{0.25}{s} \cdot \frac{1}{\pi \cdot 2}\right)} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{{\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}}{r}\right) \]
7. Step-by-step derivation

   1. associate-*l/ 8.6%

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

   2. *-commutative 8.6%

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

   3. associate-/r* 8.6%

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

   4. metadata-eval 8.6%

      \[\leadsto \frac{0.25 \cdot \frac{\color{blue}{0.5}}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{1}{r}\right) \]

8. Simplified 99.6%

   \[\leadsto \color{blue}{\frac{0.25 \cdot \frac{0.5}{\pi}}{s}} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{{\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}}{r}\right) \]

9. Taylor expanded in r around 0 8.8%

   \[\leadsto \frac{0.25 \cdot \frac{0.5}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\color{blue}{1 + -0.3333333333333333 \cdot \frac{r}{s}}}{r}\right) \]
10. Step-by-step derivation

    1. *-commutative 8.8%

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

    2. associate-/r/ 8.8%

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

11. Simplified 8.8%

    \[\leadsto \frac{0.25 \cdot \frac{0.5}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\color{blue}{1 + \frac{r}{\frac{s}{-0.3333333333333333}}}}{r}\right) \]

12. Taylor expanded in s around 0 8.8%

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

    1. associate-/r* 8.9%

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

14. Simplified 8.9%

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

15. Final simplification 8.9%

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

Alternative 11: 9.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.5%

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

5. Taylor expanded in r around 0 8.6%

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

   1. associate-/r* 8.6%

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

   2. div-inv 8.6%

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

7. Applied egg-rr 8.6%

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

8. Final simplification 8.6%

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

Alternative 12: 9.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{0.25 \cdot \frac{0.5}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{1}{r}\right) \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (* (/ (* 0.25 (/ 0.5 PI)) s) (+ (/ (exp (/ r (- s))) r) (/ 1.0 r))))

C

float code(float s, float r) {
	return ((0.25f * (0.5f / ((float) M_PI))) / s) * ((expf((r / -s)) / r) + (1.0f / r));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.25) * Float32(Float32(0.5) / Float32(pi))) / s) * Float32(Float32(exp(Float32(r / Float32(-s))) / r) + Float32(Float32(1.0) / r)))
end

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.25) * (single(0.5) / single(pi))) / s) * ((exp((r / -s)) / r) + (single(1.0) / r));
end

Derivation

1. Initial program 99.8%

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

3. Simplified 99.5%

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

5. Taylor expanded in r around 0 8.6%

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

   1. metadata-eval 8.6%

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

   2. associate-/r* 8.6%

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

   3. *-commutative 8.6%

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

   4. associate-*l* 8.6%

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

   5. associate-/l/ 8.6%

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

   6. div-inv 8.6%

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

   7. *-commutative 8.6%

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

7. Applied egg-rr 8.6%

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

   1. associate-*l/ 8.6%

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

   2. *-commutative 8.6%

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

   3. associate-/r* 8.6%

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

   4. metadata-eval 8.6%

      \[\leadsto \frac{0.25 \cdot \frac{\color{blue}{0.5}}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{1}{r}\right) \]

9. Simplified 8.6%

   \[\leadsto \color{blue}{\frac{0.25 \cdot \frac{0.5}{\pi}}{s}} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{1}{r}\right) \]

10. Final simplification 8.6%

    \[\leadsto \frac{0.25 \cdot \frac{0.5}{\pi}}{s} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{1}{r}\right) \]
11. Add Preprocessing

Alternative 13: 9.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.5%

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

5. Taylor expanded in r around 0 8.6%

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

6. Final simplification 8.6%

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

Alternative 14: 9.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.5%

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

5. Taylor expanded in r around 0 8.6%

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

6. Taylor expanded in s around 0 8.6%

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

   1. associate-*r/ 8.6%

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

   2. mul-1-neg 8.6%

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

8. Simplified 8.6%

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

9. Final simplification 8.6%

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

Alternative 15: 9.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.5%

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

5. Taylor expanded in r around 0 8.6%

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

6. Taylor expanded in r around inf 8.6%

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

   1. associate-*r/ 8.6%

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

   2. *-commutative 8.6%

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

   3. mul-1-neg 8.6%

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

   4. associate-*r* 8.6%

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

8. Simplified 8.6%

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

9. Taylor expanded in r around inf 8.6%

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

    1. associate-*r* 8.6%

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

    2. associate-*r/ 8.6%

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

    3. neg-mul-1 8.6%

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

    4. neg-mul-1 8.6%

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

    5. distribute-frac-neg 8.6%

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

    6. distribute-lft-in 8.6%

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

    7. metadata-eval 8.6%

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

11. Simplified 8.6%

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

12. Taylor expanded in s around 0 8.6%

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

13. Final simplification 8.6%

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

Alternative 16: 9.7% accurate, 15.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.5%

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

5. Taylor expanded in r around 0 8.6%

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

6. Taylor expanded in r around inf 8.6%

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

   1. associate-*r/ 8.6%

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

   2. *-commutative 8.6%

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

   3. mul-1-neg 8.6%

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

   4. associate-*r* 8.6%

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

8. Simplified 8.6%

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

9. Taylor expanded in r around inf 8.6%

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

    1. associate-*r* 8.6%

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

    2. associate-*r/ 8.6%

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

    3. neg-mul-1 8.6%

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

    4. neg-mul-1 8.6%

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

    5. distribute-frac-neg 8.6%

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

    6. distribute-lft-in 8.6%

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

    7. metadata-eval 8.6%

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

11. Simplified 8.6%

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

12. Taylor expanded in r around 0 8.6%

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

13. Final simplification 8.6%

    \[\leadsto \frac{0.125 + \frac{0.125}{\frac{r}{s} + 1}}{s \cdot \left(r \cdot \pi\right)} \]
14. Add Preprocessing

Alternative 17: 9.2% accurate, 33.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.5%

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

5. Taylor expanded in r around 0 8.6%

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

6. Taylor expanded in s around inf 8.2%

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

7. Final simplification 8.2%

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

Alternative 18: 9.2% accurate, 33.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.25) / r) / 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%

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

3. Simplified 99.5%

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

   1. add-sqr-sqrt 99.5%

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

   2. sqrt-unprod 99.5%

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

   3. pow-prod-down 99.5%

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

   4. prod-exp 99.8%

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

   5. metadata-eval 99.8%

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

6. Applied egg-rr 99.8%

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

7. Taylor expanded in s around inf 8.2%

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

   1. associate-/r* 8.2%

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

   2. associate-/r* 8.2%

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

9. Simplified 8.2%

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

10. Final simplification 8.2%

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

Reproduce

herbie shell --seed 2024021 
(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))))