Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.5%

Time: 36.6s

Alternatives: 16

Speedup: N/A×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \frac{0.25 \cdot e^{-\frac{r}{s}}}{r \cdot \left(s \cdot \left(2 \cdot \pi\right)\right)} + \frac{0.75 \cdot e^{\frac{r \cdot -0.3333333333333333}{s}}}{\mathsf{expm1}\left(\mathsf{log1p}\left(s \cdot \left(r \cdot \left(\pi \cdot 6\right)\right)\right)\right)} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/ (* 0.25 (exp (- (/ r s)))) (* r (* s (* 2.0 PI))))
  (/
   (* 0.75 (exp (/ (* r -0.3333333333333333) s)))
   (expm1 (log1p (* s (* r (* PI 6.0))))))))

C

float code(float s, float r) {
	return ((0.25f * expf(-(r / s))) / (r * (s * (2.0f * ((float) M_PI))))) + ((0.75f * expf(((r * -0.3333333333333333f) / s))) / expm1f(log1pf((s * (r * (((float) M_PI) * 6.0f))))));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.25) * exp(Float32(-Float32(r / s)))) / Float32(r * Float32(s * Float32(Float32(2.0) * Float32(pi))))) + Float32(Float32(Float32(0.75) * exp(Float32(Float32(r * Float32(-0.3333333333333333)) / s))) / expm1(log1p(Float32(s * Float32(r * Float32(Float32(pi) * Float32(6.0))))))))
end

Derivation

1. Initial program 99.5%

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

   1. expm1-log1p-u 99.4%

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r\right)\right)}} \]

   2. *-commutative 99.4%

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\left(s \cdot \left(6 \cdot \pi\right)\right)} \cdot r\right)\right)} \]

   3. associate-*l* 99.5%

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{s \cdot \left(\left(6 \cdot \pi\right) \cdot r\right)}\right)\right)} \]

   4. *-commutative 99.5%

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\mathsf{expm1}\left(\mathsf{log1p}\left(s \cdot \left(\color{blue}{\left(\pi \cdot 6\right)} \cdot r\right)\right)\right)} \]

3. Applied egg-rr 99.5%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(s \cdot \left(\left(\pi \cdot 6\right) \cdot r\right)\right)\right)}} \]

4. Taylor expanded in r around inf 99.5%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot \color{blue}{e^{-0.3333333333333333 \cdot \frac{r}{s}}}}{\mathsf{expm1}\left(\mathsf{log1p}\left(s \cdot \left(\left(\pi \cdot 6\right) \cdot r\right)\right)\right)} \]
5. Step-by-step derivation

   1. associate-*r/ 99.5%

      \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\color{blue}{\frac{-0.3333333333333333 \cdot r}{s}}}}{\mathsf{expm1}\left(\mathsf{log1p}\left(s \cdot \left(\left(\pi \cdot 6\right) \cdot r\right)\right)\right)} \]

6. Simplified 99.5%

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot \color{blue}{e^{\frac{-0.3333333333333333 \cdot r}{s}}}}{\mathsf{expm1}\left(\mathsf{log1p}\left(s \cdot \left(\left(\pi \cdot 6\right) \cdot r\right)\right)\right)} \]

7. Final simplification 99.5%

   \[\leadsto \frac{0.25 \cdot e^{-\frac{r}{s}}}{r \cdot \left(s \cdot \left(2 \cdot \pi\right)\right)} + \frac{0.75 \cdot e^{\frac{r \cdot -0.3333333333333333}{s}}}{\mathsf{expm1}\left(\mathsf{log1p}\left(s \cdot \left(r \cdot \left(\pi \cdot 6\right)\right)\right)\right)} \]

Alternative 2: 99.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \left(0.75 \cdot \frac{0.16666666666666666}{s \cdot {\left(\sqrt[3]{\pi}\right)}^{3}}\right) \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r}\right) \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (*
  (* 0.75 (/ 0.16666666666666666 (* s (pow (cbrt PI) 3.0))))
  (+ (/ (exp (/ r (- s))) r) (/ (exp (* -0.3333333333333333 (/ r s))) r))))

C

float code(float s, float r) {
	return (0.75f * (0.16666666666666666f / (s * powf(cbrtf(((float) M_PI)), 3.0f)))) * ((expf((r / -s)) / r) + (expf((-0.3333333333333333f * (r / s))) / r));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(0.75) * Float32(Float32(0.16666666666666666) / Float32(s * (cbrt(Float32(pi)) ^ Float32(3.0))))) * Float32(Float32(exp(Float32(r / Float32(-s))) / r) + Float32(exp(Float32(Float32(-0.3333333333333333) * Float32(r / s))) / r)))
end

Derivation

1. Initial program 99.5%

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

3. Simplified 99.1%

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

   1. associate-/r* 99.1%

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

   2. metadata-eval 99.1%

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

   3. associate-/r* 99.1%

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

   4. clear-num 99.1%

      \[\leadsto \color{blue}{\frac{1}{\frac{6 \cdot \left(\pi \cdot s\right)}{0.75}}} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{{\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}}{r}\right) \]

   5. associate-/r/ 99.0%

      \[\leadsto \color{blue}{\left(\frac{1}{6 \cdot \left(\pi \cdot s\right)} \cdot 0.75\right)} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{{\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}}{r}\right) \]

   6. associate-/r* 99.1%

      \[\leadsto \left(\color{blue}{\frac{\frac{1}{6}}{\pi \cdot s}} \cdot 0.75\right) \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{{\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}}{r}\right) \]

   7. metadata-eval 99.1%

      \[\leadsto \left(\frac{\color{blue}{0.16666666666666666}}{\pi \cdot s} \cdot 0.75\right) \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{{\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}}{r}\right) \]

5. Applied egg-rr 99.1%

   \[\leadsto \color{blue}{\left(\frac{0.16666666666666666}{\pi \cdot s} \cdot 0.75\right)} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{{\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}}{r}\right) \]

6. Taylor expanded in r around inf 99.5%

   \[\leadsto \left(\frac{0.16666666666666666}{\pi \cdot s} \cdot 0.75\right) \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\color{blue}{e^{-0.3333333333333333 \cdot \frac{r}{s}}}}{r}\right) \]
7. Step-by-step derivation

   1. add-cube-cbrt 99.5%

      \[\leadsto \left(\frac{0.16666666666666666}{\color{blue}{\left(\left(\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}\right) \cdot \sqrt[3]{\pi}\right)} \cdot s} \cdot 0.75\right) \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r}\right) \]

   2. pow3 99.5%

      \[\leadsto \left(\frac{0.16666666666666666}{\color{blue}{{\left(\sqrt[3]{\pi}\right)}^{3}} \cdot s} \cdot 0.75\right) \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r}\right) \]

8. Applied egg-rr 99.5%

   \[\leadsto \left(\frac{0.16666666666666666}{\color{blue}{{\left(\sqrt[3]{\pi}\right)}^{3}} \cdot s} \cdot 0.75\right) \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r}\right) \]

9. Final simplification 99.5%

   \[\leadsto \left(0.75 \cdot \frac{0.16666666666666666}{s \cdot {\left(\sqrt[3]{\pi}\right)}^{3}}\right) \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r}\right) \]

Alternative 3: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

2. Taylor expanded in s around 0 99.5%

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

3. Final simplification 99.5%

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

Alternative 4: 99.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r}\right) \cdot \left(0.75 \cdot \left(0.16666666666666666 \cdot \frac{1}{s \cdot \pi}\right)\right) \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (*
  (+ (/ (exp (/ r (- s))) r) (/ (exp (* -0.3333333333333333 (/ r s))) r))
  (* 0.75 (* 0.16666666666666666 (/ 1.0 (* s PI))))))

C

float code(float s, float r) {
	return ((expf((r / -s)) / r) + (expf((-0.3333333333333333f * (r / s))) / r)) * (0.75f * (0.16666666666666666f * (1.0f / (s * ((float) M_PI)))));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(exp(Float32(r / Float32(-s))) / r) + Float32(exp(Float32(Float32(-0.3333333333333333) * Float32(r / s))) / r)) * Float32(Float32(0.75) * Float32(Float32(0.16666666666666666) * Float32(Float32(1.0) / Float32(s * Float32(pi))))))
end

MATLAB

function tmp = code(s, r)
	tmp = ((exp((r / -s)) / r) + (exp((single(-0.3333333333333333) * (r / s))) / r)) * (single(0.75) * (single(0.16666666666666666) * (single(1.0) / (s * single(pi)))));
end

Derivation

1. Initial program 99.5%

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

3. Simplified 99.1%

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

   1. associate-/r* 99.1%

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

   2. metadata-eval 99.1%

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

   3. associate-/r* 99.1%

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

   4. clear-num 99.1%

      \[\leadsto \color{blue}{\frac{1}{\frac{6 \cdot \left(\pi \cdot s\right)}{0.75}}} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{{\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}}{r}\right) \]

   5. associate-/r/ 99.0%

      \[\leadsto \color{blue}{\left(\frac{1}{6 \cdot \left(\pi \cdot s\right)} \cdot 0.75\right)} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{{\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}}{r}\right) \]

   6. associate-/r* 99.1%

      \[\leadsto \left(\color{blue}{\frac{\frac{1}{6}}{\pi \cdot s}} \cdot 0.75\right) \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{{\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}}{r}\right) \]

   7. metadata-eval 99.1%

      \[\leadsto \left(\frac{\color{blue}{0.16666666666666666}}{\pi \cdot s} \cdot 0.75\right) \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{{\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}}{r}\right) \]

5. Applied egg-rr 99.1%

   \[\leadsto \color{blue}{\left(\frac{0.16666666666666666}{\pi \cdot s} \cdot 0.75\right)} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{{\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}}{r}\right) \]

6. Taylor expanded in r around inf 99.5%

   \[\leadsto \left(\frac{0.16666666666666666}{\pi \cdot s} \cdot 0.75\right) \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\color{blue}{e^{-0.3333333333333333 \cdot \frac{r}{s}}}}{r}\right) \]
7. Step-by-step derivation

   1. add-cube-cbrt 99.5%

      \[\leadsto \left(\frac{0.16666666666666666}{\color{blue}{\left(\left(\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}\right) \cdot \sqrt[3]{\pi}\right)} \cdot s} \cdot 0.75\right) \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r}\right) \]

   2. pow3 99.5%

      \[\leadsto \left(\frac{0.16666666666666666}{\color{blue}{{\left(\sqrt[3]{\pi}\right)}^{3}} \cdot s} \cdot 0.75\right) \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r}\right) \]

8. Applied egg-rr 99.5%

   \[\leadsto \left(\frac{0.16666666666666666}{\color{blue}{{\left(\sqrt[3]{\pi}\right)}^{3}} \cdot s} \cdot 0.75\right) \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r}\right) \]
9. Step-by-step derivation

   1. div-inv 99.5%

      \[\leadsto \left(\color{blue}{\left(0.16666666666666666 \cdot \frac{1}{{\left(\sqrt[3]{\pi}\right)}^{3} \cdot s}\right)} \cdot 0.75\right) \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r}\right) \]

   2. rem-cube-cbrt 99.5%

      \[\leadsto \left(\left(0.16666666666666666 \cdot \frac{1}{\color{blue}{\pi} \cdot s}\right) \cdot 0.75\right) \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r}\right) \]

10. Applied egg-rr 99.5%

    \[\leadsto \left(\color{blue}{\left(0.16666666666666666 \cdot \frac{1}{\pi \cdot s}\right)} \cdot 0.75\right) \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r}\right) \]

11. Final simplification 99.5%

    \[\leadsto \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r}\right) \cdot \left(0.75 \cdot \left(0.16666666666666666 \cdot \frac{1}{s \cdot \pi}\right)\right) \]

Alternative 5: 99.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r}\right) \cdot \left(0.75 \cdot \frac{0.16666666666666666}{s \cdot \pi}\right) \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (*
  (+ (/ (exp (/ r (- s))) r) (/ (exp (* -0.3333333333333333 (/ r s))) r))
  (* 0.75 (/ 0.16666666666666666 (* s PI)))))

C

float code(float s, float r) {
	return ((expf((r / -s)) / r) + (expf((-0.3333333333333333f * (r / s))) / r)) * (0.75f * (0.16666666666666666f / (s * ((float) M_PI))));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(exp(Float32(r / Float32(-s))) / r) + Float32(exp(Float32(Float32(-0.3333333333333333) * Float32(r / s))) / r)) * Float32(Float32(0.75) * Float32(Float32(0.16666666666666666) / Float32(s * Float32(pi)))))
end

MATLAB

function tmp = code(s, r)
	tmp = ((exp((r / -s)) / r) + (exp((single(-0.3333333333333333) * (r / s))) / r)) * (single(0.75) * (single(0.16666666666666666) / (s * single(pi))));
end

Derivation

1. Initial program 99.5%

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

3. Simplified 99.1%

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

   1. associate-/r* 99.1%

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

   2. metadata-eval 99.1%

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

   3. associate-/r* 99.1%

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

   4. clear-num 99.1%

      \[\leadsto \color{blue}{\frac{1}{\frac{6 \cdot \left(\pi \cdot s\right)}{0.75}}} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{{\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}}{r}\right) \]

   5. associate-/r/ 99.0%

      \[\leadsto \color{blue}{\left(\frac{1}{6 \cdot \left(\pi \cdot s\right)} \cdot 0.75\right)} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{{\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}}{r}\right) \]

   6. associate-/r* 99.1%

      \[\leadsto \left(\color{blue}{\frac{\frac{1}{6}}{\pi \cdot s}} \cdot 0.75\right) \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{{\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}}{r}\right) \]

   7. metadata-eval 99.1%

      \[\leadsto \left(\frac{\color{blue}{0.16666666666666666}}{\pi \cdot s} \cdot 0.75\right) \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{{\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}}{r}\right) \]

5. Applied egg-rr 99.1%

   \[\leadsto \color{blue}{\left(\frac{0.16666666666666666}{\pi \cdot s} \cdot 0.75\right)} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{{\left(e^{-0.3333333333333333}\right)}^{\left(\frac{r}{s}\right)}}{r}\right) \]

6. Taylor expanded in r around inf 99.5%

   \[\leadsto \left(\frac{0.16666666666666666}{\pi \cdot s} \cdot 0.75\right) \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{\color{blue}{e^{-0.3333333333333333 \cdot \frac{r}{s}}}}{r}\right) \]

7. Final simplification 99.5%

   \[\leadsto \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r}\right) \cdot \left(0.75 \cdot \frac{0.16666666666666666}{s \cdot \pi}\right) \]

Alternative 6: 99.5% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Simplified 99.1%

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

4. Taylor expanded in r around inf 99.4%

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

5. Final simplification 99.4%

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

Alternative 7: 11.8% accurate, 1.4× 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 99.5%

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

3. Simplified 99.1%

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

4. Taylor expanded in r around 0 9.2%

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

5. Taylor expanded in s around inf 8.7%

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

   1. *-commutative 8.7%

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

   2. add-sqr-sqrt 8.7%

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

   3. sqrt-unprod 8.5%

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

   4. sqr-neg 8.5%

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

   5. sqrt-unprod -0.0%

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

   6. add-sqr-sqrt 4.5%

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

   7. distribute-rgt-neg-in 4.5%

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

   8. *-commutative 4.5%

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

   9. distribute-rgt-neg-in 4.5%

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

   10. log1p-expm1-u 7.1%

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

   11. *-commutative 7.1%

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

   12. distribute-lft-neg-in 7.1%

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

   13. distribute-lft-neg-in 7.1%

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

   14. add-sqr-sqrt -0.0%

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

   15. sqrt-unprod 10.9%

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

   16. sqr-neg 10.9%

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

   17. sqrt-unprod 11.1%

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

   18. add-sqr-sqrt 11.1%

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

   19. *-commutative 11.1%

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

   20. associate-*l* 11.1%

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

   21. *-commutative 11.1%

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

7. Applied egg-rr 11.1%

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

8. Final simplification 11.1%

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

Alternative 8: 9.6% accurate, 2.0× speedup

Math

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

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(0.125) * Float32(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 99.5%

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

3. Simplified 99.1%

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

4. Taylor expanded in r around 0 9.2%

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

5. Taylor expanded in r around inf 9.2%

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

   1. mul-1-neg 9.2%

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

7. Simplified 9.2%

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

8. Final simplification 9.2%

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

Alternative 9: 9.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Simplified 99.1%

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

4. Taylor expanded in r around 0 9.2%

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

5. Taylor expanded in r around inf 9.2%

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

   1. associate-*r/ 9.2%

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

   2. *-commutative 9.2%

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

   3. times-frac 9.2%

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

   4. associate-/l/ 9.2%

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

   5. associate-*r/ 9.2%

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

   6. mul-1-neg 9.2%

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

7. Simplified 9.2%

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

8. Taylor expanded in s around 0 9.2%

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

   1. associate-/r* 9.2%

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

   2. *-commutative 9.2%

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

   3. associate-/r* 9.2%

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

   4. mul-1-neg 9.2%

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

   5. associate-*r* 9.2%

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

10. Simplified 9.2%

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

11. Final simplification 9.2%

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

Alternative 10: 9.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Simplified 99.1%

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

4. Taylor expanded in r around 0 9.2%

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

5. Taylor expanded in s around 0 9.2%

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

6. Taylor expanded in r around inf 9.2%

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

   1. associate-*r/ 9.2%

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

   2. *-commutative 9.2%

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

   3. times-frac 9.2%

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

   4. associate-*r/ 9.2%

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

   5. mul-1-neg 9.2%

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

8. Simplified 9.2%

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

9. Final simplification 9.2%

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

Alternative 11: 9.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Simplified 99.1%

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

4. Taylor expanded in r around 0 9.2%

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

5. Taylor expanded in r around inf 9.2%

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

   1. associate-*r/ 9.2%

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

   2. *-commutative 9.2%

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

   3. times-frac 9.2%

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

   4. associate-/l/ 9.2%

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

   5. associate-*r/ 9.2%

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

   6. mul-1-neg 9.2%

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

7. Simplified 9.2%

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

8. Final simplification 9.2%

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

Alternative 12: 9.1% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Simplified 99.1%

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

4. Taylor expanded in r around 0 9.2%

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

5. Taylor expanded in s around inf 8.7%

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

   1. associate-/r* 8.7%

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

7. Simplified 8.7%

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

   1. div-inv 8.7%

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

   2. *-commutative 8.7%

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

9. Applied egg-rr 8.7%

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

10. Final simplification 8.7%

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

Alternative 13: 9.1% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Simplified 99.1%

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

4. Taylor expanded in r around 0 9.2%

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

5. Taylor expanded in r around inf 9.2%

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

   1. associate-*r/ 9.2%

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

   2. *-commutative 9.2%

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

   3. times-frac 9.2%

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

   4. associate-/l/ 9.2%

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

   5. associate-*r/ 9.2%

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

   6. mul-1-neg 9.2%

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

7. Simplified 9.2%

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

8. Taylor expanded in r around 0 8.7%

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

9. Final simplification 8.7%

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

Alternative 14: 9.1% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Simplified 99.1%

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

4. Taylor expanded in r around 0 9.2%

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

5. Taylor expanded in s around inf 8.7%

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

   1. clear-num 8.7%

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

   2. inv-pow 8.7%

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

   3. associate-*r* 8.7%

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

   4. associate-/l* 8.7%

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

7. Applied egg-rr 8.7%

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

   1. unpow-1 8.7%

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

   2. associate-/l* 8.7%

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

9. Simplified 8.7%

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

10. Final simplification 8.7%

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

Alternative 15: 9.1% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.5%

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

3. Simplified 99.1%

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

4. Taylor expanded in r around 0 9.2%

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

5. Taylor expanded in s around inf 8.7%

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

6. Final simplification 8.7%

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

Alternative 16: 9.1% 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 99.5%

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

3. Simplified 99.1%

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

4. Taylor expanded in r around 0 9.2%

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

5. Taylor expanded in s around inf 8.7%

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

   1. associate-/r* 8.7%

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

7. Simplified 8.7%

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

8. Taylor expanded in r around 0 8.7%

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

   1. associate-/r* 8.7%

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

   2. *-commutative 8.7%

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

   3. associate-/r* 8.7%

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

   4. associate-*r* 8.7%

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

10. Simplified 8.7%

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

11. Final simplification 8.7%

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

Reproduce

herbie shell --seed 2023309 
(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))))