Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.6%

Time: 22.4s

Alternatives: 11

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: 99.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{r}{-s}} + {\left(e^{-0.6666666666666666}\right)}^{\left(\frac{\frac{r}{s}}{2}\right)}}{r} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (*
  (/ 0.125 (* s PI))
  (/ (+ (exp (/ r (- s))) (pow (exp -0.6666666666666666) (/ (/ r s) 2.0))) r)))

C

float code(float s, float r) {
	return (0.125f / (s * ((float) M_PI))) * ((expf((r / -s)) + powf(expf(-0.6666666666666666f), ((r / s) / 2.0f))) / r);
}

Julia

function code(s, r)
	return Float32(Float32(Float32(0.125) / Float32(s * Float32(pi))) * Float32(Float32(exp(Float32(r / Float32(-s))) + (exp(Float32(-0.6666666666666666)) ^ Float32(Float32(r / s) / Float32(2.0)))) / r))
end

MATLAB

function tmp = code(s, r)
	tmp = (single(0.125) / (s * single(pi))) * ((exp((r / -s)) + (exp(single(-0.6666666666666666)) ^ ((r / s) / single(2.0)))) / r);
end

Derivation

1. Initial program 99.4%

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

3. Simplified 99.1%

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

5. Taylor expanded in r around inf 99.4%

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

   1. pow-exp 99.1%

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

   2. sqr-pow 99.1%

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

   3. pow-prod-down 99.1%

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

   4. prod-exp 99.5%

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

   5. metadata-eval 99.5%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \frac{e^{-1 \cdot \frac{r}{s}} + {\left(e^{\color{blue}{-0.6666666666666666}}\right)}^{\left(\frac{\frac{r}{s}}{2}\right)}}{r} \]

7. Applied egg-rr 99.5%

   \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \frac{e^{-1 \cdot \frac{r}{s}} + \color{blue}{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{\frac{r}{s}}{2}\right)}}}{r} \]
8. Step-by-step derivation

   1. mul-1-neg 99.5%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\color{blue}{-\frac{r}{s}}} + {\left(e^{-0.6666666666666666}\right)}^{\left(\frac{\frac{r}{s}}{2}\right)}}{r} \]

   2. distribute-frac-neg 99.5%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\color{blue}{\frac{-r}{s}}} + {\left(e^{-0.6666666666666666}\right)}^{\left(\frac{\frac{r}{s}}{2}\right)}}{r} \]

9. Applied egg-rr 99.5%

   \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\color{blue}{\frac{-r}{s}}} + {\left(e^{-0.6666666666666666}\right)}^{\left(\frac{\frac{r}{s}}{2}\right)}}{r} \]

10. Final simplification 99.5%

    \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{r}{-s}} + {\left(e^{-0.6666666666666666}\right)}^{\left(\frac{\frac{r}{s}}{2}\right)}}{r} \]
11. Add Preprocessing

Alternative 2: 99.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{r}{-s}} + e^{\frac{\frac{r}{s}}{-3}}}{r} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

3. Simplified 99.1%

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

5. Taylor expanded in r around inf 99.4%

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

   1. metadata-eval 99.4%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \frac{e^{-1 \cdot \frac{r}{s}} + e^{\color{blue}{\frac{-1}{3}} \cdot \frac{r}{s}}}{r} \]

   2. times-frac 99.5%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \frac{e^{-1 \cdot \frac{r}{s}} + e^{\color{blue}{\frac{-1 \cdot r}{3 \cdot s}}}}{r} \]

   3. neg-mul-1 99.5%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \frac{e^{-1 \cdot \frac{r}{s}} + e^{\frac{\color{blue}{-r}}{3 \cdot s}}}{r} \]

   4. *-commutative 99.5%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \frac{e^{-1 \cdot \frac{r}{s}} + e^{\frac{-r}{\color{blue}{s \cdot 3}}}}{r} \]

   5. associate-/r* 99.5%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \frac{e^{-1 \cdot \frac{r}{s}} + e^{\color{blue}{\frac{\frac{-r}{s}}{3}}}}{r} \]

   6. frac-2neg 99.5%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \frac{e^{-1 \cdot \frac{r}{s}} + e^{\color{blue}{\frac{-\frac{-r}{s}}{-3}}}}{r} \]

   7. add-sqr-sqrt -0.0%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \frac{e^{-1 \cdot \frac{r}{s}} + e^{\frac{-\frac{\color{blue}{\sqrt{-r} \cdot \sqrt{-r}}}{s}}{-3}}}{r} \]

   8. sqrt-unprod 7.0%

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

   9. sqr-neg 7.0%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \frac{e^{-1 \cdot \frac{r}{s}} + e^{\frac{-\frac{\sqrt{\color{blue}{r \cdot r}}}{s}}{-3}}}{r} \]

   10. sqrt-unprod 7.0%

       \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \frac{e^{-1 \cdot \frac{r}{s}} + e^{\frac{-\frac{\color{blue}{\sqrt{r} \cdot \sqrt{r}}}{s}}{-3}}}{r} \]

   11. add-sqr-sqrt 7.0%

       \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \frac{e^{-1 \cdot \frac{r}{s}} + e^{\frac{-\frac{\color{blue}{r}}{s}}{-3}}}{r} \]

   12. distribute-frac-neg 7.0%

       \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \frac{e^{-1 \cdot \frac{r}{s}} + e^{\frac{\color{blue}{\frac{-r}{s}}}{-3}}}{r} \]

   13. add-sqr-sqrt -0.0%

       \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \frac{e^{-1 \cdot \frac{r}{s}} + e^{\frac{\frac{\color{blue}{\sqrt{-r} \cdot \sqrt{-r}}}{s}}{-3}}}{r} \]

   14. sqrt-unprod 99.5%

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

   15. sqr-neg 99.5%

       \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \frac{e^{-1 \cdot \frac{r}{s}} + e^{\frac{\frac{\sqrt{\color{blue}{r \cdot r}}}{s}}{-3}}}{r} \]

   16. sqrt-unprod 99.3%

       \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \frac{e^{-1 \cdot \frac{r}{s}} + e^{\frac{\frac{\color{blue}{\sqrt{r} \cdot \sqrt{r}}}{s}}{-3}}}{r} \]

   17. add-sqr-sqrt 99.5%

       \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \frac{e^{-1 \cdot \frac{r}{s}} + e^{\frac{\frac{\color{blue}{r}}{s}}{-3}}}{r} \]

   18. metadata-eval 99.5%

       \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \frac{e^{-1 \cdot \frac{r}{s}} + e^{\frac{\frac{r}{s}}{\color{blue}{-3}}}}{r} \]

7. Applied egg-rr 99.5%

   \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \frac{e^{-1 \cdot \frac{r}{s}} + e^{\color{blue}{\frac{\frac{r}{s}}{-3}}}}{r} \]

8. Final simplification 99.5%

   \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{r}{-s}} + e^{\frac{\frac{r}{s}}{-3}}}{r} \]
9. Add Preprocessing

Alternative 3: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

3. Simplified 99.1%

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

5. Taylor expanded in r around inf 99.4%

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

   1. mul-1-neg 99.5%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\color{blue}{-\frac{r}{s}}} + {\left(e^{-0.6666666666666666}\right)}^{\left(\frac{\frac{r}{s}}{2}\right)}}{r} \]

   2. distribute-frac-neg 99.5%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\color{blue}{\frac{-r}{s}}} + {\left(e^{-0.6666666666666666}\right)}^{\left(\frac{\frac{r}{s}}{2}\right)}}{r} \]

7. Applied egg-rr 99.4%

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

8. Final simplification 99.4%

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

Alternative 4: 44.2% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.4%

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

3. Simplified 99.1%

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

5. Taylor expanded in r around 0 8.1%

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

6. Taylor expanded in s around 0 8.1%

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

   1. *-commutative 8.1%

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

   2. associate-*l* 8.1%

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

   3. *-commutative 8.1%

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

8. Simplified 8.1%

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

   1. log1p-expm1-u 41.3%

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

10. Applied egg-rr 41.3%

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

11. Final simplification 41.3%

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

Alternative 5: 15.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.25}{\frac{r}{s} + 1}}{r \cdot \left(s \cdot \left(\pi \cdot 2\right)\right)} + \frac{0.75 \cdot e^{\frac{r}{s \cdot \left(-3\right)}}}{6 \cdot \left(s \cdot \left(\pi \cdot r\right)\right)} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/ (/ 0.25 (+ (/ r s) 1.0)) (* r (* s (* PI 2.0))))
  (/ (* 0.75 (exp (/ r (* s (- 3.0))))) (* 6.0 (* s (* PI r))))))

C

float code(float s, float r) {
	return ((0.25f / ((r / s) + 1.0f)) / (r * (s * (((float) M_PI) * 2.0f)))) + ((0.75f * expf((r / (s * -3.0f)))) / (6.0f * (s * (((float) M_PI) * r))));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.25) / Float32(Float32(r / s) + Float32(1.0))) / Float32(r * Float32(s * Float32(Float32(pi) * Float32(2.0))))) + Float32(Float32(Float32(0.75) * exp(Float32(r / Float32(s * Float32(-Float32(3.0)))))) / Float32(Float32(6.0) * Float32(s * Float32(Float32(pi) * r)))))
end

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.25) / ((r / s) + single(1.0))) / (r * (s * (single(pi) * single(2.0))))) + ((single(0.75) * exp((r / (s * -single(3.0))))) / (single(6.0) * (s * (single(pi) * r))));
end

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in s around 0 99.4%

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

   1. *-commutative 99.4%

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

   2. associate-*l* 99.4%

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

   3. *-commutative 99.4%

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

5. Simplified 99.4%

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

6. Taylor expanded in r around inf 99.4%

   \[\leadsto \frac{\color{blue}{0.25 \cdot e^{-1 \cdot \frac{r}{s}}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{6 \cdot \left(s \cdot \left(r \cdot \pi\right)\right)} \]
7. Step-by-step derivation

   1. neg-mul-1 99.4%

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

   2. rec-exp 99.4%

      \[\leadsto \frac{0.25 \cdot \color{blue}{\frac{1}{e^{\frac{r}{s}}}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{6 \cdot \left(s \cdot \left(r \cdot \pi\right)\right)} \]

   3. associate-*r/ 99.4%

      \[\leadsto \frac{\color{blue}{\frac{0.25 \cdot 1}{e^{\frac{r}{s}}}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{6 \cdot \left(s \cdot \left(r \cdot \pi\right)\right)} \]

   4. metadata-eval 99.4%

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

8. Simplified 99.4%

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

9. Taylor expanded in r around 0 14.7%

   \[\leadsto \frac{\frac{0.25}{\color{blue}{1 + \frac{r}{s}}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{6 \cdot \left(s \cdot \left(r \cdot \pi\right)\right)} \]

10. Final simplification 14.7%

    \[\leadsto \frac{\frac{0.25}{\frac{r}{s} + 1}}{r \cdot \left(s \cdot \left(\pi \cdot 2\right)\right)} + \frac{0.75 \cdot e^{\frac{r}{s \cdot \left(-3\right)}}}{6 \cdot \left(s \cdot \left(\pi \cdot r\right)\right)} \]
11. Add Preprocessing

Alternative 6: 10.0% accurate, 8.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

3. Simplified 99.1%

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

5. Taylor expanded in r around 0 8.6%

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

6. Taylor expanded in s around -inf 8.6%

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

7. Final simplification 8.6%

   \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{\frac{r}{s} \cdot -0.3333333333333333 + 1}{r} - \left(\frac{1 + \frac{r}{s} \cdot -0.5}{s} + \frac{-1}{r}\right)\right) \]
8. Add Preprocessing

Alternative 7: 10.2% accurate, 9.2× speedup

Math

\[\begin{array}{l} \\ \frac{0.125}{s \cdot \pi} \cdot \left(\frac{\frac{r \cdot 0.05555555555555555 + r \cdot 0.5}{s} - 1.3333333333333333}{s} + 2 \cdot \frac{1}{r}\right) \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (*
  (/ 0.125 (* s PI))
  (+
   (/ (- (/ (+ (* r 0.05555555555555555) (* r 0.5)) s) 1.3333333333333333) s)
   (* 2.0 (/ 1.0 r)))))

C

float code(float s, float r) {
	return (0.125f / (s * ((float) M_PI))) * ((((((r * 0.05555555555555555f) + (r * 0.5f)) / s) - 1.3333333333333333f) / s) + (2.0f * (1.0f / r)));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(0.125) / Float32(s * Float32(pi))) * Float32(Float32(Float32(Float32(Float32(Float32(r * Float32(0.05555555555555555)) + Float32(r * Float32(0.5))) / s) - Float32(1.3333333333333333)) / s) + Float32(Float32(2.0) * Float32(Float32(1.0) / r))))
end

MATLAB

function tmp = code(s, r)
	tmp = (single(0.125) / (s * single(pi))) * ((((((r * single(0.05555555555555555)) + (r * single(0.5))) / s) - single(1.3333333333333333)) / s) + (single(2.0) * (single(1.0) / r)));
end

Derivation

1. Initial program 99.4%

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

3. Simplified 99.1%

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

5. Taylor expanded in s around -inf 8.6%

   \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \color{blue}{\left(-1 \cdot \frac{1.3333333333333333 + -1 \cdot \frac{0.05555555555555555 \cdot r + 0.5 \cdot r}{s}}{s} + 2 \cdot \frac{1}{r}\right)} \]

6. Final simplification 8.6%

   \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{\frac{r \cdot 0.05555555555555555 + r \cdot 0.5}{s} - 1.3333333333333333}{s} + 2 \cdot \frac{1}{r}\right) \]
7. Add Preprocessing

Alternative 8: 10.2% accurate, 12.2× speedup

Math

\[\begin{array}{l} \\ \frac{0.125}{s \cdot \pi} \cdot \left(\frac{2}{r} + \frac{\frac{r \cdot 0.5555555555555556}{s} - 1.3333333333333333}{s}\right) \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (*
  (/ 0.125 (* s PI))
  (+ (/ 2.0 r) (/ (- (/ (* r 0.5555555555555556) s) 1.3333333333333333) s))))

C

float code(float s, float r) {
	return (0.125f / (s * ((float) M_PI))) * ((2.0f / r) + ((((r * 0.5555555555555556f) / s) - 1.3333333333333333f) / s));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(0.125) / Float32(s * Float32(pi))) * Float32(Float32(Float32(2.0) / r) + Float32(Float32(Float32(Float32(r * Float32(0.5555555555555556)) / s) - Float32(1.3333333333333333)) / s)))
end

MATLAB

function tmp = code(s, r)
	tmp = (single(0.125) / (s * single(pi))) * ((single(2.0) / r) + ((((r * single(0.5555555555555556)) / s) - single(1.3333333333333333)) / s));
end

Derivation

1. Initial program 99.4%

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

3. Simplified 99.1%

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

5. Taylor expanded in r around inf 99.4%

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

   1. pow-exp 99.1%

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

   2. sqr-pow 99.1%

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

   3. pow-prod-down 99.1%

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

   4. prod-exp 99.5%

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

   5. metadata-eval 99.5%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \frac{e^{-1 \cdot \frac{r}{s}} + {\left(e^{\color{blue}{-0.6666666666666666}}\right)}^{\left(\frac{\frac{r}{s}}{2}\right)}}{r} \]

7. Applied egg-rr 99.5%

   \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \frac{e^{-1 \cdot \frac{r}{s}} + \color{blue}{{\left(e^{-0.6666666666666666}\right)}^{\left(\frac{\frac{r}{s}}{2}\right)}}}{r} \]

8. Taylor expanded in s around -inf 8.6%

   \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \color{blue}{\left(-1 \cdot \frac{1.3333333333333333 + -1 \cdot \frac{0.05555555555555555 \cdot r + 0.5 \cdot r}{s}}{s} + 2 \cdot \frac{1}{r}\right)} \]
9. Step-by-step derivation

   1. +-commutative 8.6%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \color{blue}{\left(2 \cdot \frac{1}{r} + -1 \cdot \frac{1.3333333333333333 + -1 \cdot \frac{0.05555555555555555 \cdot r + 0.5 \cdot r}{s}}{s}\right)} \]

   2. mul-1-neg 8.6%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(2 \cdot \frac{1}{r} + \color{blue}{\left(-\frac{1.3333333333333333 + -1 \cdot \frac{0.05555555555555555 \cdot r + 0.5 \cdot r}{s}}{s}\right)}\right) \]

   3. unsub-neg 8.6%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \color{blue}{\left(2 \cdot \frac{1}{r} - \frac{1.3333333333333333 + -1 \cdot \frac{0.05555555555555555 \cdot r + 0.5 \cdot r}{s}}{s}\right)} \]

   4. associate-*r/ 8.6%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\color{blue}{\frac{2 \cdot 1}{r}} - \frac{1.3333333333333333 + -1 \cdot \frac{0.05555555555555555 \cdot r + 0.5 \cdot r}{s}}{s}\right) \]

   5. metadata-eval 8.6%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{\color{blue}{2}}{r} - \frac{1.3333333333333333 + -1 \cdot \frac{0.05555555555555555 \cdot r + 0.5 \cdot r}{s}}{s}\right) \]

   6. mul-1-neg 8.6%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{2}{r} - \frac{1.3333333333333333 + \color{blue}{\left(-\frac{0.05555555555555555 \cdot r + 0.5 \cdot r}{s}\right)}}{s}\right) \]

   7. unsub-neg 8.6%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{2}{r} - \frac{\color{blue}{1.3333333333333333 - \frac{0.05555555555555555 \cdot r + 0.5 \cdot r}{s}}}{s}\right) \]

   8. distribute-rgt-out 8.6%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{2}{r} - \frac{1.3333333333333333 - \frac{\color{blue}{r \cdot \left(0.05555555555555555 + 0.5\right)}}{s}}{s}\right) \]

   9. metadata-eval 8.6%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{2}{r} - \frac{1.3333333333333333 - \frac{r \cdot \color{blue}{0.5555555555555556}}{s}}{s}\right) \]

10. Simplified 8.6%

    \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \color{blue}{\left(\frac{2}{r} - \frac{1.3333333333333333 - \frac{r \cdot 0.5555555555555556}{s}}{s}\right)} \]

11. Final simplification 8.6%

    \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{2}{r} + \frac{\frac{r \cdot 0.5555555555555556}{s} - 1.3333333333333333}{s}\right) \]
12. Add Preprocessing

Alternative 9: 10.2% accurate, 12.2× speedup

Math

\[\begin{array}{l} \\ \frac{0.125}{s \cdot \pi} \cdot \left(\frac{2}{r} + \frac{r \cdot \frac{0.5555555555555556}{s} - 1.3333333333333333}{s}\right) \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (*
  (/ 0.125 (* s PI))
  (+ (/ 2.0 r) (/ (- (* r (/ 0.5555555555555556 s)) 1.3333333333333333) s))))

C

float code(float s, float r) {
	return (0.125f / (s * ((float) M_PI))) * ((2.0f / r) + (((r * (0.5555555555555556f / s)) - 1.3333333333333333f) / s));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(0.125) / Float32(s * Float32(pi))) * Float32(Float32(Float32(2.0) / r) + Float32(Float32(Float32(r * Float32(Float32(0.5555555555555556) / s)) - Float32(1.3333333333333333)) / s)))
end

MATLAB

function tmp = code(s, r)
	tmp = (single(0.125) / (s * single(pi))) * ((single(2.0) / r) + (((r * (single(0.5555555555555556) / s)) - single(1.3333333333333333)) / s));
end

Derivation

1. Initial program 99.4%

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

3. Simplified 99.1%

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

5. Taylor expanded in s around -inf 8.6%

   \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \color{blue}{\left(-1 \cdot \frac{1.3333333333333333 + -1 \cdot \frac{0.05555555555555555 \cdot r + 0.5 \cdot r}{s}}{s} + 2 \cdot \frac{1}{r}\right)} \]
6. Step-by-step derivation

   1. +-commutative 8.6%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \color{blue}{\left(2 \cdot \frac{1}{r} + -1 \cdot \frac{1.3333333333333333 + -1 \cdot \frac{0.05555555555555555 \cdot r + 0.5 \cdot r}{s}}{s}\right)} \]

   2. mul-1-neg 8.6%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(2 \cdot \frac{1}{r} + \color{blue}{\left(-\frac{1.3333333333333333 + -1 \cdot \frac{0.05555555555555555 \cdot r + 0.5 \cdot r}{s}}{s}\right)}\right) \]

   3. unsub-neg 8.6%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \color{blue}{\left(2 \cdot \frac{1}{r} - \frac{1.3333333333333333 + -1 \cdot \frac{0.05555555555555555 \cdot r + 0.5 \cdot r}{s}}{s}\right)} \]

   4. associate-*r/ 8.6%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\color{blue}{\frac{2 \cdot 1}{r}} - \frac{1.3333333333333333 + -1 \cdot \frac{0.05555555555555555 \cdot r + 0.5 \cdot r}{s}}{s}\right) \]

   5. metadata-eval 8.6%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{\color{blue}{2}}{r} - \frac{1.3333333333333333 + -1 \cdot \frac{0.05555555555555555 \cdot r + 0.5 \cdot r}{s}}{s}\right) \]

   6. mul-1-neg 8.6%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{2}{r} - \frac{1.3333333333333333 + \color{blue}{\left(-\frac{0.05555555555555555 \cdot r + 0.5 \cdot r}{s}\right)}}{s}\right) \]

   7. unsub-neg 8.6%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{2}{r} - \frac{\color{blue}{1.3333333333333333 - \frac{0.05555555555555555 \cdot r + 0.5 \cdot r}{s}}}{s}\right) \]

   8. distribute-rgt-out 8.6%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{2}{r} - \frac{1.3333333333333333 - \frac{\color{blue}{r \cdot \left(0.05555555555555555 + 0.5\right)}}{s}}{s}\right) \]

   9. metadata-eval 8.6%

      \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{2}{r} - \frac{1.3333333333333333 - \frac{r \cdot \color{blue}{0.5555555555555556}}{s}}{s}\right) \]

   10. associate-/l* 8.6%

       \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{2}{r} - \frac{1.3333333333333333 - \color{blue}{r \cdot \frac{0.5555555555555556}{s}}}{s}\right) \]

7. Simplified 8.6%

   \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \color{blue}{\left(\frac{2}{r} - \frac{1.3333333333333333 - r \cdot \frac{0.5555555555555556}{s}}{s}\right)} \]

8. Final simplification 8.6%

   \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{2}{r} + \frac{r \cdot \frac{0.5555555555555556}{s} - 1.3333333333333333}{s}\right) \]
9. Add Preprocessing

Alternative 10: 9.1% accurate, 21.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{r}{\frac{0.125}{s \cdot \pi} \cdot 2}} \end{array} \]

FPCore

(FPCore (s r) :precision binary32 (/ 1.0 (/ r (* (/ 0.125 (* s PI)) 2.0))))

C

float code(float s, float r) {
	return 1.0f / (r / ((0.125f / (s * ((float) M_PI))) * 2.0f));
}

Julia

function code(s, r)
	return Float32(Float32(1.0) / Float32(r / Float32(Float32(Float32(0.125) / Float32(s * Float32(pi))) * Float32(2.0))))
end

MATLAB

function tmp = code(s, r)
	tmp = single(1.0) / (r / ((single(0.125) / (s * single(pi))) * single(2.0)));
end

Derivation

1. Initial program 99.4%

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

3. Simplified 99.1%

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

5. Taylor expanded in r around 0 8.1%

   \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \color{blue}{\frac{2}{r}} \]
6. Step-by-step derivation

   1. associate-*r/ 8.1%

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

   2. clear-num 8.1%

      \[\leadsto \color{blue}{\frac{1}{\frac{r}{\frac{0.125}{s \cdot \pi} \cdot 2}}} \]

7. Applied egg-rr 8.1%

   \[\leadsto \color{blue}{\frac{1}{\frac{r}{\frac{0.125}{s \cdot \pi} \cdot 2}}} \]
8. Add Preprocessing

Alternative 11: 9.1% accurate, 33.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.4%

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

3. Simplified 99.1%

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

5. Taylor expanded in r around 0 8.1%

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

6. Taylor expanded in s around 0 8.1%

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

   1. *-commutative 8.1%

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

   2. associate-*l* 8.1%

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

   3. *-commutative 8.1%

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

8. Simplified 8.1%

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

9. Final simplification 8.1%

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

Reproduce

herbie shell --seed 2024114 
(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))))