Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.6%

Time: 12.1s

Alternatives: 12

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, 1.0× speedup

Math

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

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/ (/ 0.25 (exp (/ r s))) (* r (* s (* 2.0 PI))))
  (/ (* 0.75 (exp (/ r (* 3.0 (- s))))) (* r (* s (* 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 / (3.0f * -s)))) / (r * (s * (((float) M_PI) * 6.0f))));
}

Julia

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in r around inf 99.6%

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

   1. neg-mul-1 99.6%

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

   2. rec-exp 99.6%

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

   3. associate-*r/ 99.6%

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

   4. metadata-eval 99.6%

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

5. Simplified 99.6%

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

6. Final simplification 99.6%

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

Alternative 2: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Simplified 99.2%

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

   1. pow-exp 99.5%

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

   2. associate-*r/ 99.5%

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

6. Applied egg-rr 99.5%

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

   1. *-un-lft-identity 99.5%

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

   2. associate-/r* 99.6%

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

8. Applied egg-rr 99.6%

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

   1. *-un-lft-identity 99.6%

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

10. Applied egg-rr 99.6%

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

11. Final simplification 99.6%

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

Alternative 3: 99.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

float code(float s, float r) {
	return (0.125f / (s * ((float) M_PI))) * ((expf((r / -s)) / r) + (expf(((r / -3.0f) / s)) / 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(exp(Float32(Float32(r / Float32(-3.0)) / s)) / r)))
end

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Simplified 99.2%

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

   1. pow-exp 99.5%

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

   2. associate-*r/ 99.5%

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

6. Applied egg-rr 99.5%

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

   1. add-sqr-sqrt 99.4%

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

   2. sqrt-unprod 99.5%

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

   3. sqr-neg 99.5%

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

   4. sqrt-unprod -0.0%

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

   5. add-sqr-sqrt 10.1%

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

   6. distribute-rgt-neg-in 10.1%

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

   7. *-commutative 10.1%

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

   8. distribute-rgt-neg-in 10.1%

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

   9. add-sqr-sqrt 10.1%

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

   10. sqrt-unprod 10.1%

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

   11. sqr-neg 10.1%

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

   12. sqrt-unprod -0.0%

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

   13. add-sqr-sqrt 99.5%

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

   14. metadata-eval 99.5%

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

   15. metadata-eval 99.5%

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

   16. div-inv 99.5%

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

   17. frac-2neg 99.5%

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

   18. remove-double-neg 99.5%

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

   19. metadata-eval 99.5%

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

8. Applied egg-rr 99.5%

   \[\leadsto \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{e^{\frac{\color{blue}{\frac{r}{-3}}}{s}}}{r}\right) \]
9. Add Preprocessing

Alternative 4: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Simplified 99.2%

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

   1. pow-exp 99.5%

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

   2. associate-*r/ 99.5%

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

6. Applied egg-rr 99.5%

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

7. Final simplification 99.5%

   \[\leadsto \left(\frac{e^{\frac{r}{-s}}}{r} + \frac{e^{\frac{r \cdot -0.3333333333333333}{s}}}{r}\right) \cdot \frac{0.125}{s \cdot \pi} \]
8. 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(r / Float32(-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 99.6%

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

3. Simplified 99.2%

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

5. Taylor expanded in r around inf 99.5%

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

6. Final simplification 99.5%

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

Alternative 6: 43.9% 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 99.6%

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

3. Simplified 99.2%

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

5. Taylor expanded in s around inf 11.2%

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

   1. div-inv 11.2%

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

   2. *-commutative 11.2%

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

   3. *-commutative 11.2%

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

   4. associate-*l* 11.2%

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

   5. *-commutative 11.2%

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

7. Applied egg-rr 11.2%

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

   1. associate-*r/ 11.2%

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

   2. metadata-eval 11.2%

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

9. Simplified 11.2%

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

10. Taylor expanded in r around 0 11.2%

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

    1. *-commutative 11.2%

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

    2. associate-*l* 11.2%

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

12. Simplified 11.2%

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

    1. log1p-expm1-u 44.8%

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

14. Applied egg-rr 44.8%

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

15. Final simplification 44.8%

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

Alternative 7: 15.2% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in r around inf 99.6%

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

   1. neg-mul-1 99.6%

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

   2. rec-exp 99.6%

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

   3. associate-*r/ 99.6%

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

   4. metadata-eval 99.6%

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

5. Simplified 99.6%

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

6. Taylor expanded in r around 0 18.6%

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

7. Final simplification 18.6%

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

Alternative 8: 9.9% accurate, 1.8× speedup

Math

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

FPCore

(FPCore (s r)
 :precision binary32
 (*
  (/ (/ 0.125 s) PI)
  (+
   (/ (exp (/ (* r -0.3333333333333333) s)) 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)) * ((expf(((r * -0.3333333333333333f) / s)) / r) + (((-1.0f - ((r / s) * -0.5f)) / s) + (1.0f / r)));
}

Julia

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

Derivation

1. Initial program 99.6%

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

3. Simplified 99.2%

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

   1. pow-exp 99.5%

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

   2. associate-*r/ 99.5%

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

6. Applied egg-rr 99.5%

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

   1. *-un-lft-identity 99.5%

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

   2. associate-/r* 99.6%

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

8. Applied egg-rr 99.6%

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

   1. *-un-lft-identity 99.6%

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

10. Applied egg-rr 99.6%

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

11. Taylor expanded in s around -inf 12.8%

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

12. Final simplification 12.8%

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

Alternative 9: 9.9% accurate, 1.8× speedup

Math

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

FPCore

(FPCore (s r)
 :precision binary32
 (*
  (/ 0.125 (* s PI))
  (+
   (/ (exp (/ (* r -0.3333333333333333) s)) 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))) * ((expf(((r * -0.3333333333333333f) / s)) / 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(exp(Float32(Float32(r * Float32(-0.3333333333333333)) / s)) / 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))) * ((exp(((r * single(-0.3333333333333333)) / s)) / r) + (((single(-1.0) - ((r / s) * single(-0.5))) / s) + (single(1.0) / r)));
end

Derivation

1. Initial program 99.6%

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

3. Simplified 99.2%

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

   1. pow-exp 99.5%

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

   2. associate-*r/ 99.5%

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

6. Applied egg-rr 99.5%

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

7. Taylor expanded in s around -inf 12.7%

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

8. Final simplification 12.7%

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

Alternative 10: 9.9% accurate, 11.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.25}{r \cdot \pi} - \frac{\frac{\frac{r}{\pi} \cdot -0.06944444444444445}{s} + \frac{0.16666666666666666}{\pi}}{s}}{s} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (/
  (-
   (/ 0.25 (* r PI))
   (/
    (+ (/ (* (/ r PI) -0.06944444444444445) s) (/ 0.16666666666666666 PI))
    s))
  s))

C

float code(float s, float r) {
	return ((0.25f / (r * ((float) M_PI))) - (((((r / ((float) M_PI)) * -0.06944444444444445f) / s) + (0.16666666666666666f / ((float) M_PI))) / s)) / s;
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.25) / Float32(r * Float32(pi))) - Float32(Float32(Float32(Float32(Float32(r / Float32(pi)) * Float32(-0.06944444444444445)) / s) + Float32(Float32(0.16666666666666666) / Float32(pi))) / s)) / s)
end

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.25) / (r * single(pi))) - (((((r / single(pi)) * single(-0.06944444444444445)) / s) + (single(0.16666666666666666) / single(pi))) / s)) / s;
end

Derivation

1. Initial program 99.6%

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

   1. +-commutative 99.6%

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

   2. times-frac 99.6%

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

   3. fma-define 99.7%

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

   4. associate-*l* 99.6%

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

   5. associate-/r* 99.6%

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

   6. metadata-eval 99.6%

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

   7. *-commutative 99.6%

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

   8. neg-mul-1 99.6%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\frac{\color{blue}{-1 \cdot r}}{3 \cdot s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   9. times-frac 99.5%

      \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\color{blue}{\frac{-1}{3} \cdot \frac{r}{s}}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   10. metadata-eval 99.5%

       \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\color{blue}{-0.3333333333333333} \cdot \frac{r}{s}}}{r}, \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]

   11. times-frac 99.5%

       \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r}, \color{blue}{\frac{0.25}{\left(2 \cdot \pi\right) \cdot s} \cdot \frac{e^{\frac{-r}{s}}}{r}}\right) \]

3. Simplified 99.5%

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

5. Taylor expanded in s around -inf 12.7%

   \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{-1 \cdot \frac{-0.0625 \cdot \frac{r}{\pi} + -0.006944444444444444 \cdot \frac{r}{\pi}}{s} - 0.16666666666666666 \cdot \frac{1}{\pi}}{s} - 0.25 \cdot \frac{1}{r \cdot \pi}}{s}} \]
6. Step-by-step derivation

   1. mul-1-neg 12.7%

      \[\leadsto \color{blue}{-\frac{-1 \cdot \frac{-1 \cdot \frac{-0.0625 \cdot \frac{r}{\pi} + -0.006944444444444444 \cdot \frac{r}{\pi}}{s} - 0.16666666666666666 \cdot \frac{1}{\pi}}{s} - 0.25 \cdot \frac{1}{r \cdot \pi}}{s}} \]

7. Simplified 12.7%

   \[\leadsto \color{blue}{-\frac{\left(-\frac{\left(-\frac{\frac{r}{\pi} \cdot -0.06944444444444445}{s}\right) - \frac{0.16666666666666666}{\pi}}{s}\right) - \frac{0.25}{r \cdot \pi}}{s}} \]

8. Final simplification 12.7%

   \[\leadsto \frac{\frac{0.25}{r \cdot \pi} - \frac{\frac{\frac{r}{\pi} \cdot -0.06944444444444445}{s} + \frac{0.16666666666666666}{\pi}}{s}}{s} \]
9. Add Preprocessing

Alternative 11: 8.9% accurate, 17.8× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.25}{r \cdot \pi} - \frac{0.16666666666666666}{s \cdot \pi}}{s} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (/ (- (/ 0.25 (* r PI)) (/ 0.16666666666666666 (* s PI))) s))

C

float code(float s, float r) {
	return ((0.25f / (r * ((float) M_PI))) - (0.16666666666666666f / (s * ((float) M_PI)))) / s;
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.25) / Float32(r * Float32(pi))) - Float32(Float32(0.16666666666666666) / Float32(s * Float32(pi)))) / s)
end

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.25) / (r * single(pi))) - (single(0.16666666666666666) / (s * single(pi)))) / s;
end

Derivation

1. Initial program 99.6%

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

3. Simplified 99.2%

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

5. Taylor expanded in s around inf 11.4%

   \[\leadsto \color{blue}{\frac{0.25 \cdot \frac{1}{r \cdot \pi} - 0.16666666666666666 \cdot \frac{1}{s \cdot \pi}}{s}} \]
6. Step-by-step derivation

   1. associate-*r/ 11.4%

      \[\leadsto \frac{0.25 \cdot \frac{1}{r \cdot \pi} - \color{blue}{\frac{0.16666666666666666 \cdot 1}{s \cdot \pi}}}{s} \]

   2. metadata-eval 11.4%

      \[\leadsto \frac{0.25 \cdot \frac{1}{r \cdot \pi} - \frac{\color{blue}{0.16666666666666666}}{s \cdot \pi}}{s} \]

   3. associate-*r/ 11.4%

      \[\leadsto \frac{\color{blue}{\frac{0.25 \cdot 1}{r \cdot \pi}} - \frac{0.16666666666666666}{s \cdot \pi}}{s} \]

   4. metadata-eval 11.4%

      \[\leadsto \frac{\frac{\color{blue}{0.25}}{r \cdot \pi} - \frac{0.16666666666666666}{s \cdot \pi}}{s} \]

7. Simplified 11.4%

   \[\leadsto \color{blue}{\frac{\frac{0.25}{r \cdot \pi} - \frac{0.16666666666666666}{s \cdot \pi}}{s}} \]
8. Add Preprocessing

Alternative 12: 8.9% 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 99.6%

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

3. Simplified 99.2%

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

5. Taylor expanded in s around inf 11.2%

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

Reproduce

herbie shell --seed 2024170 
(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))))