Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.5%

Time: 9.3s

Alternatives: 10

Speedup: N/A×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

   1. +-commutative 99.7%

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

   2. times-frac 99.7%

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

   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* 99.7%

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

   5. associate-/r* 99.7%

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

   6. metadata-eval 99.7%

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

   7. *-commutative 99.7%

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

   8. neg-mul-1 99.7%

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

   9. times-frac 99.7%

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

   10. metadata-eval 99.7%

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

   11. times-frac 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in s around 0 99.7%

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

   1. distribute-lft-out 99.7%

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

   2. associate-/l* 99.7%

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

   3. mul-1-neg 99.7%

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

   4. distribute-neg-frac 99.7%

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

   5. *-commutative 99.7%

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

   6. associate-*l/ 99.7%

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

   7. associate-*r/ 99.7%

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

7. Simplified 99.7%

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

8. Taylor expanded in r around inf 99.7%

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

   1. mul-1-neg 99.7%

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

   2. associate-*r/ 99.7%

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

   3. associate-*l/ 99.7%

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

   4. *-commutative 99.7%

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

10. Simplified 99.7%

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

    1. associate-*r/ 99.7%

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

    2. *-commutative 99.7%

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

12. Applied egg-rr 99.7%

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

    1. div-inv 99.7%

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

14. Applied egg-rr 99.7%

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

15. Final simplification 99.7%

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

Alternative 2: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

   1. +-commutative 99.7%

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

   2. times-frac 99.7%

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

   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* 99.7%

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

   5. associate-/r* 99.7%

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

   6. metadata-eval 99.7%

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

   7. *-commutative 99.7%

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

   8. neg-mul-1 99.7%

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

   9. times-frac 99.7%

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

   10. metadata-eval 99.7%

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

   11. times-frac 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in s around 0 99.7%

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

   1. distribute-lft-out 99.7%

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

   2. associate-/l* 99.7%

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

   3. mul-1-neg 99.7%

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

   4. distribute-neg-frac 99.7%

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

   5. *-commutative 99.7%

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

   6. associate-*l/ 99.7%

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

   7. associate-*r/ 99.7%

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

7. Simplified 99.7%

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

8. Taylor expanded in r around inf 99.7%

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

   1. mul-1-neg 99.7%

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

   2. associate-*r/ 99.7%

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

   3. associate-*l/ 99.7%

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

   4. *-commutative 99.7%

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

10. Simplified 99.7%

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

11. Final simplification 99.7%

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

Alternative 3: 11.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.7%

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

   1. +-commutative 99.7%

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

   2. times-frac 99.7%

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

   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* 99.7%

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

   5. associate-/r* 99.7%

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

   6. metadata-eval 99.7%

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

   7. *-commutative 99.7%

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

   8. neg-mul-1 99.7%

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

   9. times-frac 99.7%

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

   10. metadata-eval 99.7%

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

   11. times-frac 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in s around inf 8.3%

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

   1. log1p-expm1-u 10.9%

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

   2. associate-*r* 10.9%

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

7. Applied egg-rr 10.9%

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

8. Final simplification 10.9%

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

Alternative 4: 9.9% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

   1. +-commutative 99.7%

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

   2. times-frac 99.7%

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

   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* 99.7%

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

   5. associate-/r* 99.7%

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

   6. metadata-eval 99.7%

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

   7. *-commutative 99.7%

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

   8. neg-mul-1 99.7%

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

   9. times-frac 99.7%

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

   10. metadata-eval 99.7%

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

   11. times-frac 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in s around 0 99.7%

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

   1. distribute-lft-out 99.7%

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

   2. associate-/l* 99.7%

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

   3. mul-1-neg 99.7%

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

   4. distribute-neg-frac 99.7%

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

   5. *-commutative 99.7%

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

   6. associate-*l/ 99.7%

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

   7. associate-*r/ 99.7%

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

7. Simplified 99.7%

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

8. Taylor expanded in r around inf 99.7%

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

   1. mul-1-neg 99.7%

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

   2. associate-*r/ 99.7%

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

   3. associate-*l/ 99.7%

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

   4. *-commutative 99.7%

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

10. Simplified 99.7%

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

11. Taylor expanded in s around -inf 9.0%

    \[\leadsto 0.125 \cdot \frac{\color{blue}{-1 \cdot \frac{-1 \cdot \frac{0.05555555555555555 \cdot \frac{{r}^{2}}{\pi} + 0.5 \cdot \frac{{r}^{2}}{\pi}}{s} + \left(0.3333333333333333 \cdot \frac{r}{\pi} + \frac{r}{\pi}\right)}{s} + 2 \cdot \frac{1}{\pi}}}{r \cdot s} \]
12. Step-by-step derivation

    1. +-commutative 9.0%

       \[\leadsto 0.125 \cdot \frac{\color{blue}{2 \cdot \frac{1}{\pi} + -1 \cdot \frac{-1 \cdot \frac{0.05555555555555555 \cdot \frac{{r}^{2}}{\pi} + 0.5 \cdot \frac{{r}^{2}}{\pi}}{s} + \left(0.3333333333333333 \cdot \frac{r}{\pi} + \frac{r}{\pi}\right)}{s}}}{r \cdot s} \]

    2. mul-1-neg 9.0%

       \[\leadsto 0.125 \cdot \frac{2 \cdot \frac{1}{\pi} + \color{blue}{\left(-\frac{-1 \cdot \frac{0.05555555555555555 \cdot \frac{{r}^{2}}{\pi} + 0.5 \cdot \frac{{r}^{2}}{\pi}}{s} + \left(0.3333333333333333 \cdot \frac{r}{\pi} + \frac{r}{\pi}\right)}{s}\right)}}{r \cdot s} \]

    3. unsub-neg 9.0%

       \[\leadsto 0.125 \cdot \frac{\color{blue}{2 \cdot \frac{1}{\pi} - \frac{-1 \cdot \frac{0.05555555555555555 \cdot \frac{{r}^{2}}{\pi} + 0.5 \cdot \frac{{r}^{2}}{\pi}}{s} + \left(0.3333333333333333 \cdot \frac{r}{\pi} + \frac{r}{\pi}\right)}{s}}}{r \cdot s} \]

    4. associate-*r/ 9.0%

       \[\leadsto 0.125 \cdot \frac{\color{blue}{\frac{2 \cdot 1}{\pi}} - \frac{-1 \cdot \frac{0.05555555555555555 \cdot \frac{{r}^{2}}{\pi} + 0.5 \cdot \frac{{r}^{2}}{\pi}}{s} + \left(0.3333333333333333 \cdot \frac{r}{\pi} + \frac{r}{\pi}\right)}{s}}{r \cdot s} \]

    5. metadata-eval 9.0%

       \[\leadsto 0.125 \cdot \frac{\frac{\color{blue}{2}}{\pi} - \frac{-1 \cdot \frac{0.05555555555555555 \cdot \frac{{r}^{2}}{\pi} + 0.5 \cdot \frac{{r}^{2}}{\pi}}{s} + \left(0.3333333333333333 \cdot \frac{r}{\pi} + \frac{r}{\pi}\right)}{s}}{r \cdot s} \]

13. Simplified 9.0%

    \[\leadsto 0.125 \cdot \frac{\color{blue}{\frac{2}{\pi} - \frac{1.3333333333333333 \cdot \frac{r}{\pi} - \frac{{r}^{2}}{\pi} \cdot \frac{0.5555555555555556}{s}}{s}}}{r \cdot s} \]

14. Final simplification 9.0%

    \[\leadsto 0.125 \cdot \frac{\frac{2}{\pi} + \frac{\frac{{r}^{2}}{\pi} \cdot \frac{0.5555555555555556}{s} - 1.3333333333333333 \cdot \frac{r}{\pi}}{s}}{r \cdot s} \]
15. Add Preprocessing

Alternative 5: 9.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{r}{\pi} \cdot 0.06944444444444445 + s \cdot \left(0.25 \cdot \frac{s}{r \cdot \pi} + 0.16666666666666666 \cdot \frac{-1}{\pi}\right)}{{s}^{3}} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (/
  (+
   (* (/ r PI) 0.06944444444444445)
   (* s (+ (* 0.25 (/ s (* r PI))) (* 0.16666666666666666 (/ -1.0 PI)))))
  (pow s 3.0)))

C

float code(float s, float r) {
	return (((r / ((float) M_PI)) * 0.06944444444444445f) + (s * ((0.25f * (s / (r * ((float) M_PI)))) + (0.16666666666666666f * (-1.0f / ((float) M_PI)))))) / powf(s, 3.0f);
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(r / Float32(pi)) * Float32(0.06944444444444445)) + Float32(s * Float32(Float32(Float32(0.25) * Float32(s / Float32(r * Float32(pi)))) + Float32(Float32(0.16666666666666666) * Float32(Float32(-1.0) / Float32(pi)))))) / (s ^ Float32(3.0)))
end

MATLAB

function tmp = code(s, r)
	tmp = (((r / single(pi)) * single(0.06944444444444445)) + (s * ((single(0.25) * (s / (r * single(pi)))) + (single(0.16666666666666666) * (single(-1.0) / single(pi)))))) / (s ^ single(3.0));
end

Derivation

1. Initial program 99.7%

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

   1. +-commutative 99.7%

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

   2. times-frac 99.7%

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

   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* 99.7%

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

   5. associate-/r* 99.7%

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

   6. metadata-eval 99.7%

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

   7. *-commutative 99.7%

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

   8. neg-mul-1 99.7%

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

   9. times-frac 99.7%

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

   10. metadata-eval 99.7%

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

   11. times-frac 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in s around 0 99.7%

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

   1. distribute-lft-out 99.7%

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

   2. associate-/l* 99.7%

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

   3. mul-1-neg 99.7%

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

   4. distribute-neg-frac 99.7%

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

   5. *-commutative 99.7%

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

   6. associate-*l/ 99.7%

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

   7. associate-*r/ 99.7%

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

7. Simplified 99.7%

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

8. Taylor expanded in r around inf 99.7%

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

   1. mul-1-neg 99.7%

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

   2. associate-*r/ 99.7%

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

   3. associate-*l/ 99.7%

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

   4. *-commutative 99.7%

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

10. Simplified 99.7%

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

11. Taylor expanded in s around -inf 9.0%

    \[\leadsto 0.125 \cdot \frac{\color{blue}{-1 \cdot \frac{-1 \cdot \frac{0.05555555555555555 \cdot \frac{{r}^{2}}{\pi} + 0.5 \cdot \frac{{r}^{2}}{\pi}}{s} + \left(0.3333333333333333 \cdot \frac{r}{\pi} + \frac{r}{\pi}\right)}{s} + 2 \cdot \frac{1}{\pi}}}{r \cdot s} \]
12. Step-by-step derivation

    1. +-commutative 9.0%

       \[\leadsto 0.125 \cdot \frac{\color{blue}{2 \cdot \frac{1}{\pi} + -1 \cdot \frac{-1 \cdot \frac{0.05555555555555555 \cdot \frac{{r}^{2}}{\pi} + 0.5 \cdot \frac{{r}^{2}}{\pi}}{s} + \left(0.3333333333333333 \cdot \frac{r}{\pi} + \frac{r}{\pi}\right)}{s}}}{r \cdot s} \]

    2. mul-1-neg 9.0%

       \[\leadsto 0.125 \cdot \frac{2 \cdot \frac{1}{\pi} + \color{blue}{\left(-\frac{-1 \cdot \frac{0.05555555555555555 \cdot \frac{{r}^{2}}{\pi} + 0.5 \cdot \frac{{r}^{2}}{\pi}}{s} + \left(0.3333333333333333 \cdot \frac{r}{\pi} + \frac{r}{\pi}\right)}{s}\right)}}{r \cdot s} \]

    3. unsub-neg 9.0%

       \[\leadsto 0.125 \cdot \frac{\color{blue}{2 \cdot \frac{1}{\pi} - \frac{-1 \cdot \frac{0.05555555555555555 \cdot \frac{{r}^{2}}{\pi} + 0.5 \cdot \frac{{r}^{2}}{\pi}}{s} + \left(0.3333333333333333 \cdot \frac{r}{\pi} + \frac{r}{\pi}\right)}{s}}}{r \cdot s} \]

    4. associate-*r/ 9.0%

       \[\leadsto 0.125 \cdot \frac{\color{blue}{\frac{2 \cdot 1}{\pi}} - \frac{-1 \cdot \frac{0.05555555555555555 \cdot \frac{{r}^{2}}{\pi} + 0.5 \cdot \frac{{r}^{2}}{\pi}}{s} + \left(0.3333333333333333 \cdot \frac{r}{\pi} + \frac{r}{\pi}\right)}{s}}{r \cdot s} \]

    5. metadata-eval 9.0%

       \[\leadsto 0.125 \cdot \frac{\frac{\color{blue}{2}}{\pi} - \frac{-1 \cdot \frac{0.05555555555555555 \cdot \frac{{r}^{2}}{\pi} + 0.5 \cdot \frac{{r}^{2}}{\pi}}{s} + \left(0.3333333333333333 \cdot \frac{r}{\pi} + \frac{r}{\pi}\right)}{s}}{r \cdot s} \]

13. Simplified 9.0%

    \[\leadsto 0.125 \cdot \frac{\color{blue}{\frac{2}{\pi} - \frac{1.3333333333333333 \cdot \frac{r}{\pi} - \frac{{r}^{2}}{\pi} \cdot \frac{0.5555555555555556}{s}}{s}}}{r \cdot s} \]

14. Taylor expanded in s around 0 9.0%

    \[\leadsto \color{blue}{\frac{0.06944444444444445 \cdot \frac{r}{\pi} + s \cdot \left(0.25 \cdot \frac{s}{r \cdot \pi} - 0.16666666666666666 \cdot \frac{1}{\pi}\right)}{{s}^{3}}} \]

15. Final simplification 9.0%

    \[\leadsto \frac{\frac{r}{\pi} \cdot 0.06944444444444445 + s \cdot \left(0.25 \cdot \frac{s}{r \cdot \pi} + 0.16666666666666666 \cdot \frac{-1}{\pi}\right)}{{s}^{3}} \]
16. Add Preprocessing

Alternative 6: 9.9% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.16666666666666666 \cdot \frac{-1}{\pi} - \frac{\frac{r}{\pi} \cdot -0.0625 + \frac{r}{\pi} \cdot -0.006944444444444444}{s}}{s} + 0.25 \cdot \frac{1}{r \cdot \pi}}{s} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (/
  (+
   (/
    (-
     (* 0.16666666666666666 (/ -1.0 PI))
     (/ (+ (* (/ r PI) -0.0625) (* (/ r PI) -0.006944444444444444)) s))
    s)
   (* 0.25 (/ 1.0 (* r PI))))
  s))

C

float code(float s, float r) {
	return ((((0.16666666666666666f * (-1.0f / ((float) M_PI))) - ((((r / ((float) M_PI)) * -0.0625f) + ((r / ((float) M_PI)) * -0.006944444444444444f)) / s)) / s) + (0.25f * (1.0f / (r * ((float) M_PI))))) / s;
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(Float32(Float32(0.16666666666666666) * Float32(Float32(-1.0) / Float32(pi))) - Float32(Float32(Float32(Float32(r / Float32(pi)) * Float32(-0.0625)) + Float32(Float32(r / Float32(pi)) * Float32(-0.006944444444444444))) / s)) / s) + Float32(Float32(0.25) * Float32(Float32(1.0) / Float32(r * Float32(pi))))) / s)
end

MATLAB

function tmp = code(s, r)
	tmp = ((((single(0.16666666666666666) * (single(-1.0) / single(pi))) - ((((r / single(pi)) * single(-0.0625)) + ((r / single(pi)) * single(-0.006944444444444444))) / s)) / s) + (single(0.25) * (single(1.0) / (r * single(pi))))) / s;
end

Derivation

1. Initial program 99.7%

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

   1. +-commutative 99.7%

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

   2. times-frac 99.7%

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

   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* 99.7%

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

   5. associate-/r* 99.7%

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

   6. metadata-eval 99.7%

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

   7. *-commutative 99.7%

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

   8. neg-mul-1 99.7%

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

   9. times-frac 99.7%

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

   10. metadata-eval 99.7%

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

   11. times-frac 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in s around -inf 9.0%

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

6. Final simplification 9.0%

   \[\leadsto \frac{\frac{0.16666666666666666 \cdot \frac{-1}{\pi} - \frac{\frac{r}{\pi} \cdot -0.0625 + \frac{r}{\pi} \cdot -0.006944444444444444}{s}}{s} + 0.25 \cdot \frac{1}{r \cdot \pi}}{s} \]
7. Add Preprocessing

Alternative 7: 9.9% accurate, 11.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

   1. +-commutative 99.7%

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

   2. times-frac 99.7%

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

   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* 99.7%

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

   5. associate-/r* 99.7%

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

   6. metadata-eval 99.7%

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

   7. *-commutative 99.7%

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

   8. neg-mul-1 99.7%

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

   9. times-frac 99.7%

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

   10. metadata-eval 99.7%

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

   11. times-frac 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in s around -inf 9.0%

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

   1. associate-*r/ 9.0%

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

7. Simplified 9.0%

   \[\leadsto \color{blue}{\frac{\frac{\frac{-0.16666666666666666}{\pi} - \frac{r}{\pi} \cdot \frac{-0.06944444444444445}{s}}{s} + \frac{0.25}{r \cdot \pi}}{s}} \]
8. Step-by-step derivation

   1. clear-num 9.0%

      \[\leadsto \frac{\frac{\frac{-0.16666666666666666}{\pi} - \color{blue}{\frac{1}{\frac{\pi}{r}}} \cdot \frac{-0.06944444444444445}{s}}{s} + \frac{0.25}{r \cdot \pi}}{s} \]

   2. frac-times 9.0%

      \[\leadsto \frac{\frac{\frac{-0.16666666666666666}{\pi} - \color{blue}{\frac{1 \cdot -0.06944444444444445}{\frac{\pi}{r} \cdot s}}}{s} + \frac{0.25}{r \cdot \pi}}{s} \]

   3. metadata-eval 9.0%

      \[\leadsto \frac{\frac{\frac{-0.16666666666666666}{\pi} - \frac{\color{blue}{-0.06944444444444445}}{\frac{\pi}{r} \cdot s}}{s} + \frac{0.25}{r \cdot \pi}}{s} \]

9. Applied egg-rr 9.0%

   \[\leadsto \frac{\frac{\frac{-0.16666666666666666}{\pi} - \color{blue}{\frac{-0.06944444444444445}{\frac{\pi}{r} \cdot s}}}{s} + \frac{0.25}{r \cdot \pi}}{s} \]

10. Final simplification 9.0%

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

Alternative 8: 9.9% accurate, 11.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

   1. +-commutative 99.7%

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

   2. times-frac 99.7%

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

   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* 99.7%

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

   5. associate-/r* 99.7%

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

   6. metadata-eval 99.7%

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

   7. *-commutative 99.7%

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

   8. neg-mul-1 99.7%

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

   9. times-frac 99.7%

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

   10. metadata-eval 99.7%

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

   11. times-frac 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in s around -inf 9.0%

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

   1. associate-*r/ 9.0%

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

7. Simplified 9.0%

   \[\leadsto \color{blue}{\frac{\frac{\frac{-0.16666666666666666}{\pi} - \frac{r}{\pi} \cdot \frac{-0.06944444444444445}{s}}{s} + \frac{0.25}{r \cdot \pi}}{s}} \]

8. Final simplification 9.0%

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

Alternative 9: 9.0% accurate, 33.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.7%

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

3. Simplified 99.6%

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

5. Taylor expanded in r around 0 8.7%

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

6. Taylor expanded in s around inf 8.3%

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

   1. *-commutative 8.3%

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

   2. associate-/r* 8.3%

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

   3. *-commutative 8.3%

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

   4. associate-/r* 8.3%

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

8. Simplified 8.3%

   \[\leadsto \color{blue}{\frac{\frac{\frac{0.25}{\pi}}{s}}{r}} \]
9. Add Preprocessing

Alternative 10: 9.0% 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 99.7%

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

   1. +-commutative 99.7%

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

   2. times-frac 99.7%

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

   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* 99.7%

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

   5. associate-/r* 99.7%

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

   6. metadata-eval 99.7%

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

   7. *-commutative 99.7%

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

   8. neg-mul-1 99.7%

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

   9. times-frac 99.7%

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

   10. metadata-eval 99.7%

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

   11. times-frac 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in s around inf 8.3%

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

Reproduce

herbie shell --seed 2024095 
(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))))