Disney BSSRDF, PDF of scattering profile

Percentage Accurate: 99.6% → 99.6%

Time: 13.7s

Alternatives: 19

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, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

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

5. Taylor expanded in r around inf 99.7%

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

   1. mul-1-neg 99.7%

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

7. Applied egg-rr 99.7%

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

8. Final simplification 99.7%

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

Alternative 2: 44.2% 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 99.5%

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

5. Taylor expanded in s around inf 8.8%

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

   1. *-commutative 8.8%

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

   2. associate-*l* 8.8%

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

   3. *-commutative 8.8%

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

7. Simplified 8.8%

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

   1. log1p-expm1-u 41.4%

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

9. Applied egg-rr 41.4%

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

Alternative 3: 11.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

function code(s, r)
	return Float32(Float32(0.25) / log1p(expm1(Float32(r * Float32(s * 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 99.5%

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

5. Taylor expanded in s around inf 8.8%

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

   1. log1p-expm1-u 12.0%

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

7. Applied egg-rr 12.0%

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

Alternative 4: 10.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \frac{0.125 \cdot \left(\frac{e^{\frac{r}{-s}}}{r \cdot \pi} + \left(\frac{0.3333333333333333 \cdot \frac{-1}{\pi} - -0.05555555555555555 \cdot \frac{r}{s \cdot \pi}}{s} + \frac{1}{r \cdot \pi}\right)\right)}{s} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (/
  (*
   0.125
   (+
    (/ (exp (/ r (- s))) (* r PI))
    (+
     (/
      (-
       (* 0.3333333333333333 (/ -1.0 PI))
       (* -0.05555555555555555 (/ r (* s PI))))
      s)
     (/ 1.0 (* r PI)))))
  s))

C

float code(float s, float r) {
	return (0.125f * ((expf((r / -s)) / (r * ((float) M_PI))) + ((((0.3333333333333333f * (-1.0f / ((float) M_PI))) - (-0.05555555555555555f * (r / (s * ((float) M_PI))))) / s) + (1.0f / (r * ((float) M_PI)))))) / s;
}

Julia

function code(s, r)
	return Float32(Float32(Float32(0.125) * Float32(Float32(exp(Float32(r / Float32(-s))) / Float32(r * Float32(pi))) + Float32(Float32(Float32(Float32(Float32(0.3333333333333333) * Float32(Float32(-1.0) / Float32(pi))) - Float32(Float32(-0.05555555555555555) * Float32(r / Float32(s * Float32(pi))))) / s) + Float32(Float32(1.0) / Float32(r * Float32(pi)))))) / s)
end

MATLAB

function tmp = code(s, r)
	tmp = (single(0.125) * ((exp((r / -s)) / (r * single(pi))) + ((((single(0.3333333333333333) * (single(-1.0) / single(pi))) - (single(-0.05555555555555555) * (r / (s * single(pi))))) / s) + (single(1.0) / (r * single(pi)))))) / s;
end

Derivation

1. Initial program 99.6%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
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 99.6%

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

   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 99.6%

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

   10. metadata-eval 99.6%

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

   11. times-frac 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in s around 0 99.6%

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

   1. distribute-lft-out 99.6%

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

   2. neg-mul-1 99.6%

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

   3. distribute-neg-frac2 99.6%

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

   4. associate-*r/ 99.6%

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

7. Simplified 99.6%

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

8. Taylor expanded in s around -inf 9.7%

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

9. Final simplification 9.7%

   \[\leadsto \frac{0.125 \cdot \left(\frac{e^{\frac{r}{-s}}}{r \cdot \pi} + \left(\frac{0.3333333333333333 \cdot \frac{-1}{\pi} - -0.05555555555555555 \cdot \frac{r}{s \cdot \pi}}{s} + \frac{1}{r \cdot \pi}\right)\right)}{s} \]
10. Add Preprocessing

Alternative 5: 10.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ 0.125 \cdot \frac{\frac{e^{\frac{r}{-s}}}{r} + \left(\frac{1}{r} - \frac{0.3333333333333333 + \frac{r}{s} \cdot -0.05555555555555555}{s}\right)}{s \cdot \pi} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (*
  0.125
  (/
   (+
    (/ (exp (/ r (- s))) r)
    (-
     (/ 1.0 r)
     (/ (+ 0.3333333333333333 (* (/ r s) -0.05555555555555555)) s)))
   (* s PI))))

C

float code(float s, float r) {
	return 0.125f * (((expf((r / -s)) / r) + ((1.0f / r) - ((0.3333333333333333f + ((r / s) * -0.05555555555555555f)) / s))) / (s * ((float) M_PI)));
}

Julia

function code(s, r)
	return Float32(Float32(0.125) * Float32(Float32(Float32(exp(Float32(r / Float32(-s))) / r) + Float32(Float32(Float32(1.0) / r) - Float32(Float32(Float32(0.3333333333333333) + Float32(Float32(r / s) * Float32(-0.05555555555555555))) / s))) / Float32(s * Float32(pi))))
end

MATLAB

function tmp = code(s, r)
	tmp = single(0.125) * (((exp((r / -s)) / r) + ((single(1.0) / r) - ((single(0.3333333333333333) + ((r / s) * single(-0.05555555555555555))) / s))) / (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 99.5%

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

5. Taylor expanded in s around 0 99.6%

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

   1. neg-mul-1 99.6%

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

   2. distribute-neg-frac2 99.6%

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

   3. associate-*r/ 99.6%

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

7. Simplified 99.6%

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

8. Taylor expanded in s around -inf 9.7%

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

9. Final simplification 9.7%

   \[\leadsto 0.125 \cdot \frac{\frac{e^{\frac{r}{-s}}}{r} + \left(\frac{1}{r} - \frac{0.3333333333333333 + \frac{r}{s} \cdot -0.05555555555555555}{s}\right)}{s \cdot \pi} \]
10. Add Preprocessing

Alternative 6: 9.2% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
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 99.6%

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

   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 99.6%

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

   10. metadata-eval 99.6%

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

   11. times-frac 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in s around 0 99.6%

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

   1. distribute-lft-out 99.6%

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

   2. neg-mul-1 99.6%

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

   3. distribute-neg-frac2 99.6%

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

   4. associate-*r/ 99.6%

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

7. Simplified 99.6%

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

8. Taylor expanded in s around inf 9.5%

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

9. Final simplification 9.5%

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

Alternative 7: 9.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
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 99.6%

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

   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 99.6%

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

   10. metadata-eval 99.6%

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

   11. times-frac 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in s around 0 99.6%

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

   1. distribute-lft-out 99.6%

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

   2. neg-mul-1 99.6%

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

   3. distribute-neg-frac2 99.6%

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

   4. associate-*r/ 99.6%

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

7. Simplified 99.6%

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

8. Taylor expanded in s around inf 9.5%

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

   1. associate-*r/ 9.5%

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

   2. metadata-eval 9.5%

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

10. Simplified 9.5%

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

Alternative 8: 9.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

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

5. Taylor expanded in s around 0 99.6%

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

   1. neg-mul-1 99.6%

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

   2. distribute-neg-frac2 99.6%

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

   3. associate-*r/ 99.6%

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

7. Simplified 99.6%

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

8. Taylor expanded in s around inf 9.5%

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

   1. associate-*r/ 9.5%

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

   2. metadata-eval 9.5%

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

10. Simplified 9.5%

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

Alternative 9: 9.5% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.125 \cdot \frac{0.05555555555555555 \cdot \frac{r}{\pi} + \frac{r}{\pi} \cdot 0.5}{s} + 0.16666666666666666 \cdot \frac{-1}{\pi}}{s} + 0.25 \cdot \frac{\frac{1}{\pi}}{r}}{s} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (/
  (+
   (/
    (+
     (* 0.125 (/ (+ (* 0.05555555555555555 (/ r PI)) (* (/ r PI) 0.5)) s))
     (* 0.16666666666666666 (/ -1.0 PI)))
    s)
   (* 0.25 (/ (/ 1.0 PI) r)))
  s))

C

float code(float s, float r) {
	return ((((0.125f * (((0.05555555555555555f * (r / ((float) M_PI))) + ((r / ((float) M_PI)) * 0.5f)) / s)) + (0.16666666666666666f * (-1.0f / ((float) M_PI)))) / s) + (0.25f * ((1.0f / ((float) M_PI)) / r))) / s;
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(Float32(Float32(0.125) * Float32(Float32(Float32(Float32(0.05555555555555555) * Float32(r / Float32(pi))) + Float32(Float32(r / Float32(pi)) * Float32(0.5))) / s)) + Float32(Float32(0.16666666666666666) * Float32(Float32(-1.0) / Float32(pi)))) / s) + Float32(Float32(0.25) * Float32(Float32(Float32(1.0) / Float32(pi)) / r))) / s)
end

MATLAB

function tmp = code(s, r)
	tmp = ((((single(0.125) * (((single(0.05555555555555555) * (r / single(pi))) + ((r / single(pi)) * single(0.5))) / s)) + (single(0.16666666666666666) * (single(-1.0) / single(pi)))) / s) + (single(0.25) * ((single(1.0) / single(pi)) / r))) / 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 99.5%

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

5. Taylor expanded in s around -inf 9.5%

   \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{0.125 \cdot \frac{0.05555555555555555 \cdot \frac{r}{\pi} + 0.5 \cdot \frac{r}{\pi}}{s} - 0.16666666666666666 \cdot \frac{1}{\pi}}{s} - 0.25 \cdot \frac{1}{r \cdot \pi}}{s}} \]
6. Step-by-step derivation

   1. inv-pow 9.5%

      \[\leadsto -1 \cdot \frac{-1 \cdot \frac{0.125 \cdot \frac{0.05555555555555555 \cdot \frac{r}{\pi} + 0.5 \cdot \frac{r}{\pi}}{s} - 0.16666666666666666 \cdot \frac{1}{\pi}}{s} - 0.25 \cdot \color{blue}{{\left(r \cdot \pi\right)}^{-1}}}{s} \]

   2. *-commutative 9.5%

      \[\leadsto -1 \cdot \frac{-1 \cdot \frac{0.125 \cdot \frac{0.05555555555555555 \cdot \frac{r}{\pi} + 0.5 \cdot \frac{r}{\pi}}{s} - 0.16666666666666666 \cdot \frac{1}{\pi}}{s} - 0.25 \cdot {\color{blue}{\left(\pi \cdot r\right)}}^{-1}}{s} \]

   3. unpow-prod-down 9.4%

      \[\leadsto -1 \cdot \frac{-1 \cdot \frac{0.125 \cdot \frac{0.05555555555555555 \cdot \frac{r}{\pi} + 0.5 \cdot \frac{r}{\pi}}{s} - 0.16666666666666666 \cdot \frac{1}{\pi}}{s} - 0.25 \cdot \color{blue}{\left({\pi}^{-1} \cdot {r}^{-1}\right)}}{s} \]

   4. inv-pow 9.4%

      \[\leadsto -1 \cdot \frac{-1 \cdot \frac{0.125 \cdot \frac{0.05555555555555555 \cdot \frac{r}{\pi} + 0.5 \cdot \frac{r}{\pi}}{s} - 0.16666666666666666 \cdot \frac{1}{\pi}}{s} - 0.25 \cdot \left(\color{blue}{\frac{1}{\pi}} \cdot {r}^{-1}\right)}{s} \]

   5. inv-pow 9.4%

      \[\leadsto -1 \cdot \frac{-1 \cdot \frac{0.125 \cdot \frac{0.05555555555555555 \cdot \frac{r}{\pi} + 0.5 \cdot \frac{r}{\pi}}{s} - 0.16666666666666666 \cdot \frac{1}{\pi}}{s} - 0.25 \cdot \left(\frac{1}{\pi} \cdot \color{blue}{\frac{1}{r}}\right)}{s} \]

7. Applied egg-rr 9.4%

   \[\leadsto -1 \cdot \frac{-1 \cdot \frac{0.125 \cdot \frac{0.05555555555555555 \cdot \frac{r}{\pi} + 0.5 \cdot \frac{r}{\pi}}{s} - 0.16666666666666666 \cdot \frac{1}{\pi}}{s} - 0.25 \cdot \color{blue}{\left(\frac{1}{\pi} \cdot \frac{1}{r}\right)}}{s} \]
8. Step-by-step derivation

   1. un-div-inv 9.5%

      \[\leadsto -1 \cdot \frac{-1 \cdot \frac{0.125 \cdot \frac{0.05555555555555555 \cdot \frac{r}{\pi} + 0.5 \cdot \frac{r}{\pi}}{s} - 0.16666666666666666 \cdot \frac{1}{\pi}}{s} - 0.25 \cdot \color{blue}{\frac{\frac{1}{\pi}}{r}}}{s} \]

9. Applied egg-rr 9.5%

   \[\leadsto -1 \cdot \frac{-1 \cdot \frac{0.125 \cdot \frac{0.05555555555555555 \cdot \frac{r}{\pi} + 0.5 \cdot \frac{r}{\pi}}{s} - 0.16666666666666666 \cdot \frac{1}{\pi}}{s} - 0.25 \cdot \color{blue}{\frac{\frac{1}{\pi}}{r}}}{s} \]

10. Final simplification 9.5%

    \[\leadsto \frac{\frac{0.125 \cdot \frac{0.05555555555555555 \cdot \frac{r}{\pi} + \frac{r}{\pi} \cdot 0.5}{s} + 0.16666666666666666 \cdot \frac{-1}{\pi}}{s} + 0.25 \cdot \frac{\frac{1}{\pi}}{r}}{s} \]
11. Add Preprocessing

Alternative 10: 9.5% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.125 \cdot \frac{0.05555555555555555 \cdot \frac{r}{\pi} + \frac{r}{\pi} \cdot 0.5}{s} - \frac{0.16666666666666666}{\pi}}{s} + 0.25 \cdot \frac{1}{r \cdot \pi}}{s} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (/
  (+
   (/
    (-
     (* 0.125 (/ (+ (* 0.05555555555555555 (/ r PI)) (* (/ r PI) 0.5)) s))
     (/ 0.16666666666666666 PI))
    s)
   (* 0.25 (/ 1.0 (* r PI))))
  s))

C

float code(float s, float r) {
	return ((((0.125f * (((0.05555555555555555f * (r / ((float) M_PI))) + ((r / ((float) M_PI)) * 0.5f)) / s)) - (0.16666666666666666f / ((float) M_PI))) / s) + (0.25f * (1.0f / (r * ((float) M_PI))))) / s;
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(Float32(Float32(0.125) * Float32(Float32(Float32(Float32(0.05555555555555555) * Float32(r / Float32(pi))) + Float32(Float32(r / Float32(pi)) * Float32(0.5))) / s)) - Float32(Float32(0.16666666666666666) / Float32(pi))) / s) + Float32(Float32(0.25) * Float32(Float32(1.0) / Float32(r * Float32(pi))))) / s)
end

MATLAB

function tmp = code(s, r)
	tmp = ((((single(0.125) * (((single(0.05555555555555555) * (r / single(pi))) + ((r / single(pi)) * single(0.5))) / s)) - (single(0.16666666666666666) / single(pi))) / s) + (single(0.25) * (single(1.0) / (r * 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 99.5%

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

5. Taylor expanded in s around -inf 9.5%

   \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{0.125 \cdot \frac{0.05555555555555555 \cdot \frac{r}{\pi} + 0.5 \cdot \frac{r}{\pi}}{s} - 0.16666666666666666 \cdot \frac{1}{\pi}}{s} - 0.25 \cdot \frac{1}{r \cdot \pi}}{s}} \]
6. Step-by-step derivation

   1. un-div-inv 9.5%

      \[\leadsto -1 \cdot \frac{-1 \cdot \frac{0.125 \cdot \frac{0.05555555555555555 \cdot \frac{r}{\pi} + 0.5 \cdot \frac{r}{\pi}}{s} - \color{blue}{\frac{0.16666666666666666}{\pi}}}{s} - 0.25 \cdot \frac{1}{r \cdot \pi}}{s} \]

7. Applied egg-rr 9.5%

   \[\leadsto -1 \cdot \frac{-1 \cdot \frac{0.125 \cdot \frac{0.05555555555555555 \cdot \frac{r}{\pi} + 0.5 \cdot \frac{r}{\pi}}{s} - \color{blue}{\frac{0.16666666666666666}{\pi}}}{s} - 0.25 \cdot \frac{1}{r \cdot \pi}}{s} \]

8. Final simplification 9.5%

   \[\leadsto \frac{\frac{0.125 \cdot \frac{0.05555555555555555 \cdot \frac{r}{\pi} + \frac{r}{\pi} \cdot 0.5}{s} - \frac{0.16666666666666666}{\pi}}{s} + 0.25 \cdot \frac{1}{r \cdot \pi}}{s} \]
9. Add Preprocessing

Alternative 11: 9.5% accurate, 8.6× speedup

Math

\[\begin{array}{l} \\ \frac{0.25 \cdot \frac{1}{r \cdot \pi} + \frac{0.125 \cdot \frac{\frac{r \cdot 0.5555555555555556}{\pi}}{s} + 0.16666666666666666 \cdot \frac{-1}{\pi}}{s}}{s} \end{array} \]

FPCore

(FPCore (s r)
 :precision binary32
 (/
  (+
   (* 0.25 (/ 1.0 (* r PI)))
   (/
    (+
     (* 0.125 (/ (/ (* r 0.5555555555555556) PI) s))
     (* 0.16666666666666666 (/ -1.0 PI)))
    s))
  s))

C

float code(float s, float r) {
	return ((0.25f * (1.0f / (r * ((float) M_PI)))) + (((0.125f * (((r * 0.5555555555555556f) / ((float) M_PI)) / s)) + (0.16666666666666666f * (-1.0f / ((float) M_PI)))) / s)) / s;
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.25) * Float32(Float32(1.0) / Float32(r * Float32(pi)))) + Float32(Float32(Float32(Float32(0.125) * Float32(Float32(Float32(r * Float32(0.5555555555555556)) / Float32(pi)) / s)) + Float32(Float32(0.16666666666666666) * Float32(Float32(-1.0) / Float32(pi)))) / s)) / s)
end

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.25) * (single(1.0) / (r * single(pi)))) + (((single(0.125) * (((r * single(0.5555555555555556)) / single(pi)) / s)) + (single(0.16666666666666666) * (single(-1.0) / single(pi)))) / s)) / s;
end

Derivation

1. Initial program 99.6%

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

3. Simplified 99.5%

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

5. Taylor expanded in s around -inf 9.5%

   \[\leadsto \color{blue}{-1 \cdot \frac{-1 \cdot \frac{0.125 \cdot \frac{0.05555555555555555 \cdot \frac{r}{\pi} + 0.5 \cdot \frac{r}{\pi}}{s} - 0.16666666666666666 \cdot \frac{1}{\pi}}{s} - 0.25 \cdot \frac{1}{r \cdot \pi}}{s}} \]

6. Taylor expanded in r around 0 9.5%

   \[\leadsto -1 \cdot \frac{-1 \cdot \frac{0.125 \cdot \frac{\color{blue}{0.5555555555555556 \cdot \frac{r}{\pi}}}{s} - 0.16666666666666666 \cdot \frac{1}{\pi}}{s} - 0.25 \cdot \frac{1}{r \cdot \pi}}{s} \]
7. Step-by-step derivation

   1. *-commutative 9.5%

      \[\leadsto -1 \cdot \frac{-1 \cdot \frac{0.125 \cdot \frac{\color{blue}{\frac{r}{\pi} \cdot 0.5555555555555556}}{s} - 0.16666666666666666 \cdot \frac{1}{\pi}}{s} - 0.25 \cdot \frac{1}{r \cdot \pi}}{s} \]

   2. associate-*l/ 9.5%

      \[\leadsto -1 \cdot \frac{-1 \cdot \frac{0.125 \cdot \frac{\color{blue}{\frac{r \cdot 0.5555555555555556}{\pi}}}{s} - 0.16666666666666666 \cdot \frac{1}{\pi}}{s} - 0.25 \cdot \frac{1}{r \cdot \pi}}{s} \]

8. Simplified 9.5%

   \[\leadsto -1 \cdot \frac{-1 \cdot \frac{0.125 \cdot \frac{\color{blue}{\frac{r \cdot 0.5555555555555556}{\pi}}}{s} - 0.16666666666666666 \cdot \frac{1}{\pi}}{s} - 0.25 \cdot \frac{1}{r \cdot \pi}}{s} \]

9. Final simplification 9.5%

   \[\leadsto \frac{0.25 \cdot \frac{1}{r \cdot \pi} + \frac{0.125 \cdot \frac{\frac{r \cdot 0.5555555555555556}{\pi}}{s} + 0.16666666666666666 \cdot \frac{-1}{\pi}}{s}}{s} \]
10. Add Preprocessing

Alternative 12: 9.5% accurate, 10.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 99.6%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
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 99.6%

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

   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 99.6%

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

   10. metadata-eval 99.6%

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

   11. times-frac 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in s around inf 9.4%

   \[\leadsto \color{blue}{\frac{\left(0.006944444444444444 \cdot \frac{r}{{s}^{2} \cdot \pi} + \left(0.0625 \cdot \frac{r}{{s}^{2} \cdot \pi} + 0.25 \cdot \frac{1}{r \cdot \pi}\right)\right) - \frac{0.16666666666666666}{s \cdot \pi}}{s}} \]
6. Step-by-step derivation

7. Simplified 9.4%

   \[\leadsto \color{blue}{\frac{\frac{r}{{s}^{2} \cdot \pi} \cdot 0.06944444444444445 + \left(\frac{0.25}{r \cdot \pi} + \frac{-0.16666666666666666}{s \cdot \pi}\right)}{s}} \]
8. Step-by-step derivation

   1. unpow2 9.4%

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

9. Applied egg-rr 9.4%

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

10. Final simplification 9.4%

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

Alternative 13: 9.5% accurate, 11.0× speedup

Math

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

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(Float32(r / Float32(pi)) * 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 99.6%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
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 99.6%

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

   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 99.6%

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

   10. metadata-eval 99.6%

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

   11. times-frac 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in s around -inf 9.5%

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

   1. mul-1-neg 9.5%

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

7. Simplified 9.5%

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

8. Final simplification 9.5%

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

Alternative 14: 8.6% accurate, 12.2× speedup

Math

\[\begin{array}{l} \\ \frac{0.125 \cdot \frac{\frac{r}{s \cdot \pi} \cdot -1.3333333333333333 + \frac{1}{\pi} \cdot 2}{r}}{s} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = (single(0.125) * ((((r / (s * single(pi))) * single(-1.3333333333333333)) + ((single(1.0) / single(pi)) * single(2.0))) / r)) / 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 99.6%

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

   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 99.6%

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

   10. metadata-eval 99.6%

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

   11. times-frac 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in s around 0 99.6%

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

   1. distribute-lft-out 99.6%

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

   2. neg-mul-1 99.6%

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

   3. distribute-neg-frac2 99.6%

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

   4. associate-*r/ 99.6%

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

7. Simplified 99.6%

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

8. Taylor expanded in s around inf 9.5%

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

   1. associate-*r/ 9.5%

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

   2. metadata-eval 9.5%

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

10. Simplified 9.5%

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

11. Taylor expanded in r around 0 8.8%

    \[\leadsto \frac{0.125 \cdot \color{blue}{\frac{-1.3333333333333333 \cdot \frac{r}{s \cdot \pi} + 2 \cdot \frac{1}{\pi}}{r}}}{s} \]

12. Final simplification 8.8%

    \[\leadsto \frac{0.125 \cdot \frac{\frac{r}{s \cdot \pi} \cdot -1.3333333333333333 + \frac{1}{\pi} \cdot 2}{r}}{s} \]
13. Add Preprocessing

Alternative 15: 8.6% 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 99.5%

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

5. Taylor expanded in s around inf 8.8%

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

   1. associate-*r/ 8.8%

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

   2. metadata-eval 8.8%

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

   3. associate-*r/ 8.8%

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

   4. metadata-eval 8.8%

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

7. Simplified 8.8%

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

Alternative 16: 8.6% accurate, 25.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = ((single(1.0) / single(pi)) / s) * (single(0.25) / 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 99.5%

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

5. Taylor expanded in s around inf 8.8%

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

   1. associate-/r* 8.8%

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

7. Simplified 8.8%

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

   1. clear-num 8.8%

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

   2. associate-/r/ 8.8%

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

   3. *-commutative 8.8%

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

   4. associate-/r* 8.8%

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

9. Applied egg-rr 8.8%

   \[\leadsto \color{blue}{\frac{\frac{1}{\pi}}{s} \cdot \frac{0.25}{r}} \]
10. Add Preprocessing

Alternative 17: 8.6% accurate, 33.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(s, r)
	tmp = single(0.25) / (single(pi) * (r * 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 99.5%

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

5. Taylor expanded in s around inf 8.8%

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

   1. *-commutative 8.8%

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

   2. associate-*l* 8.8%

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

   3. *-commutative 8.8%

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

7. Simplified 8.8%

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

   1. pow1 8.8%

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

   2. associate-*r* 8.8%

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

9. Applied egg-rr 8.8%

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

    1. pow1 8.8%

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

    2. *-commutative 8.8%

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

11. Applied egg-rr 8.8%

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

12. Final simplification 8.8%

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

Alternative 18: 8.6% accurate, 33.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 99.6%

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

3. Simplified 99.5%

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

5. Taylor expanded in s around inf 8.8%

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

   1. *-commutative 8.8%

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

   2. associate-*l* 8.8%

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

   3. *-commutative 8.8%

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

7. Simplified 8.8%

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

Alternative 19: 8.6% 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 99.5%

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

5. Taylor expanded in s around inf 8.8%

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

Reproduce

herbie shell --seed 2024131 
(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))))